The Physics of Guitar Tone

For a string of sounding length L displaced to height h at relative position R = a/L and released, initial modal amplitude is C_n ∝ (2h/n²π²R(1−R))·sin(nπR): 1/n² is the corner's roll-off, 1/[R(1−R)] a scalar gain, sin(nπR) the comb that zeroes every m-th harmonic when L/a = m; the first spectral maximum sits at f0/2R with later maxima at odd multiples. Measured on an electric guitar's magnetic pickup (13th harmonic absent at a 14.8 cm pluck on a 64.3 cm string) and on an acoustic string across 21 positions × 5 trials (harmonic amplitude tracks standing-wave amplitude at the pluck point). Bounds: the exciter has width so notches never reach zero; bridge nonlinearity regenerates missing modes over about 1 s so the comb is an attack-window property; the pickup is a second spatial sampler at x_pu whose factor sin(nπx_pu/L) multiplies the hand's comb (derivation). Because the string is transversely stiffer near an end (F ≈ Th(1/a + 1/(L−a))), 'same force' and 'same displacement' plucks are different experiments and 'near the bridge is brighter' is a normalized-spectrum statement, not a level statement. Population: physical measurements on single instruments (one electric, one acoustic, one robot); outcome: measured harmonic amplitudes; class: MODEL confirmed by INTERVENTION.

A triangular initial shape has a spatial Fourier series whose coefficients vanish for modes with a node at the kink; finite contact width convolves the kink and fills the null; the quadratic nonlinearity at a moving termination couples modes and refills them; a magnetic pickup senses string velocity at one point, so its own sin(nπx_pu/L) factor multiplies the launched spectrum.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 7 cited, 6 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S1 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S1 and S13 give the modal law and the F_C1 = f0/2R comb; S19 (via the anthropic leg) the pickup null at n = 13; S18 the 21-position microphone sweep and the finite-width floor; S16 the ~1 s regeneration citing Legge and Fletcher 1984; S21 the robot's position sweep; S5 machine plucks.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)opened, the number was not on the pageMODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S5Carral, Plucking the String: The Excitation Mechanism of the Guitar, ASA 160th meeting lay-language paper, 2010 (reporting Carral and Paset, JASA 123(5), 2008)opened, quote confirmedINTERVENTION: machine plucks with 0.46 mm and 2 mm plectra; amplitude and spectral centroid differed and the centroid ordering reversed after about one second; gradual release favors the fundamental, sudden release excites high modes.
• S13Traube and D'Alessandro, Vocal synthesis and graphical representation of the phonetic gestures underlying guitar timbre description, DAFx-05, 2005opened, quote confirmedMODEL: the ideal plucked-string spectral envelope is comb-shaped with first maximum F_C1 = f0/2R and later maxima at odd multiples.
• S16Fletcher, The nonlinear physics of musical instruments, Rep. Prog. Phys. 62:723 to 764, 1999 (text opened 2026-10-05)opened, quote confirmedMODEL + cited measurement: modes missing at excitation are regenerated by nonlinear effects at the bridge over a time of order 1 s (citing Legge and Fletcher, JASA 1984).
• S18Aldea, Plucking Location and Timbre of a Guitar, International Scholastic Journal of Science 15(1), 2021 (text opened 2026-10-05)opened, quote confirmedINTERVENTION: an acoustic guitar E4 string plucked at 21 positions, 5 trials each, microphone at 100,000 samples/s; harmonic amplitudes follow the standing-wave amplitude at the pluck point and never reach zero because the exciter has width.
• S19Perov, Johnson, Perova-Mello, The physics of guitar string vibrations, American Journal of Physics 84:38, 2016 (Crossref-verified; opened by the anthropic leg via ResearchGate, publisher page 403 for this writer)opened, quote confirmedINTERVENTION: with L = 64.3 cm and a pluck at 14.8 cm the 13th harmonic was absent from the magnetic pickup's FFT, the predicted sin(nπa/L) = 0 null (as quoted by the anthropic leg).
• S21Wiciak, Małecki, Tokarczyk, Impact of a guitar string pluck on the instrument's tone, Tecniacústica Oporto, 2016opened, quote confirmedINTERVENTION: a guitar-playing robot with a movable plucking module varied pluck position and pick angle because human repeatability was too low; position changed the spectral bands observed.

The page may say the pick's position chooses which harmonics are launched; it may not say the hands set the harmonic content independent of where the pickup sits or of the bridge.

Where the pick lands on the string decides which harmonics are launched: the pluck point multiplies every mode by a sine comb, so a pluck one third of the way along silences every third harmonic. The gate found that the Woodhouse Guitar I PDF the legs leaned on does not carry the triangular-pluck formula attributed to it, so the law now rests on Traube and D'Alessandro, Perov's missing thirteenth harmonic on a magnetic pickup, and Aldea's twenty-one-position sweep; the physics is unchanged, the citation must be re-pinned before the page quotes it. Three bounds travel with it: the exciter has width so no notch reaches zero, the bridge refills the missing modes in about a second so the comb is an attack-window property, and the pickup sits on the same string as a second comb (2.3). For the page this means 'the angle of attack' is one of two spatial filters on the same string, and the hand cannot restore a harmonic the pickup sits on a node of.

Pluta, Jasiński, Tokarczyk and Grygiel (Vibrations in Physical Systems 36(2), 2025; arXiv 2606.24356, 2026-06-23) moved a robotic plucker toward an acoustic guitar in 192 μm attack-depth steps with plectra of six materials and measured loudness, timbre descriptors, harmonic content and decay. At low depth the string is not fully excited and the sound is weak and markedly altered; the width of that band depends on plectrum material; past it each depth step raises loudness, lowers inharmonicity and noisiness and moves timbre toward fuller low frequencies and more roughness while decay shape stays similar. The transfer from gesture to sound is therefore gated and qualitatively different below threshold. Boundary: one acoustic guitar, one robot, a microphone chain; transfer to a solidbody with a magnetic pickup is UNVERIFIED; the authors warn remounting plectra broke cross-pick comparison and the arXiv referee note leaves open whether depth was isolated from speed or angle. Outcome: measured loudness, inharmonicity, noisiness, timbre descriptors; class: INTERVENTION (controlled physical measurement) with stated confounds.

A plectrum must deflect the string past the point where it slips off; below that depth the release is partial and the initial displacement and velocity distributions are small and irregular; above it, larger displacement raises mean tension (B ∝ 1/T), lowering measured inharmonicity, and increases low-mode energy.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Refuted by the adversary on single source · Sources: 4 cited, 4 opened and confirmed

• PROBATION: 1 misquote flag elsewhere in this vector (below the kill line of 2); this axiom's own sources were clean -> capped at PROBABLE
• adversary REFUTED the claim
• citations: refetched 100% (>=90%), resolved 96% (>=80%) -> OK
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both

S10 is the journal paper (the openai leg quotes its remounting caveat); S25 the arXiv version with the referee note (anthropic leg); S3 and S24 supply the tension-modulation reading of the inharmonicity trend (an inference across sources).

• S3Bank, Energy-based synthesis of tension modulation in strings, DAFx-09, 2009opened, quote confirmedMODEL: ΔT ≈ (EA/2L)∫|u_x|² tension modulation, the geometric nonlinearity behind the onset pitch glide.
• S10Pluta, Jasiński, Tokarczyk, Grygiel, The effect of micro-changes in the pluck trajectory on the sound of an acoustic guitar, Vibrations in Physical Systems 36(2):2025205, 2025opened, quote confirmedINTERVENTION: robotic plucks at 192 μm attack-depth steps with six plectrum materials on an acoustic guitar; below a material-dependent depth the string is incompletely excited; above it loudness rises, inharmonicity and noisiness fall, timbre fills in lows and roughens; remounting plectra broke cross-pick comparison.
• S24Tolonen, Välimäki, Karjalainen, Modeling of tension modulation nonlinearity in plucked strings, IEEE Trans. Speech and Audio Processing, 2000-05 (DOI 10.1109/89.841212)opened, quote confirmedMODEL: tension modulation produces an exponentially decaying pitch glide, longitudinal modes and phantom partials; the glide is an important timbral feature of the electric guitar among others.
• S25Pluta, Jasiński, Tokarczyk, Grygiel, arXiv 2606.24356, submitted 2026-06-23 (arXiv version of S10)opened, quote confirmedINTERVENTION (same study as S10); the public referee note leaves open whether depth was isolated from pick speed or angle.

The page may say a soft pick stroke can under-excite the string; it may not attribute the break-up of a hard note to the string itself.

The one opened experiment that sweeps how deep the pick goes, Pluta's robot on an acoustic guitar in steps of a fifth of a millimetre, found a threshold below which the string is not fully excited and above which each step raises loudness and lowers inharmonicity and noisiness. The adversary's objection stands as written in the axiom itself: one instrument, one robot, a microphone chain, and the authors' own warning that remounting picks broke cross-pick comparison. It matters to the page because at the string, digging in first completes the excitation and cleans the partials; the harmonics that appear when a tube chain is dug into belong to the amplifier's first stage (3.1), not to the string.

A real pluck leaves the string with both a displacement and a velocity distribution at release, and those, not peak force, determine the free motion. Carral's machine plucks (ASA 2010, reporting Carral and Paset JASA 2008) with 0.46 mm and 2 mm plectra: the thin plectrum bent more, the thick one displaced the string farther, amplitude and spectral centroid differed and the centroid ordering reversed after about one second; release timing depends on plectrum elasticity and friction so the same depth gives different initial amplitude and modes present; gradual release enhances the fundamental, sudden release excites higher modes. 'Pick angle' must be operationalized as edge orientation, travel direction, and the string's displacement and velocity direction at release, which are not interchangeable. The AGH robot (2016) varied angle and position repeatably. The anthropic leg quotes an acoustic-guitar angle sweep (little change until a steep fall near the parallel limit; perpendicular plucks strong and fast-decaying, parallel weaker and longer) that this writer could not pin to an opened URL, so its angle and decibel figures carry no number here and the envelope half rests on A4. Cuzzucoli and Lombardo's model parameters (plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 to 5 mm) are measured on one classical guitar, not population values. Class: INTERVENTION (machine plucks) + MODEL (contact release).

The string's free motion is set by its state at the instant contact ends; a release that unloads over time τ suppresses energy above about 1/τ, while a sudden slip or recontact injects a broadband transient; a stiff, perpendicular plectrum minimizes τ and a flexible, slanted one maximizes it.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 5 cited, 4 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S1 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S5 for the two-plectrum machine experiment and the gradual-versus-sudden release mechanism; S21 for the robot's angle and position plucks; S1, S6, S23 for the contact-release model and its measured parameters.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)opened, the number was not on the pageMODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S5Carral, Plucking the String: The Excitation Mechanism of the Guitar, ASA 160th meeting lay-language paper, 2010 (reporting Carral and Paset, JASA 123(5), 2008)opened, quote confirmedINTERVENTION: machine plucks with 0.46 mm and 2 mm plectra; amplitude and spectral centroid differed and the centroid ordering reversed after about one second; gradual release favors the fundamental, sudden release excites high modes.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009opened, quote confirmedMODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S21Wiciak, Małecki, Tokarczyk, Impact of a guitar string pluck on the instrument's tone, Tecniacústica Oporto, 2016opened, quote confirmedINTERVENTION: a guitar-playing robot with a movable plucking module varied pluck position and pick angle because human repeatability was too low; position changed the spectral bands observed.
• S23Cuzzucoli and Lombardo, Physical model of the plucking process in the classical guitar (ICMC 1997; CMJ 1999)opened, quote confirmedMODEL with empirically measured parameters: plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 mm (piano) to 5 mm (forte) on one classical guitar.

The page may say the angle of attack changes the attack's spectrum through how the string is released; it may not describe it as a smooth dial, and it may not quote decibel figures for an electric.

How the string leaves the pick, gradual or sudden, sets the attack's high-frequency content more than how hard it was pressed; Carral's two machine plectra are the measured case. The angle sweep the legs quote, a dead band then a steep fall near the parallel limit, was never pinned to a page any writer could open, so the axiom carries no angle and no decibel figure, and the gate's S1 tripwire lowered it further. The page may say the angle of attack shapes the attack through the release it produces; it may not describe that as a smooth dial or quote a number for an electric guitar.

Every string mode is a pair, one polarization normal to the body and one parallel, at slightly different frequencies with different damping because the termination's admittance differs by direction; the player sets the initial mixture by the pluck's plane. Where vertical bridge admittance exceeds horizontal, energy leaves the vertical polarization faster and the sum shows a fast onset decay then a long sustain, with beating where the pair's frequencies differ. Woodhouse's necessary condition (JASA 150(6):4375, 2021): both polarizations share the same intrinsic damping, so a strong contrast in total damping requires one to lose energy through the bridge more than the other by at least the intrinsic damping. Pick plane therefore controls the decay envelope if and only if the termination is that anisotropic; on an isotropic high-impedance termination the same gesture changes spectrum and leaves the envelope unchanged. Measured: Paté, Le Carrou and Fabre (2014) resolved both polarizations of electric-guitar strings on a rigid fixture with two optical sensors and separated intrinsic loss from structural coupling; an Acta Acustica 2020 rig measured polarization coupling through a half-round saddle with two laser Doppler vibrometers; Le Carrou's group (2019) measured that the player's grasp, strap and thigh contact change the electric guitar's modal damping and frequencies. Intrinsic damping varies by an order of magnitude across the audible range. The solidbody loss partition (intrinsic, contact, structure) is UNVERIFIED. Population: single electric instruments and fixtures; outcome: measured per-polarization decay rates; class: MODEL (theorem) confirmed by INTERVENTION.

A termination with direction-dependent admittance drains energy from the two orthogonal polarizations at different rates; the observed envelope is the sum of two exponentials whose separation equals the admittance contrast minus the shared intrinsic loss, so anisotropy below intrinsic loss yields a single decay whatever the pluck plane.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on unverifiable-unsupported · Sources: 8 cited, 7 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S1 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S1, S17, S27 for the two-polarization mechanism and gesture link; S15 for the necessary condition; S2 and S14 for polarization-resolved measurements; S9 for the player-dependent termination; S26 for the order-of-magnitude damping variation.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)opened, the number was not on the pageMODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S2Paté, Le Carrou, Fabre, Predicting the decay time of solid body electric guitar tones, JASA 135(5):3045 to 3055, 2014opened, quote confirmedINTERVENTION: polarization-resolved decay of electric-guitar strings on a rigid fixture with two optical sensors, separating intrinsic string loss from structural coupling.
• S9Le Carrou, Paté, Chomette, Influence of the player on the dynamics of the electric guitar, JASA, published 2019-11-06opened, quote confirmedINTERVENTION: the player's grasp, strap and thigh contact change the electric guitar's modal frequencies and add modal damping, depending on body part and mode shape.
• S14Acta Acustica 2020, article aacus200019 (polarization coupling through a half-round saddle; title not verified by this writer, publisher page 403 on 2026-10-05)opened, quote confirmedINTERVENTION: coupling of the two transverse polarizations through a plain half-round saddle measured with two laser Doppler vibrometers and a reproducible plectrum pluck (as quoted by the anthropic leg).
• S15Woodhouse, A necessary condition for double-decay envelopes in stringed instruments, JASA 150(6):4375, 2021-12-01 (DOI 10.1121/10.0009012, Crossref-verified)opened, quote confirmedMODEL (theorem): a strong contrast in damping between the two polarizations requires one to lose energy through the bridge more than the other by at least the intrinsic damping.
• S17Woodhouse, Euphonics 7.3 Multiple strings and double decaysopened, quote confirmedMODEL: each string mode is a pair of polarizations with different frequencies and damping; unequal bridge admittance gives a two-stage envelope; the player sets the initial mixture by the gesture.
• S26Woodhouse, Euphonics: Guitar IIopened, quote confirmedMODEL + measurement: intrinsic string damping varies by an order of magnitude across the audible range, so no single decay constant describes a gesture's effect.
• S27Smith, Physical Audio Signal Processing: String Coupling Effects (online book)opened, quote confirmedMODEL: vertical and horizontal polarizations couple to the bridge with different admittance, giving fast then slow decay and beating.

The page may say the hand chooses how the note's energy is split between two decays; it must add that the bridge decides whether the two decays differ at all.

The pluck's plane splits the note between two polarizations, and whether that split is audible as a two-stage decay depends on the bridge, not the hand: Woodhouse's theorem requires one polarization to lose energy through the termination by at least the string's own damping before the envelopes separate. Paté's polarization-resolved decays on electric strings and Le Carrou's finding that how the player holds the guitar changes the structure's damping make the termination a moving target. The decision for the page is a required addition, not a cut: the hand chooses how the energy is split, and the bridge decides whether the two decays differ at all.

With axially immobile ends, small transverse slopes add length Δℓ ≈ ½∫|u_x|² dx and tension ΔT ≈ (EA/2L)∫|u_x|² dx; large-amplitude onset motion raises mean tension and pitch, which glides down as amplitude decays (Bank DAFx-09 2009; Tolonen, Välimäki, Karjalainen IEEE TSAP 2000, who name the glide an important timbral feature of the electric guitar). For a dominant mode with A(t) ∝ e^(−γt) the offset falls as e^(−2γt). For a triangular pluck of height h at distance p from the bridge, Δf/f ≈ (1/2ε0)·h²/[2p(L−p)] with ε0 = T/EA, so the offset is quadratic in depth and larger near the bridge at equal depth: force and position interact on pitch (anthropic-leg derivation). The same geometric nonlinearity produces longitudinal modes, phantom partials and mode coupling; the material stays Hookean. Magnitudes: Legge and Fletcher's nylon result that a 10 mm deflection is reachable only by pulling bounds the effect as small at ordinary classical dynamics; for steel electric strings no opened source measured the glide at a stated depth, so the circulating figures (a blog model's 26 cents at 4.7 mm; a derived 8 to 9 cent upper bound at h = 3 mm, p = L/5, L = 0.635 m, ε0 ≈ 0.007) are MODEL outputs only. Friction can partition tension across nut or saddle, qualifying a uniform-tension model. Class: MODEL; the only measurements are model fits to recorded tones.

Transverse displacement lengthens the string by a second-order amount, which through the axial stiffness EA raises tension in proportion to the squared slope integral; wave speed √(T/μ) rises with it, so pitch tracks the squared amplitude and relaxes twice as fast as amplitude.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 5 cited, 4 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S1 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S3 for the tension-modulation law and its energy-based synthesis; S24 for the exponentially decaying glide, longitudinal modes and phantom partials; S1 for the triangular geometry behind the h²/[p(L−p)] scaling; S4 for friction partitioning tension across nut and saddle.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)opened, the number was not on the pageMODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S3Bank, Energy-based synthesis of tension modulation in strings, DAFx-09, 2009opened, quote confirmedMODEL: ΔT ≈ (EA/2L)∫|u_x|² tension modulation, the geometric nonlinearity behind the onset pitch glide.
• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21opened, quote confirmedINTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S16Fletcher, The nonlinear physics of musical instruments, Rep. Prog. Phys. 62:723 to 764, 1999 (text opened 2026-10-05)opened, quote confirmedMODEL + cited measurement: modes missing at excitation are regenerated by nonlinear effects at the bridge over a time of order 1 s (citing Legge and Fletcher, JASA 1984).
• S24Tolonen, Välimäki, Karjalainen, Modeling of tension modulation nonlinearity in plucked strings, IEEE Trans. Speech and Audio Processing, 2000-05 (DOI 10.1109/89.841212)opened, quote confirmedMODEL: tension modulation produces an exponentially decaying pitch glide, longitudinal modes and phantom partials; the glide is an important timbral feature of the electric guitar among others.

The page may say a hard attack starts sharp and settles; it may not quote a cent figure.

A hard pluck stretches the string, raises its tension and starts the note sharp, with the offset falling as the square of the amplitude envelope; that is tension modulation, a geometric nonlinearity Bank and Tolonen modelled against recorded tones. No opened source measured the glide at a stated depth on a steel electric string, so the circulating cent figures (a blog's twenty-six cents, a derived eight to nine) are model outputs and do not enter the page. The page may say a hard attack starts sharp and settles; it may not quote a cent figure, and it may not call this 'breaking into harmonics', since the string adds only the phantom partials of its own nonlinearity.

Regime A, contact: the force to bring a string to the fret is F ≈ T1 sin θ1 + T2 sin θ2, set by action, finger position, tension and fret geometry; action dominates and higher frets need more force (Grimes 2014). Regime B, hold: a fret is a unilateral contact (g ≥ 0, N ≥ 0, gN = 0), so static touch does not guarantee the boundary during a large pluck; detachment, recontact and collision with the next fret are nonlinear and the threshold depends on excitation amplitude and direction (Evangelista and Eckerholm 2009; Evangelista 2011); the anthropic leg writes F_hold > T(d/a + d/b) + κTh(1/p + 1/(L−p)), so fretting pressure adequate for soft picking chokes under hard picking: the two hands couple through the boundary (derivation). Regime C, excess force: pressing past contact lengthens the path behind the fret and sharpens the note only insofar as the extension raises tension in the sounding segment, with fret friction and anchor compliance governing transmission; the capstan law with a fraction-of-a-degree wrap at a fret equalizes tension almost freely while the nut's larger wrap resists slip, so excess-force sharpening is near-reversible and bend redistribution across the nut is not (derivation). The unavoidable contact stretch that action imposes on every fretted note is a setup constant removed by saddle compensation, not a gesture; the player's excess force is a few cents by geometry for half a millimeter of over-press between supports 8 and 25 mm apart (derivation). Circulating figures (14 to 15 cents from forums; 'a quarter tone' from an instructional site) are ARGUMENT and carry no number. A force-sensing fretboard (Ren 2023) reads 0 to 25 N at 20 Hz on compliant frets and cannot resolve attack mechanics. Finger-applied force is not the fret reaction: skin compression and fingerboard contact carry part of the load. Class: MODEL with S7's bend leg measured; instrumentation CORRELATIONAL.

A fret pushes but cannot pull, so the boundary holds only while the string's transverse force at the fret stays below the finger's normal load; pressing past contact adds path length quadratically in depression and raises tension only where friction lets the segments equalize.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a step in the reasoning · Sources: 5 cited, 5 opened and confirmed

• PROBATION: 1 misquote flag elsewhere in this vector (below the kill line of 2); this axiom's own sources were clean -> capped at PROBABLE
• adversary REFUTED the claim
• citations: refetched 100% (>=90%), resolved 96% (>=80%) -> OK
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S7 for the fretting-force equation and action dominance; S6 and S8 for the unilateral contact, collision and friction model; S4 for friction limiting tension equalization at nut and saddle; S12 for the force-sensing fretboard's range and limits.

• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21opened, quote confirmedINTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009opened, quote confirmedMODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S7Grimes, String Theory: The Physics of String-Bending and Other Electric Guitar Techniques, PLOS ONE 9(7):e102088, 2014-07-23opened, quote confirmedINTERVENTION (bend pitches measured at fixed displacements on an electric guitar) + MODEL (fretting-force equation, action dominance, radial and axial vibrato simulation, whammy-bar tension).
• S8Evangelista, Physical model of the string-fret interaction, DAFx-11, 2011opened, quote confirmedMODEL: the fret clamps the vertical polarization while the horizontal stays free under friction; the finger adds damping and clamps the trailing segment.
• S12Ren et al., force-sensing fretboard, arXiv 2304.06521, 2023opened, quote confirmedCORRELATIONAL (instrumentation): per-fret contact force read 0 to 25 N at 20 Hz on compliant sensing frets; does not preserve an ordinary fret boundary or resolve attack mechanics.

The page may say fingertip pressure past the fret bends pitch a little and that a hard pick stroke demands more fretting force; it may not say pressure shapes tone independently of the instrument's setup.

'Fingertip pressure' is three quantities the page runs together: the force to reach the fret, which action and geometry set; the force to hold the fret through a hard pick stroke, which rises with pick depth and couples the two hands; and the excess force past contact, which sharpens the note only insofar as friction lets the added tension reach the sounding segment. Grimes's bend measurements are the one measured leg; the hold threshold is a derivation the adversary rightly called unsupported by any fetched source, and the fourteen-to-fifteen-cent figures in circulation are forum arithmetic and were cut. The page may say pressure past the fret bends pitch a little and that hard picking demands more fretting force; it may not say pressure shapes tone independently of the instrument's setup, which is where gemini's one overreach sat.

A stationary dashpot c_f at x_f adds to mode n a decay-rate increment Δγ_n = (c_f/μL)·sin²(nπx_f/L): a mode with a node at the contact is barely touched, one with an antinode strongly damped (weak-contact model; the finger as a spring-mass-damper in Evangelista and Eckerholm 2009 and Cuzzucoli and Lombardo). Because damping force is proportional to velocity and higher modes move faster at a given amplitude, the finger removes high-frequency energy faster than a rigid boundary, but the sin² factor makes the filter position-dependent and comb-like, and real fingertips have finite patches whose stiffness and dissipation change with pressure. The one damping figure in the legs (about 2 Ns/m at 2 N, flesh stiffness 41.6 to 237 N/m, Hajian and Howe 1997 via a McGill thesis) is a haptics measurement of a fingertip against a probe, second-hand and unopened, entering only as an order of magnitude. A finger behind an established fret couples through that boundary; lifting the string off the fret and leaving flesh contact removes the hard node and the string dies fast. On a solidbody the soundboard loss channel is largely absent, so intrinsic and contact damping rise in relative weight, the structural reason fretting-hand damping is more audible on an electric (inference; the partition is UNVERIFIED). Intrinsic damping already varies by an order of magnitude across the band, so no single constant describes the added loss, and a changed composite envelope does not prove changed damping without partial-by-partial decay measurement. Class: MODEL; no string measurement of fingertip damping was opened.

A viscous point load extracts power proportional to the squared velocity of the string at the contact, which for mode n scales with sin²(nπx_f/L) and with frequency; where the structure drains little energy, the contact becomes a leading loss term.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed PROBABLE) · Refuted by the adversary on a source said less than the claim · Sources: 6 cited, 5 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S1 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed PROBABLE -> granted SIGNAL (gate lowers, never raises)

S6 and S23 for the finger as a damped spring-mass contact; S20 for the second-hand fingertip impedance figures; S26 for the frequency dependence of intrinsic damping; S2 for the partial-by-partial decay method that separates damping from excitation weighting.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)opened, the number was not on the pageMODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S2Paté, Le Carrou, Fabre, Predicting the decay time of solid body electric guitar tones, JASA 135(5):3045 to 3055, 2014opened, quote confirmedINTERVENTION: polarization-resolved decay of electric-guitar strings on a rigid fixture with two optical sensors, separating intrinsic string loss from structural coupling.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009opened, quote confirmedMODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S20Kry, PhD thesis, McGill (reports Hajian and Howe 1997, J. Biomech. Eng. 119(1), fingertip impedance)opened, quote confirmedARGUMENT (second-hand): fingertip damping about 2 Ns/m at a 2 N interaction force and stiffness 41.6 to 237 N/m, scaling nonlinearly with force; the primary record was not opened.
• S23Cuzzucoli and Lombardo, Physical model of the plucking process in the classical guitar (ICMC 1997; CMJ 1999)opened, quote confirmedMODEL with empirically measured parameters: plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 mm (piano) to 5 mm (forte) on one classical guitar.
• S26Woodhouse, Euphonics: Guitar IIopened, quote confirmedMODEL + measurement: intrinsic string damping varies by an order of magnitude across the audible range, so no single decay constant describes a gesture's effect.

The page may say the fretting hand shapes decay, and that this matters more on an electric; it should not call the finger a filter with a single character.

A fingertip touching a vibrating string is a damper whose bite on each mode scales with that mode's motion at the contact point, so it is mode-selective and position-dependent rather than a single low-pass character. Everything here is model inference; the one damping figure is a haptics measurement of a fingertip against a probe, quoted second-hand, and the solidbody loss partition that would say how audible this is on an electric is unverified. The useful structural point for the page is that the electric's missing soundboard loss is why fretting-hand damping is a more audible control on an electric than on an acoustic, and that this stays an inference until partial-by-partial decays are measured.

Bending lengthens the sounding span slightly (lowering pitch alone) but raises tension by EA(1/cos θ − 1), and tension wins, so the fundamental f ≈ (1/2L)√{[T + EA(1/cos θ − 1)]/(μ0 cos θ)} rises for a bend up or down alike; only magnitude matters. Grimes (2014) measured this on an electric guitar with nails driven into the neck holding strings at fixed bend angles across several strings. Near straight the path-length gain is quadratic, Δℓ ≈ (b²/2)(1/ℓ1 + 1/ℓ2), so with f − f0 ≈ Cb² a hand motion b0 + b1 sin Ωt yields a component at Ω proportional to 2b0b1 and one at 2Ω: around the unbent note (b0 = 0) pitch rises twice per hand cycle and never falls below f0; around a pre-bend the response is near-linear and bipolar at the hand rate. Hand rate, pitch-modulation rate and pitch slew are different quantities and the modulation is not sinusoidal in general. Axial vibrato (rolling along the string at zero bend angle) modulates tension directly and can raise and lower pitch. Frequency modulation puts sidebands around partials that are not new modes, and co-varying fingertip damping adds amplitude modulation. Friction at nut and saddle strands tension imbalances after a bend (Kemp 2017), so the same finger position need not reproduce the same pitch. Benito and McPherson's TMR sensor (2021) reads horizontal displacement on an electric guitar and calibrates it to bend pitch empirically, disambiguating up from down bends that sound alike. Grimes's vibrato treatment is a simulation and players differ widely in width and rate; circulating ranges (4 to 7 Hz, 40 to 140 cents) come from an unrefereed calculator and carry no number. Class: INTERVENTION (fixed-displacement bend pitches) + MODEL (vibrato), CORRELATIONAL sensing.

Lateral displacement stretches the string against its axial stiffness EA, and because frequency goes as √T the gain in tension outruns the loss from added length; the stretch is even in displacement, so an oscillation centered on zero is rectified to twice the drive frequency.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Survived the adversary's attack · Sources: 4 cited, 4 opened and confirmed

• PROBATION: 1 misquote flag elsewhere in this vector (below the kill line of 2); this axiom's own sources were clean -> capped at PROBABLE
• survived a grounded cross-lab adversary (google vs authors ["anthropic"])
• citations: refetched 100% (>=90%), resolved 96% (>=80%) -> OK
• EARNED FORGED: survived a grounded cross-lab adversary AND primary sources confirmed
• PROBATION CAP: earned FORGED -> PROBABLE (one misquote elsewhere in the vector)
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S7 for the bend equation, the nailed-string measurement, axial vibrato and the simulation's limits; S4 for friction-stranded tension after a bend; S22 for the displacement-to-pitch sensor calibration and up/down disambiguation; S11 for gesture-capture methods.

• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21opened, quote confirmedINTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S7Grimes, String Theory: The Physics of String-Bending and Other Electric Guitar Techniques, PLOS ONE 9(7):e102088, 2014-07-23opened, quote confirmedINTERVENTION (bend pitches measured at fixed displacements on an electric guitar) + MODEL (fretting-force equation, action dominance, radial and axial vibrato simulation, whammy-bar tension).
• S11Perez-Carrillo, Finger-String Interaction Analysis in Guitar Playing With Optical Motion Capture, Frontiers in Computer Science, 2019-11-15opened, quote confirmedCORRELATIONAL (gesture capture): 240 Hz optical motion capture plus audio estimates contact, pressure and release phases, position, velocity, direction and displacement; no force measured.
• S22Benito Temprano and McPherson, A TMR Angle Sensor for Gesture Acquisition and Disambiguation on the Electric Guitar, Audio Mostly 2021 (DOI 10.1145/3478384.3478427)opened, quote confirmedCORRELATIONAL (sensing): horizontal string displacement sensed on an electric guitar and calibrated empirically to bend pitch; upward and downward bends that sound alike are disambiguated.

The page may say a player's vibrato is a pitch trajectory the hands write through tension; it may not quote a rate or depth for any player or say the pitch follows the hand one-to-one.

A bend raises pitch whichever way it goes because tension gain beats length gain, and because pitch is quadratic in displacement near straight, a symmetric vibrato about an unbent note rises twice per hand cycle and never falls below the note, while vibrato about a pre-bend moves at the hand rate in both directions. This is the one the hands on the string vector axiom that survived the adversary unrefuted on Grimes's nailed-bend measurements and the McPherson sensor calibration. For the page's 'vibrato timing' it is the sharpest correction in the run: the hand's rate and the pitch's rate are the same only on a pre-bend, nut friction leaves hysteresis so the same finger position need not give the same pitch, and no measured width or rate for any named player, SRV included, exists in the opened record.

Trace the forward signal in a passive guitar into a classic tube amplifier (the Fender 5F1 schematic, opened). The joints are: hand to string (mechanical); string to magnetic field (a change of flux linkage across an air gap, no conductor); field to coil (Faraday induction); coil through pots, jack and cable to the first grid (one conductive loop, A); grid to plate at each tube (an electrostatic field controlling a cathode-to-plate stream powered by the amplifier's own supply; the pickup's current never becomes the plate current); plate to the next grid through a coupling capacitor (displacement current across a dielectric, DC blocked); power tube to transformer primary (conductive loop B); primary to secondary (mutual induction, no copper path); secondary to voice coil (conductive loop C); voice coil to cone (Lorentz force F = Bl·i); cone to air to ear (acoustic). At least five joints have no ohmic path. What runs nearly end to end is the ground reference: strings and bridge bonded to the guitar common, common down the cable shield to the chassis, chassis to the speaker return; the 5F1 also carries a 22 kΩ feedback resistor from the secondary to a preamp cathode, so the amplifier is not 'isolated by the transformer' either. A continuity meter from a string to a speaker terminal may read a closed path; that is the reference route, not the audio's route. Bounds: holds for every passive-pickup, output-transformer tube chain with a grounded secondary; an OTL amplifier removes one magnetic boundary and keeps the vacuum and dielectric ones; active pickups and pedals add powered stages and more non-conductive joints, never fewer. Evidence class: MODEL (physics derivation) plus an opened schematic; population: a physical artifact, not a human sample; outcome: a topology count.

Faraday's law generates EMF from changing flux linkage with no conductor between string and coil; Maxwell's displacement-current term carries AC across a capacitor's dielectric while charges never cross it; mutual induction moves energy between transformer windings through shared flux, not shared copper; a triode's plate current is a locally powered stream controlled by the grid-cathode field. Energy is not conserved along the chain (each stage injects supply energy); the continuous-valued waveform is. 'Analog' names continuity of the signal's value in time and amplitude, never continuity of a wire.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a step in the reasoning · Sources: 8 cited, 8 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S1 (displacement current and induction), S2 (transformer mutual induction and reflected impedance), S3 (5F1 schematic: coupling caps, output transformer, grounded secondary, 22 kΩ feedback), S4 (ECC83S: a tube's grid controls a locally powered plate current), S5 (capacitor and transformer coupling block DC), S6 (the bonded ground network in a passive guitar), S7 (galvanic isolation defined), S8 (the OTL case). Opened 2026-10-04; S3 is a 1950s drawing, S1, S2, S4 current.

• S1OpenStax University Physics Vol. 2, 16.1 Maxwell's Equations and Electromagnetic Wavesopened, quote confirmedDisplacement current carries AC across a capacitor's dielectric with no conduction; Faraday induction needs changing flux linkage, not a conductor
• S2OpenStax University Physics Vol. 2, 15.6 Transformersopened, quote confirmedTransformer secondary voltage arises from shared changing flux, not conduction; reflected impedance scales with the square of the turns ratio
• S3Fender Champ 5F1 schematicopened, quote confirmedA minimal tube chain: two triode stages, interstage coupling capacitors, power tube, output transformer with grounded secondary and a 22 kΩ feedback path to a preamp cathode
• S4JJ Electronic ECC83S datasheetopened, quote confirmedTransconductance, plate resistance and interelectrode capacitances (1.6 pF grid-cathode, 1.7 pF grid-anode) used for the Miller input-capacitance estimate
• S5All About Circuits, Input and Output Couplingopened, quote confirmedCapacitor and transformer coupling pass AC and block DC between amplifier stages and between an amplifier and its load
• S6StewMac, Understanding Guitar Wiring Part 8: Grounding and Shieldingopened, quote confirmedIn a passive guitar the bridge, strings, pot cases, jack sleeve and shielding are bonded to one common ground
• S7Texas Instruments, Isolation technologiesopened, quote confirmedGalvanic isolation defined: no direct conduction path, signal exchanged by capacitive, inductive or optical coupling
• S8Wikipedia, Output transformerlessopened, quote confirmedThe OTL case: removing the output transformer removes one magnetic boundary but not the vacuum and dielectric ones

The page sentence 'The signal path from string to speaker is an unbroken electrical circuit' is REFUTED as written and must be rewritten; 'All-analog signal chain... no digital modeling' is true and may stay. The corrected line: the signal is continuous in value from string to speaker but crosses at least five non-conductive joints (an air gap, vacuums, dielectrics, a transformer core, the air) and three separately powered circuits; what runs unbroken is the ground.

Traced joint by joint through the Fender 5F1 schematic, the forward signal crosses at least five places with no conductor: the air gap between string and coil, the vacuum of every tube, each coupling capacitor's dielectric, the output transformer's core, and the air to the ear; each gain stage draws its energy from the wall, not from the string. What runs unbroken is the ground reference, which is why a continuity meter from string to speaker terminal would mislead. The adversary's refutation turned on the word 'galvanically separate' sitting beside 'shared ground', a real wording tension the page need not inherit: the sentence the page may keep is that the waveform's value is continuous and never sampled from string to speaker, and the sentence it must drop is that the path is one unbroken electrical circuit.

With a stationary coil the induced EMF is e(t) = −dλ/dt, λ the linked flux; the vibrating ferromagnetic string modulates λ, and the alternating flux is about 1 percent of the static field even for strong vibration (Lemme). Expanding λ about rest gives a leading term proportional to string velocity projected onto the pickup's sensitivity direction plus higher-order terms scaling with position times velocity. So: (a) the pickup reports velocity, displacement differentiated, weighting every partial by its own frequency, a +6 dB per octave tilt relative to a displacement sensor that no winding, magnet or pot undoes (confirmed from the other side by a position-sensing pickup patent claiming 6 dB per octave more low end); (b) because flux versus pole-to-string distance follows a hyperbolic curve, sensitivity changes with instantaneous position, modelled as a static nonlinearity followed by a derivative (DAFx 2018; Paiva, Pakarinen, Välimäki), so the chain's first distortion is the magnetic field's shape, not a tube; (c) the Guadagnin et al. bench, a controlled harmonic-motion rig driving a string segment past a pickup, reports velocity-sensitive behaviour and greatest sensitivity to motion normal to the pickup face in the tested arrangement, INTERVENTION-class evidence for that rig, not a universal constant. Height is a gain and a filter: near 3 mm, output falls about 3 dB per millimetre of added magnet-to-string distance (Zollner, measured on the single coils he swept). Population: physical pickups on a bench; outcome: measured voltage and spectrum; design: controlled physical measurement and physical model.

Faraday's law makes EMF proportional to the time derivative of linked flux; a pole piece's field falls off roughly hyperbolically with distance, so the displacement-to-flux map is curved, and differentiating a curved map of a sinusoid yields harmonics (a symmetric lateral map yields a term at twice the motion frequency). Lenz's law makes the transducer bilateral: the static magnet pulls on the string and loads its vibration.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 7 cited, 6 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S9 (Guadagnin et al., AES: controlled string-motion bench, velocity-sensitive, z-polarisation), S10 (Lemme: 1 percent alternating flux, hyperbolic flux curve, magnet pull), S11 (DAFx 2018 paper 39: static nonlinearity then derivative), S12 (Paiva, Pakarinen, Välimäki: distance-dependent nonlinearity), S13 (Zollner 5.4.1: the 3 dB per mm height rule), S14 (US 9,679,549: 6 dB per octave more low end from position sensing), S15 (Horton and Moore: the string modifying the field). Dates: S10 about 2003, S11 2018, S13 2014 (English 2019), S14 granted 2017; all fetched 2026-10-04.

• S9Guadagnin et al., AES e-library paper 19201 (controlled string-motion bench and pickup transduction model)did not openVelocity-sensitive behaviour and greatest sensitivity to z-polarised motion measured on a controlled harmonic-motion rig
• S10Helmuth Lemme, The Secrets of Electric Guitar Pickupsopened, quote confirmedAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S11DAFx 2018 paper 39, nonlinear guitar pickup modellingopened, quote confirmedMagneto-electric conversion modelled as a static nonlinearity followed by a derivative; output proportional to velocity times the gradient of the nonlinear flux map
• S12Paiva, Pakarinen, Välimäki, Acoustics and Modeling of Pickupsopened, quote confirmedDistance-dependent magnetic nonlinearity; the tone branch interacting with pickup inductance to form a lower resonance; exciter-core loading caution
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019opened, quote confirmedAbout 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required
• S14US Patent 9,679,549 (position-sensing string pickup)opened, quote confirmedA position-sensing design claims 6 dB per octave more low-frequency response than a velocity pickup, confirming the velocity tilt from the other direction
• S15Horton and Moore, Rollins College, modelling the electric guitar pickupopened, quote confirmedExperimental modelling of how the ferromagnetic string modifies the pickup's field and flux linkage

The page may say the pickup hears the string's velocity, not its position, tilting every note toward its upper partials before the amplifier, and that the magnet's field shape is the chain's first source of harmonics. It must not say the pickup 'captures the string's sound' as if it were a flat microphone.

A magnetic pickup hears string velocity, not position, so every partial is tilted up by six decibels an octave relative to what the string is doing, and the curved flux-versus-distance map adds the chain's first harmonics before any tube. The adversary caught a real inconsistency the page should not repeat: Lemme calls the fall-off hyperbolic while Zollner's chapter states an inverse-square law, so the shape of the curve is disputed between the two bench authors even though its curvature, and the distortion it produces, is not. Guadagnin's rig finding that sensitivity is greatest normal to the pickup face is one rig and one arrangement, not a constant. The page may say the pickup tilts and already distorts the note; it may not describe the pickup as a flat microphone on the string.

For an ideal fixed-end string of vibrating length ℓ, a linear pickup with sensitivity kernel g(x) outputs e(t) = −∫ g(x) u̇(x,t) dx, so partial n carries three separable factors: A_n, what the hand put into the mode; ω_n, the velocity tilt; and ∫ g(x) sin(nπx/ℓ) dx, where the pickup looks. For a point sensor at x_p the factor is sin(nπx_p/ℓ): every partial with a node over the pickup is absent from the linear output, and the nulls sit at F_node = (L_scale / X_pickup) · F_open, the pitch of the string fretted at the pickup location (Tillman). Worked on a 25.5 inch Stratocaster, low E 82.4 Hz: neck pickup at 6.375 inches from the bridge, first null 330 Hz (the 4th partial) and every multiple; middle at 3.875 inches, first null 542 Hz; bridge at 1.625 inches, first null 1274 Hz. On a 24.75 inch Les Paul, neck at 6.25 inches nulls first near 326 Hz, bridge at 1.5 inches near 1360 Hz (same formula). A finite aperture of width w adds a sinc(nπw/2ℓ) factor, a second higher comb; Zollner measures the magnetic window at about 1 inch for a thin single coil and 2.5 inches for a humbucker, and a series humbucker samples at two points, adding a third comb by the shift theorem. Bounds: Zollner's caution that because a single coil's electrical resonance limits transmission to about 5 kHz the aperture term has little influence on single coils, so position and electrical resonance dominate there; fretting changes ℓ so a fixed pickup does not keep its node relationships; an electrical resonance cannot recreate a partial the linear spatial sampling cancelled. Evidence class: MODEL (analytic, with checkable null predictions) plus Zollner's measured apertures; population: physical instruments.

A standing wave has fixed nodes; a sensor integrating motion over a region weights each mode by the mode shape's integral over that region. Sampling theory makes a point sample a sine comb and a rectangular window a sinc low-pass, and the two factors separate (Tillman, Zollner). The position comb and the plucking-point comb are the same mathematics at two points on the string.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 5 cited, 5 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S16 (Tillman 2002: comb derivation, fretted-pitch rule, Strat worked values), S17 (Tillman demo), S18 (Zollner 5.4.4: aperture widths, the 5 kHz caution, 'not a measuring device'), S19 (Tillman, pickup mixing: two sampling points), S20 (AES 16551: aperture and polarisation weighting). Fetched 2026-10-04; S16 is 2002, S18 is 2014 (English 2019).

• S16J. Donald Tillman, Response Effects of Guitar Pickup Position and Width (2000, updated 2002)opened, quote confirmedPosition comb derivation; nulls at the fretted-at-the-pickup pitch; Strat worked values (neck first null 330 Hz, middle 542 Hz, bridge 1274 Hz); width as a second separable comb
• S17Tillman, Pickup Response Demoopened, quote confirmedInteractive demonstration of the position and width comb responses
• S18Zollner, Physik der Elektrogitarre ch. 5.4.4 (the magnetic aperture), English 2019opened, quote confirmedAperture about 1 inch for a thin single coil and 2.5 inches for a wide humbucker; aperture has little influence on single-coil transmission because the electrical resonance limits it to about 5 kHz; a pickup is not a flat measuring device
• S19Tillman, Pickup Mixingopened, quote confirmedThe two-pickup and two-coil sampling case and its comb
• S20AES e-library paper 16551 (pickup sensing aperture and polarisation weighting)opened, quote confirmedThe pickup's magnetic sensitivity occupies a region along the string and weights the two transverse polarisations differently

The page may say that where the pickup sits decides which partials it can hear at all and that bridge and neck sounds are geometry, not electronics; the 'angle of attack' line should be paired with the position comb so the reader sees the hand's input and the pickup's sample point as two separate filters on one string.

Where the pickup sits under the string is a second spatial comb with nulls at the pitch the string would have if fretted over the pickup, which is the whole bridge-versus-neck sound without a word about magnets or windings; Tillman's worked Stratocaster nulls and Zollner's measured apertures carry it. The adversary's point that Zollner describes the aperture window as Gaussian-like rather than rectangular is correct and changes only the exact low-pass kernel, not the direction; Zollner's own caution that for a single coil the electrical resonance dominates the aperture term is kept. This is the same mathematics as the pluck-point comb (1.1), and the page should pair the two so a reader sees the hand's input and the pickup's sample point as two separate filters on one string.

Electrically a passive pickup is a source EMF behind R_s + sL feeding a parallel load Y = 1/R_V + 1/Z_in + s(C_p + C_c) + the tone branch, so H(s) = 1 / (1 + (R_s + sL) Y(s)). With one parallel R_L and one total C it is a second-order low-pass: flat below, 12 dB per octave above, a peak near 1/(2π√(LC)) of height Q ≈ R_L √(C/L) (voltage peak slightly below the LC frequency, present only when Q > 1/√2). Measured magnitudes: L 1 to 10 H (2 to 3 H Strat single coil, 4 to 8 H humbucker); coil C about 80 to 200 pF (90 to 120 pF Strat, about 50 pF PAF, 250 to 350 pF single-coil-sized humbucker); cable C 300 to 1,000 pF for 10 to 20 feet at 70 to 150 pF per metre (Belden 9778: 45 pF per foot, 148 pF per metre, so 20 feet is 900 pF; page revised 2026-02-20). The cable's C is typically three to ten times the coil's, so the cable, not the pickup, usually sets the resonant frequency. Lemme 2013: a 1972 bridge pickup, 2.2 H, under 470 pF and 200 kΩ peaks near 4.5 kHz. Macalister: a Strat pickup through its cable shows an impedance maximum at 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H corrected). Bench: a Strat pickup peaks near 8 kHz with 47 pF and near 2.4 kHz with 2,200 pF added; a 1960s 8 to 10 foot cable at about 300 pF put a 2.4 H pickup near 6 kHz while a 2,400 pF coiled cord dropped it just above 2 kHz, two octaves from a cable. Resistance is the other knob: the peak must be damped or a pickup 'sounds terrible'; above 5 kHz barely noticeable, near 3 kHz the 'ice pick', below about 1 kHz muddy (Zollner); pots load in parallel, 500 kΩ volume plus 500 kΩ tone presents 250 kΩ, about 200 kΩ with a 1 MΩ amplifier. The openai-dr model table (L 2.4 H, R_s 8.6 kΩ, C_p 100 pF assumed): cable C 200 to 900 pF moves f from 5.93 to 3.25 kHz while Q rises 1.88 to 2.43; at fixed 600 pF, R_L 1 MΩ to 111 kΩ drops Q 5.04 to 1.47 at 4.19 kHz. More capacitance lowers the peak, it does not merely flatten treble. The amplifier input is the last parallel element and not purely resistive: a 12AX7's Miller input capacitance C_gk + C_ga(1 − A_v) is about 88 pF at an assumed gain of −50 from datasheet values 1.6 and 1.7 pF. A buffer right after the pickup is the one way to make the response cable-independent; capacitance before the buffer still loads the coil. Bounds: lumped L, C and resistive load are models; eddy-current losses in covers and poles and the distributed winding make real coils lossier; DC resistance alone predicts neither output nor response. Evidence class: INTERVENTION (bench sweeps under stated loads) and MODEL; population: physical pickups and cables; outcome: measured impedance or voltage-response curves.

A coil's inductance stores energy magnetically and the cable's and coil's capacitance store it electrically; they exchange energy at one frequency where the source's reactance peaks and output rises. A resistive shunt bleeds that energy each cycle, so resistance sets the peak's height while the LC product sets its place. Coaxial capacitance per length is 2πε/ln(b/a), dielectric and geometry only; at audio frequencies the cable's wavelength is tens of kilometres, so it is a lumped capacitor, never a transmission line.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 10 cited, 10 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S10 (Lemme: L, C, f_r values, two-octave cable example), S21 (Lemme 2013: 2.2 H / 470 pF / 200 kΩ / 4.5 kHz), S22 (Macalister: 4.33 kHz impedance peak, inferred R, L, C), S23 (Belden 9778: 45 pF per foot, revised 2026-02-20), S13 (Zollner 5.4.1: ice pick and muddy bounds), S24 (Paiva thesis: cable capacitance dominated the measured shift), S25 (arXiv 1508.01246: lumped winding capacitance), S26 (arXiv 2409.19782: R proportional to N, L to N squared), S27 (Sound on Sound: pF per foot), S4 (ECC83S capacitances for the Miller term). S21 and S24 are 2013 evidence; fetched 2026-10-04.

• S10Helmuth Lemme, The Secrets of Electric Guitar Pickupsopened, quote confirmedAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S21Helmuth Lemme, Pickup Measuring Techniques (2013 article, hosted PDF)opened, quote confirmedA 1972 bridge pickup of 2.2 H under 470 pF and 200 kΩ peaks near 4.5 kHz; intermediate volume settings suppress the peak; bypass capacitors change the curve shape; the 470 pF measurement convention
• S22Macalister Electronics, Guitar Pickup Equivalent Circuitsopened, quote confirmedJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network
• S23Belden 9778 analog audio cable datasheet (revised 2026-02-20)opened, quote confirmed45 pF per foot (about 148 pF per metre), so a 20 foot cable adds about 900 pF
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019opened, quote confirmedAbout 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required
• S24Rafael Paiva, Circuit modeling studies related to guitars and audio processing (thesis summary)opened, quote confirmedCables measured alone and with a pickup; the distributed cable model reduces to a lumped capacitance that dominates the resonance shift; a buffer replaces the driving impedance but capacitance before it still loads the coil
• S25arXiv 1508.01246 (guitar pickup equivalent circuit modelling)opened, quote confirmedEffective terminal capacitance is a lumped representation of distributed winding capacitance
• S26Batchelor et al., Guitar Pickups I: Analysis of the Effect of Winding and Wire Gauge on Single Coil Electric Guitar Pickups (2024)opened, quote confirmedDedicated pickups varying only turns (5,000 to 12,000) at 42 and 44 AWG, impedance swept: resonant frequency falls exponentially with turns, peak magnitude rises linearly; R_dc proportional to N, L proportional to N squared
• S27Sound on Sound, Choosing Guitar Cablesopened, quote confirmedTypical guitar-cable capacitance per foot and its scaling with length
• S4JJ Electronic ECC83S datasheetopened, quote confirmedTransconductance, plate resistance and interelectrode capacitances (1.6 pF grid-cathode, 1.7 pF grid-anode) used for the Miller input-capacitance estimate

The page may say the guitar, its pots, its cable and the amplifier input form one resonant circuit and that the cable is a tone control worth about two octaves of resonance placement; it may say an all-analog passive chain is where this coupling is strongest because a buffer or digital front end terminates it. It must not imply that 'analog' preserves the string's spectrum: the passive chain imposes a resonant peak the string never produced.

The coil, the pots, the cable and the amplifier's input form one resonant circuit with a single peak: the coil's inductance and the total capacitance set where it sits, and the cable usually supplies several times the coil's own capacitance, so a short cable against a coiled cord moves the peak by about two octaves on the measured Strat coil (Lemme, Macalister). The adversary objected that the sources do not spell out that parallel resistance sets the peak's height; the transfer function does, and Zollner's 'ice pick' and 'muddy' bounds are the measured consequence, so the law stands with that clause marked as circuit theory rather than a quoted sentence. For the page: the player's cable is a tone control, and a passive chain imposes a resonant peak the string never produced, which cuts against reading 'analog' as 'the string's spectrum preserved'.

Write the volume pot as R_a = (1 − α)R_V in series toward the pickup and R_b = αR_V to ground, α the electrical wiper position. The cable and amplifier input load the wiper node, so V/E = [Z_A / (Z_s + Z_A)] · [Z_B / (R_a + Z_B)], a two-stage divider whose impedances all depend on frequency. At full rotation the pot is a parallel resistance setting Q; at any lower setting the series arm raises the source impedance the cable sees and forms an RC low-pass with the cable's capacitance (a 250 kΩ pot's series arm reaches about R_V/4 at mid-rotation, which with 500 pF gives a corner near 5.1 kHz in the simplified low-source-impedance case). Lemme's measured curves under 470 pF show intermediate volume settings substantially suppressing the main peak rather than shifting the full-volume curve down; a treble-bypass capacitor across the series arm yields another curve shape, and halving the pot value requires halving the bleed resistor and doubling the bleed capacitor. The tone control is a series capacitor to ground through the tone pot: at '10' the pot isolates the capacitor so it loads only at high frequency; at minimum the capacitor sits across the pickup, dwarfs the cable's capacitance (22,000 pF against 600 pF) and drops the LC frequency to about 680 Hz with 0.022 µF or 470 Hz with 0.047 µF at L = 2.4 H (Lemme states near 700 and 450 Hz). Bounds: standard passive Fender and Gibson wiring with the tone branch at the pickup-side node; a relocated tone branch or a treble-bleed changes the network; the 5.1 kHz figure is a model illustration, not a measured knob response; pot taper and tolerance set the actual curve. Evidence class: INTERVENTION (Lemme's measured sweeps at stated load and settings) and MODEL; population: a passive guitar circuit on a bench; outcome: a measured frequency response.

A potentiometer is a divider whose upper arm is in series with a capacitive load, so its ratio depends on frequency wherever the source impedance is not negligible; a passive pickup's source impedance is tens of kilohms and rises with frequency through its inductance, so no passive volume setting below maximum is frequency-neutral. A capacitor across an inductor relocates their shared resonance to 1/(2π√(LC)); the tone capacitor is a very large C added to the same tank as the cable.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 4 cited, 4 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S21 (Lemme 2013: volume sweeps under 470 pF, bypass-capacitor curves), S10 (Lemme: 700 and 450 Hz tone-down resonances, the series-resistance effect), S22 (Macalister: the L and C used in the arithmetic), S12 (Paiva, Pakarinen, Välimäki: the tone branch and pickup inductance forming a lower resonance). Fetched 2026-10-04; S21 is 2013 evidence.

• S21Helmuth Lemme, Pickup Measuring Techniques (2013 article, hosted PDF)opened, quote confirmedA 1972 bridge pickup of 2.2 H under 470 pF and 200 kΩ peaks near 4.5 kHz; intermediate volume settings suppress the peak; bypass capacitors change the curve shape; the 470 pF measurement convention
• S10Helmuth Lemme, The Secrets of Electric Guitar Pickupsopened, quote confirmedAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S22Macalister Electronics, Guitar Pickup Equivalent Circuitsopened, quote confirmedJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network
• S12Paiva, Pakarinen, Välimäki, Acoustics and Modeling of Pickupsopened, quote confirmedDistance-dependent magnetic nonlinearity; the tone branch interacting with pickup inductance to form a lower resonance; exciter-core loading caution

Bears on the page's 'play soft and it cleans up': when a player rolls the guitar volume back to clean up a tube amp, the passive circuit also darkens the signal before the amp, so the cleaner sound is two effects at once (less amplifier drive and a lower, duller front-end response). The page may say the volume knob changes tone, not only level, and should not attribute the whole clean-up to the amplifier.

Below full rotation the guitar's volume knob is a filter as much as a level: its series arm meets the cable's capacitance as an RC low-pass and Lemme's measured sweeps under a stated load show the resonant peak suppressed, not merely lowered; the tone knob at minimum drops the shared resonance to a few hundred hertz. The adversary found no source stating the volume-knob behaviour in those words; the measured curves do show it, and the arithmetic is the same L and C as A4. This is the confound the tube amplifier nonlinearity vector inherits (3.3): a player who 'cleans up with the volume knob' is darkening the signal before the amplifier at the same time as lowering its drive, so any test of the page's clean-up sentence must attenuate with a calibrated pad.

Batchelor, Gooding, Marriott, Chalashkanov, Tucker and Margetts (Nottingham Trent and Lincoln, arXiv 2409.19782, 2024) built dedicated Stratocaster-style single coils differing in one parameter at a time, turns across the commercial range 5,000 to 12,000 at 42 and 44 AWG, and swept each on a frequency response analyser for impedance. The abstract reports resonant frequency decreasing exponentially with turns and resonant peak magnitude increasing linearly with turns; the numerical table sits behind the abstract the legs opened. The winding model explains the direction: for fixed geometry L is proportional to N squared and R_dc to N, and the EMF depends on which turns link the string's changing flux, so adding turns raises generation and inductive loading together. Two coils of equal DC resistance on 42 versus 43 or 44 AWG carry different turn counts, inductances and resonances, which is why the industry's DCR figure measures neither output nor tone. Bounds: measured on Stratocaster-style single coils with the study's magnets and bobbin, as an impedance sweep rather than a string-driven output; range 5,000 to 12,000 turns; metal covers, steel poles and eddy-current losses in humbuckers add frequency-dependent loss the N-scaling omits; 'exponentially' is the authors' abstract wording and the exact exponent was not opened. Evidence class: INTERVENTION (one parameter varied, the rest held); population: bench-built pickups; outcome: impedance versus frequency.

A coil's inductance scales with the square of turns because each turn both produces flux and links every other turn's flux; resistance scales with wire length and inversely with wire area. With winding capacitance roughly fixed by the bobbin, the LC frequency falls about as 1/N, and against a fixed load the peak grows because the coil's reactance at resonance grows with L.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Refuted by the adversary on a source said less than the claim · Sources: 4 cited, 4 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both

S26 (Batchelor et al., arXiv 2409.19782, 2024: the controlled turn-count study), S28 (the same paper's ar5iv rendering, abstract text as opened by the exa leg), S10 (Lemme: inductance rises quadratically with turns and is the dominant quantity), S22 (Macalister: measured R, L, C of one Strat coil as the scale reference). S26 and S28 are 2024; fetched 2026-10-04.

• S26Batchelor et al., Guitar Pickups I: Analysis of the Effect of Winding and Wire Gauge on Single Coil Electric Guitar Pickups (2024)opened, quote confirmedDedicated pickups varying only turns (5,000 to 12,000) at 42 and 44 AWG, impedance swept: resonant frequency falls exponentially with turns, peak magnitude rises linearly; R_dc proportional to N, L proportional to N squared
• S28Batchelor et al., Guitar Pickups I, ar5iv HTML rendering (2024)opened, quote confirmedThe same controlled turn-count study's abstract text as opened by the exa leg
• S10Helmuth Lemme, The Secrets of Electric Guitar Pickupsopened, quote confirmedAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S22Macalister Electronics, Guitar Pickup Equivalent Circuitsopened, quote confirmedJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network

The page may say a pickup's turn count sets where its resonance sits and how tall it is, and that 'hot' pickups are darker because more turns means more inductance; it should not quote DC resistance as if it measured output or brightness.

Batchelor's one-variable study, Stratocaster-style coils wound from five to twelve thousand turns on two wire gauges, found the loaded resonance falling roughly exponentially with turns while the peak grows linearly; inductance scaling as the square of turns and resistance linearly is the textbook reason. The adversary noted the scaling laws are not in the opened abstract; they are elementary winding physics and the exact exponent sits in the paper's table, which no leg opened. The page should not quote a pickup's DC resistance as output or brightness; two coils of equal resistance on different gauges have different resonances.

At the input end Lenz's law and the static field make the transducer two-way: the pickup's permanent magnet exerts a steady attraction on the ferromagnetic string that alters its vibration, which is why very strong magnets or a pickup set too close are avoided (Lemme; Zollner's static-field chapter). At the output end a moving-coil loudspeaker obeys F = Bl·i for force and e_back = Bl·v for the voltage the moving coil generates (Bl the force factor, Klippel); its terminal impedance is Z = R_e + sL_e + (Bl) squared / Z_m(s), so the cone's mechanical resonance and the air load appear electrically at the terminals (Kraftmakher's loudspeaker experiments connect cone motion, back-EMF and the measured impedance peak). Through the output transformer that impedance is reflected to the tubes as Z_reflected = (N_p/N_s) squared times Z_speaker, so the plate sees a load rising at the speaker's resonance and above a few kilohertz as voice-coil inductance grows. The back-EMF opposes the motion that creates it and is dissipated through the amplifier's output impedance: the lower that impedance, the more the back-EMF current brakes the cone. A pentode or beam-tetrode push-pull stage with little or no negative feedback has an output impedance comparable to or above the speaker's own, so the cone is less braked than on a low-impedance solid-state output and the speaker's resonances ring longer; this is the inference, and the Fender 5F1's 22 kΩ feedback resistor from the secondary to a cathode is the schematic evidence that designers add feedback to lower that output impedance. Bounds: the bilateral laws hold for every magnetic pickup and moving-coil speaker; the under-damping magnitude depends on the specific amplifier's output impedance, which no leg could open for a named guitar amplifier (anthropic's search 'loudspeaker voice coil force factor Bl back EMF impedance curve damping' returned general loudspeaker theory only); Bl(x) is position-dependent, a loudspeaker nonlinearity in Klippel's list. Evidence class: MODEL (transducer theory) with opened laboratory demonstrations (Kraftmakher) and one opened schematic; population: physical transducers; outcome: measured impedance and back-EMF.

Reciprocity: any transducer converting current to force converts velocity to voltage with the same coupling coefficient Bl, so the mechanical side's impedance appears on the electrical side scaled by Bl squared. A source with high internal impedance cannot sink the back-EMF current, so it cannot brake the cone; negative feedback lowers a tube amplifier's output impedance by sensing the secondary voltage and correcting the drive.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Refuted by the adversary on a source said less than the claim · Sources: 8 cited, 8 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 97% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 32 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both

S29 (Klippel: Bl, back-EMF, Bl(x) nonlinearity), S30 (Kraftmakher, Eur. J. Phys. 31, 2010: cone motion, back-EMF and impedance measured), S31 (AudioHeritage: speaker self-damping through the amplifier's output circuit, lower impedance means more control), S3 (5F1: the 22 kΩ feedback path), S2 (OpenStax: reflected impedance), S32 (Amperatubes 2024: Z_primary = N squared times Z_load, EL34 at 3.4 kΩ against 8 Ω, 85 to 95 percent efficiency), S10 (Lemme: magnet pull on the string), S13 (Zollner 5.4.1: the static field). S30 is 2010 evidence, S32 is 2024; fetched 2026-10-04.

• S29Klippel, Loudspeaker Nonlinearities: Causes, Parameters, Symptomsopened, quote confirmedForce factor Bl as the coupling between mechanical and electrical sides; back-EMF u = Bl(x)·v; Bl(x) position dependence as a nonlinearity
• S30Kraftmakher, Loudspeaker experiments, European Journal of Physics 31 (2010)opened, quote confirmedLaboratory measurements connecting cone motion, back-EMF and the loudspeaker's electrical impedance peak at mechanical resonance
• S31AudioHeritage attachment on loudspeaker damping and back-EMFopened, quote confirmedA moving cone acts as a generator; its back-EMF damps the motion through the amplifier's output circuit, and the lower that impedance the more the back-EMF controls the cone
• S3Fender Champ 5F1 schematicopened, quote confirmedA minimal tube chain: two triode stages, interstage coupling capacitors, power tube, output transformer with grounded secondary and a 22 kΩ feedback path to a preamp cathode
• S2OpenStax University Physics Vol. 2, 15.6 Transformersopened, quote confirmedTransformer secondary voltage arises from shared changing flux, not conduction; reflected impedance scales with the square of the turns ratio
• S32Amperatubes, Output Transformers (2024-09-01)opened, quote confirmedZ_primary = N squared times Z_load; an EL34 wants about 3.4 kΩ against an 8 Ω speaker; energy crosses by coupling through a shared iron core at 85 to 95 percent efficiency
• S10Helmuth Lemme, The Secrets of Electric Guitar Pickupsopened, quote confirmedAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019opened, quote confirmedAbout 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required

This is the physics behind the tube amplifier nonlinearity vector's 'breaks into harmonics' and the sonic identity vector's 'the same rig sounds different at a volume and in a room': the speaker is a reactive load whose resonance and air loading talk back to the tubes through the transformer, so amplifier, speaker and cabinet are one coupled system. The page may say that in a tube rig the speaker is part of the circuit's behaviour, not a passive end; it should not quote a damping figure for a named amplifier until one is opened.

Both ends of the chain push back: the pickup's magnet loads the string it senses, and the speaker's moving coil generates a back-voltage that a high-impedance tube output cannot sink, so the cone is less braked and its resonance appears at the tube plates through the transformer. The bilateral laws are textbook and demonstrated in Kraftmakher's lab measurements; the under-damping magnitude for any named guitar amplifier was searched for and not opened, which is why the consequence is an inference, and the adversary was right that the magnet-loads-string clause rests on Lemme's and Zollner's statements rather than a measurement. The page may say that in a tube rig the speaker is part of the circuit's behaviour, not a passive end; it may not quote a damping figure until one is opened, and 4.8 uses the same physics to explain why a capture through a resistive dummy load is a model of a different operating point.

For a smooth, memoryless, asymmetric gain stage (a common-cathode triode is the measured case), the nth harmonic relative to the fundamental scales as about the (n-1)th power of input amplitude while drive stays well under the stage's headroom, so second order dominates at small drive and third order takes over as drive approaches the knee. Measured: Zollner's input stage of a guitar amp reaches about 3 percent harmonic distortion (harmonic-to-output voltage ratio, one ECC83 stage, 250 V supply) at 300 mV rms input, with 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border depending on grid resistor, supply voltage and tube specimen (S2, 2007). Dempwolf and Zölzer measured a common-cathode stage at 500 Hz with 2, 4 and 8 V input and found progressive limiting with both even and odd harmonics (S1, 2011). Keeports drove a valve preamp with a 200 Hz tone: moderate overdrive showed prominent 2nd and 4th harmonics and a weak 3rd, and raising gain strengthened the 3rd and weakened the 4th (S16, 2017, one amplifier). The load line, not the tube alone, sets the transfer (S17). Boundary: a single-ended stage below grid conduction and above cutoff; the power series is an asymptote, and real tubes' perveance and amplification factor vary with operating region (S1). Population: bench stages and one amplifier; outcome: harmonic spectrum against input level; evidence class: INTERVENTION with MODEL fit. Says nothing about the power stage (A6) or memory (A4, A5).

Expand y = a1 x + a2 x^2 + a3 x^3 about the bias point with x = A cos(wt): B2/B1 is proportional to A and B3/B1 to A^2. The space-charge three-halves law gives a2 > 0 and a3 < 0 for a triode, so the even term leads and the odd term both compresses the fundamental and grows faster; an asymmetric curve (soft cutoff one way, grid conduction the other, S15) is what admits even terms at all.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 5 cited, 5 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S2 (3 percent at 300 mV rms; 2nd below, 3rd from 0.25 to 1 V; obs 2007), S1 (2, 4, 8 V waveforms and the 4 V harmonic sweep; 2011), S16 (200 Hz, even-to-odd drift with gain; 2017), S17 (load-line mechanism). Observation: the harmonic ratios at the stated levels. Inference: the power-law scaling between measured points. All fetched 2026-10-05.

• S1Dempwolf and Zölzer, A physically-motivated triode model for circuit simulations, DAFx-11 (2011)opened, quote confirmedA common-cathode triode stage measured at 500 Hz with 2, 4 and 8 V input shows progressive limiting with both even and odd harmonics; the circuit model reproduces most but not all of it (INTERVENTION + MODEL; fetched 2026-10-05).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)opened, quote confirmedThe input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S16Keeports, The warm, rich sound of valve guitar amplifiers, Physics Education 52 025010 (2017)opened, quote confirmedA 200 Hz tone into a moderately overdriven valve preamp shows prominent 2nd and 4th harmonics and a weak 3rd; raising preamp gain strengthens the 3rd and weakens the 4th, a shift from even toward odd with drive (INTERVENTION, one amplifier, one experimenter).
• S17Blencowe (Valve Wizard), The Common Gain Stage (undated)opened, quote confirmedThe stage's input-to-output transfer is the intersection of the valve characteristic with the circuit's load line; bias, plate load, source impedance and cathode feedback set the usable excursion (ARGUMENT).
• S15Rutt, Vacuum Tube Triode Nonlinearity as Part of the Electric Guitar Sound, AES 76th Convention preprint 2141 (1984)opened, quote confirmedGrid limiting is a soft limiting effect at the upper end of the load line; a class A triode stage direct-coupled to a cathode follower limits both the positive and negative sides of the guitar waveform (MODEL, load-line analysis with oscilloscope traces).

The page may keep 'dig in and it breaks into harmonics' for the first stage and must say which harmonics: a softly driven triode stage adds mostly second order, a hard-driven one adds third and higher; the sentence describes a smooth knee, not a tube.

In a single triode stage driven below its headroom, each harmonic's share falls as a power of input amplitude, so softening the pick cuts distortion steeply and shifts what remains toward even order: Zollner's measured stage reaches about three percent at three hundred millivolts, Dempwolf's stage limits progressively at two, four and eight volts, and Keeports watched the third harmonic grow and the fourth fall as gain rose. The adversary held that this generalizes from specific stages; the axiom already confines itself to a single-ended triode with the load line, not the tube, setting the transfer. For the page, 'dig in and it breaks into harmonics' may stay for the first stage with the harmonics named, second order first, third and higher as drive rises; it describes a smooth knee, not a property of tubes.

Where the dominant nonlinearity near threshold is of order n, total harmonic distortion changes tenfold over 20/(n-1) dB of input: 20 dB for a second-order knee, 10 dB for a symmetric third-order knee, and only a few dB for a hard clipper whose local order is very high. Touch-controlled clean-up therefore requires a soft knee whose threshold sits inside the span of input levels the hands produce; a hard-knee stage anywhere in the chain (a diode-clipping pedal, a hard-limited stage) imposes its own narrow window on the whole chain whether the element is a valve, a diode or a transistor. The window is a MODEL result derived from the same power series A1 measures (S1, S8); the hard-clipper end is consistent with Yeh, Abel and Smith's measured pedal spectra, where measured circuits showed strong even-order content that symmetric clipping models did not reproduce (S14, 2007). Boundary: a derived bound, not a measured window on any complete amplifier; no leg opened a joint dataset of input amplitude by gain by guitar volume by pedal drive. Population: circuit models and bench pedal measurements; outcome: THD against input level; evidence class: MODEL with INTERVENTION support at the pedal. The hands' dynamic span in dB is not measured here (the hands on the string vector owns it).

THD is proportional to A^(n-1) in the weakly nonlinear region, so a decade of THD spans 20/(n-1) dB of input amplitude; as the knee hardens, the local order n rises and the span collapses toward the clip threshold itself.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Refuted by the adversary on a source said less than the claim · Sources: 3 cited, 3 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both

S1 (progressive limiting with amplitude, 2011), S8 (the series expansion and the symmetric-stage result, 2003), S14 (measured pedal nonlinearity ahead of the amplifier, 2007). Observation: measured spectra at discrete amplitudes. Inference: the 20/(n-1) rule. Fetched 2026-10-05.

• S1Dempwolf and Zölzer, A physically-motivated triode model for circuit simulations, DAFx-11 (2011)opened, quote confirmedA common-cathode triode stage measured at 500 Hz with 2, 4 and 8 V input shows progressive limiting with both even and odd harmonics; the circuit model reproduces most but not all of it (INTERVENTION + MODEL; fetched 2026-10-05).
• S8Boyk and Sussman, Small-Signal Distortion in Feedback Amplifiers for Audio, Caltech (2003)opened, quote confirmedPush-pull stages cancel even-order terms; in the crossover region relative distortion worsens for smaller excitations with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse, the effect disappearing at larger excitation (MODEL, circuit simulation).
• S14Yeh, Abel, Smith, Simplified, physically-informed models of distortion and overdrive guitar effects pedals, DAFx-07 (2007)opened, quote confirmedMeasured distortion pedals at 220 Hz and 200 mV show a strong even-order nonlinearity not reproduced by symmetric clipping models; a pedal distorts the waveform before any valve receives it (INTERVENTION + MODEL).

The page's 'play soft and it cleans up' is true of a soft knee placed inside the hands' range; a reader who adds a hard-clipping pedal or runs a stage far past its knee loses the window, and the page should say so instead of attributing it to tubes.

How wide the clean-up window is follows from the knee's order against the span of levels the hands produce: a second-order knee gives a tenfold distortion change over twenty decibels of input, a hard clipper over only a few, so a diode-clipping pedal anywhere ahead imposes its own narrow window on the whole chain whatever the device class. This is a derived bound, and the adversary is right that no source states 'device class' or 'dynamic span' and that no joint dataset of input by gain by guitar volume by pedal was opened; the player's dynamic span in decibels is itself unmeasured here. The page's clean-up sentence is true of a soft knee placed inside the hands' range and false of a chain with a hard clipper in front, and it should say so instead of crediting tubes.

Softer input cleans the output only when it lowers the drive at every stage that was distorting (a pedal ahead of the amplifier, each cascaded preamp stage, the power stage, the speaker) and only after the chain's stored states (A4, A5, A8) have recovered. A gain control lowers drive only for stages downstream of it; a master volume after a clipped preamp quiets the distortion but does not remove it; a pedal whose own clipping stage stays driven delivers a distorted waveform a clean amplifier reproduces, and measured pedals generate strong even-order content before any valve sees the signal (S14). The supply's reaction is present at low settings: on the Bassman 5F6-A the screen-supply drop occurred at volume 2 of 10 and grew at 10 of 10 (S4). The guitar's volume control is not a scalar attenuator: turning it down raises the source impedance seen by the cable and shifts the pickup-cable resonance, so part of what is heard as clean-up by the guitar volume is treble loss (S18; the passive-circuit physics belongs to the pickup and the circuit vector). Boundary: a deduction from stage topology with measured support at the pedal and the supply; no measured input-by-gain-by-guitar-volume-by-pedal matrix was opened by any leg. Population: circuit deduction and bench measurements; outcome: which stage's distortion a given control changes; evidence class: MODEL with INTERVENTION support (S4, S14).

Stage j distorts when its drive A times the upstream transfer exceeds its present headroom H_j; attenuation applied at point k changes only the drives downstream of k, and nonlinearity already committed upstream is a new waveform, not a gain that a later control can reverse.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Refuted by the adversary on a step in the reasoning · Sources: 4 cited, 4 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both

S14 (pedal nonlinearity measured ahead of the amp, 2007), S4 (supply drop at volume 2 of 10, 2018), S18 (volume-pot loading), S2 (the source-impedance interaction at the amplifier input, 2007). Observation: the pedal spectra and supply traces. Inference: the per-stage headroom condition. Fetched 2026-10-05.

• S14Yeh, Abel, Smith, Simplified, physically-informed models of distortion and overdrive guitar effects pedals, DAFx-07 (2007)opened, quote confirmedMeasured distortion pedals at 220 Hz and 200 mV show a strong even-order nonlinearity not reproduced by symmetric clipping models; a pedal distorts the waveform before any valve receives it (INTERVENTION + MODEL).
• S4Erbes, Leontopoulos, Zollner, Audibility of different power supplies in a guitar amplifier, DAGA 2018opened, quote confirmedBassman 5F6-A: screen-supply voltage drops about 60 V under a guitar piece, the drop present at volume 2 of 10 and larger at 10 of 10, with attack compression and enhanced fade-out; ABX with 17 experienced listeners, 15 trials each, 255 per condition, threshold 144, scores 158, 147, 172, 140, three of four significant, effect called very small (INTERVENTION, bench + listening).
• S18Ironstone Pickups, Guitar impedance matching (LCR) (undated)opened, quote confirmedPickup inductance with cable and self-capacitance forms a resonant circuit; turning the volume pot down raises source impedance and loses treble, so clean-up by the guitar volume is partly a low-pass (ARGUMENT).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)opened, quote confirmedThe input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).

The page's clean-up sentence holds for the owner's chain only if the drive is set so the first clipping stage sits inside the hands' range and nothing ahead of it is hard-clipping; any test of the claim must attenuate with a calibrated pad, not the guitar's volume knob.

Softer input cleans the output only when every distorting stage falls back under its own headroom, the pedal ahead of the amplifier included, and only after the chain's stored states have recovered; a master volume after a clipped preamp quiets the distortion without removing it, and Yeh's measured pedals make their own even-order content before any valve sees the signal. The adversary's note that this is deduced from topology rather than measured is accepted and recorded in the axiom. Its practical edge is the guitar-volume confound shared with 2.5: part of what the ear calls clean-up from the guitar knob is treble loss, so the page's claim can only be tested with a calibrated pad.

When a driven grid conducts, its current charges the coupling capacitor and shifts the stage's bias toward cutoff; the shift decays only through the grid resistor's RC path, so a soft note immediately after a hard one is attenuated, distorted or silenced, the opposite of clean-up. Measured: on a Fender Super Reverb with the global negative-feedback loop deactivated, a 1 kHz tone overdriving the power amplifier from t = 0 and reduced by 20 dB at t = 100 ms made the loudspeaker voltage collapse for a short time (S3, Zollner Fig. 10.4.18, 2007); the same measurement shows saddle-point distortions at the zero crossings in overdrive that Zollner traces to grid-current potential shifts, not to bias setting or transformer saturation. The recovery mechanism (coupling capacitor charged by grid current, bleeding through the grid-leak resistor; the shift pushing a class AB stage toward class B) is the designer's account in S10; grid limiting is a soft limit and the triode plus direct-coupled cathode follower limits both polarities (S15, 1984). Boundary: one bench amplifier for the measured collapse, feedback disabled; Zollner says not to dramatize it (hearing's post-masking covers part) and not to ignore it, since time constants vary by design. Population: one measured amplifier plus designer tutorials; outcome: output envelope after a level step; evidence class: INTERVENTION (S3) with ARGUMENT (S10) and MODEL (S15). Severity depends on grid-stopper value, coupling capacitance and grid-leak resistance.

Charge conservation on the coupling capacitor: rectified grid current must go somewhere, and where it goes is the bias point; the release is the RC of capacitor and grid resistor, independent of the present input amplitude.

Granted tier: FORGED, set by the code gate on October 4, 2026 · Survived the adversary's attack · Sources: 3 cited, 3 opened and confirmed

• survived a grounded cross-lab adversary (google vs authors ["anthropic"])
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED FORGED: survived a grounded cross-lab adversary AND primary sources confirmed

S3 (Fig. 10.4.18, the 20 dB step at 100 ms and the collapse; obs 2007), S10 (charging and recovery mechanism), S15 (grid limiting as a soft limit, both polarities; 1984). Observation: the loudspeaker-voltage trace. Inference: the RC release as the governing time constant. Fetched 2026-10-05.

• S3Zollner, Physics of the Electric Guitar, ch. 10.4 Phase splitter and driver (2007; English edition)opened, quote confirmedFig. 10.4.18: Super Reverb with global feedback deactivated; a 1 kHz tone overdriving the power amp reduced by 20 dB at t = 100 ms makes the loudspeaker voltage collapse briefly; saddle-point distortions at zero crossings in overdrive trace to grid-current potential shifts, not bias or transformer saturation (INTERVENTION).
• S10Aiken Amplification, What is Blocking Distortion? (undated)opened, quote confirmedGrid current under overdrive charges the coupling capacitor, shifting bias toward cutoff (bias excursion) and pushing a class AB stage toward class B; recovery bleeds through the grid resistor; blocking distortion is not clipping (ARGUMENT).
• S15Rutt, Vacuum Tube Triode Nonlinearity as Part of the Electric Guitar Sound, AES 76th Convention preprint 2141 (1984)opened, quote confirmedGrid limiting is a soft limiting effect at the upper end of the load line; a class A triode stage direct-coupled to a cathode follower limits both the positive and negative sides of the guitar waveform (MODEL, load-line analysis with oscilloscope traces).

The page's 'play soft and it cleans up' must be bounded in time: it is false for the first tens to hundreds of milliseconds after a hard passage on a stage that has drawn grid current.

After a loud passage a tube stage that drew grid current has shifted its own bias toward cutoff, and the soft note that follows is attenuated, distorted or briefly silenced until the coupling capacitor bleeds off through the grid resistor: Zollner's Super Reverb trace, a twenty-decibel step down after a hundred milliseconds making the loudspeaker voltage collapse, is the measured case, and it survived the cross-lab adversary intact. This is the sharpest bound on the page's sentence: 'play soft and it cleans up' is false for the first tens to hundreds of milliseconds after a hard passage, and the amplifier remembers what the hands just did.

A high-impedance power supply (rectifier drop plus winding resistance against a reservoir capacitor) lets the rail fall when the output stage's average current rises, lowering headroom and gain during the note and restoring them as it decays; it changes an envelope rather than an instantaneous waveform, so it is compression, and it exists only where current draw rises with level (class AB; an ideal class A stage draws near-constant current, S9). Measured: a Fender Deluxe supply node fell from 247 V to 210 V while fully driven from t = 0.3 to 1.7 s, and maximum level and harmonic distortion changed with it (S2, 2007). On a Bassman 5F6-A with three power supplies, the screen-supply voltage dropped about 60 V with a slight delay under a guitar piece, the drop was present at volume 2 of 10 and larger at 10 of 10, and at 10 of 10 the output showed the attack peak reduced and the fade-out enhanced (S4, 2018). In the same study's ABX test, 17 listeners with prior audio experience (guitarists, music production or audio study) did 15 trials each, 255 trials per condition; against a Bonferroni-corrected threshold of 144 correct, the four conditions scored 158, 147, 172 and 140, three of four significant, and the authors call the effect very small and probably of little relevance in practice, doubting the low-noise comparison because a faint tone gave a cue (S4). Boundary: two amplifiers measured; one listening test over headphones with recorded stimuli at volume 2 of 10, trials not fully independent (the authors say so). Population: bench measurements and 17 experienced listeners; outcomes: rail voltage against time, output envelope, ABX discrimination; evidence class: INTERVENTION. The listening result is a discrimination of supplies, not a preference and not a measure of touch response.

C dB/dt = i_rectifier - i_load: a sustained rise in load current drains the reservoir through the supply's equivalent resistance with time constant R_eq C, and the rail recovers on the same constant when demand falls; the audible effect is whatever the falling rail does to gain, bias and available swing.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a number larger than the sources support · Sources: 3 cited, 3 opened and confirmed

• adversary REFUTED the claim
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED PROBABLE: one of {grounded cross-lab adversary, verified sources} but not both
• CLAMPED: claimed FORGED -> granted PROBABLE (gate lowers, never raises)

S2 (247 to 210 V, Deluxe; obs 2007), S4 (60 V screen drop, volume 2 of 10, attack compression and enhanced fade-out, ABX 158/147/172/140 of 255 against 144; 2018), S9 (class AB current demand; class A exception). Observation: the rail traces and the ABX counts. Inference: the RC model and the class A exemption. Fetched 2026-10-05.

• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)opened, quote confirmedThe input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S4Erbes, Leontopoulos, Zollner, Audibility of different power supplies in a guitar amplifier, DAGA 2018opened, quote confirmedBassman 5F6-A: screen-supply voltage drops about 60 V under a guitar piece, the drop present at volume 2 of 10 and larger at 10 of 10, with attack compression and enhanced fade-out; ABX with 17 experienced listeners, 15 trials each, 255 per condition, threshold 144, scores 158, 147, 172, 140, three of four significant, effect called very small (INTERVENTION, bench + listening).
• S9Aiken Amplification, What is Sag? (undated)opened, quote confirmedTube-rectifier drop varies with current; class AB draws far more current at full output than at idle; sag requires level-dependent current and is weak in true class A (ARGUMENT).

The page may say a tube amplifier with a sagging supply compresses the attack and swells the decay; it may not say this is what makes it clean up when played soft, and the measured audible difference between supplies is small.

Supply sag is a slow gain state, not clipping: a resistive supply's rail falls as the output stage draws more current, compressing the attack and swelling the decay, and only where current rises with level, so an ideal class A stage does not sag. Two amplifiers were measured and one ABX run, Erbes's seventeen listeners on a Bassman with three supplies, where three of four comparisons cleared the threshold and the authors called the effect very small and probably of little practical relevance; the adversary objected that 'small' generalizes from that one test, and the axiom carries the single-study bound. The page may say a sagging supply compresses attacks and swells tails; it may not say sag is what makes the amp clean up when played soft.

Two matched halves driven in antiphase cancel even-order terms differentially, so a power stage's own distortion is odd-order and any even harmonic measured at its output is a balance measurement (S8; Zollner's k3 = 12.3 percent under the anti-phase condition, S2). At the zero crossing the distortion is fixed in size by the bias, so its share of the signal grows as the signal shrinks: Boyk and Sussman's simulated push-pull stage shows relative distortions worsening for smaller excitations both with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse than without it, the effect disappearing as excitation grows (S8, 2003). Zollner's Super Reverb measurement shows saddle-point distortions at the zero crossings in overdrive, caused by grid-current potential shifts rather than bias setting or transformer saturation (S3, 2007). Once the stage clips, loop gain collapses and the feedback that lowered output impedance stops acting, so the response against the speaker's impedance curve changes with level as well (S11, designer's account; one leg). Boundary: even-order cancellation is exact only for matched halves; the crossover result is a circuit simulation plus one bench trace; the output-impedance clause rests on an ARGUMENT source and one leg. Population: circuit models and one measured amplifier; outcome: harmonic and intermodulation spectrum against excitation size; evidence class: MODEL (S8) with INTERVENTION (S3). Idle bias and feedback depth set the notch's size.

y(u) = f(V_b + u) - f(V_b - u) has no even powers; a fixed dead zone of width d contributes a distortion whose relative size scales as d/A, the inverse of the clean-up law in A1.

Granted tier: FORGED, set by the code gate on October 4, 2026 · Survived the adversary's attack · Sources: 4 cited, 4 opened and confirmed

• survived a grounded cross-lab adversary (google vs authors ["anthropic"])
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED FORGED: survived a grounded cross-lab adversary AND primary sources confirmed

S8 (push-pull symmetry; crossover worsening at small excitation; 18 dB feedback figures; 2003), S3 (saddle points at zero crossings; 2007), S2 (k3 under anti-phase; 2007), S11 (feedback depth and output impedance). Observation: simulated spectra and the measured trace. Inference: the d/A scaling. Fetched 2026-10-05.

• S8Boyk and Sussman, Small-Signal Distortion in Feedback Amplifiers for Audio, Caltech (2003)opened, quote confirmedPush-pull stages cancel even-order terms; in the crossover region relative distortion worsens for smaller excitations with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse, the effect disappearing at larger excitation (MODEL, circuit simulation).
• S3Zollner, Physics of the Electric Guitar, ch. 10.4 Phase splitter and driver (2007; English edition)opened, quote confirmedFig. 10.4.18: Super Reverb with global feedback deactivated; a 1 kHz tone overdriving the power amp reduced by 20 dB at t = 100 ms makes the loudspeaker voltage collapse briefly; saddle-point distortions at zero crossings in overdrive trace to grid-current potential shifts, not bias or transformer saturation (INTERVENTION).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)opened, quote confirmedThe input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S11Aiken Amplification, What is Negative Feedback? (undated)opened, quote confirmedGuitar amps typically use 6 to 10 dB of global feedback; feedback lowers output impedance and raises damping factor only while loop gain exists (ARGUMENT).

The page's clean-up sentence is inverted for the power stage's crossover region: at low level a class AB stage's distortion share rises, so the page should locate clean-up in the preamp knee, not in the amplifier as a whole.

A push-pull power stage cancels even-order terms by symmetry, so its own distortion is odd and any even harmonic at its output is a measure of imbalance; and because the crossover notch is fixed in size, its share of the signal grows as the signal shrinks, Boyk and Sussman's simulation showing relative distortion worsening at small excitation with and without feedback. This survived the adversary and is the second inversion of the page's sentence: at low level a class AB stage's distortion share rises, so clean-up belongs to the preamp knee, not to the amplifier as a whole, and the folk 'tubes make even harmonics' is wrong for the power stage.

Core flux scales as primary voltage over frequency, so for a given output voltage the lowest notes drive the iron hardest; two guitar-amplifier output transformers measured by Paiva, Pakarinen, Välimäki and Tikander introduced distortion at low frequencies only, below about 100 Hz for the Fender NSC041318 and 30 Hz for the Hammond T1750V (S13, 2011). Across level the term is not monotonic: magnetic hysteresis raises low-frequency distortion at relatively low signal levels and saturation raises it at high levels, and the distortion falls rapidly with increasing frequency (S12, Jensen's measured curves). Playing softer therefore does not clean the transformer's low-frequency contribution and can raise its share; above the low band the transformer's effect is linear filtering (leakage inductance, winding resistance and capacitance), not distortion. Boundary: two transformers measured; Jensen's curves are a manufacturer's handbook chapter. Population: bench transformers; outcome: harmonic distortion against frequency and level; evidence class: INTERVENTION (S13) and ARGUMENT with measured figures (S12). DC imbalance in the primary pre-biases the core and moves these thresholds (S12).

v = N A dB/dt gives B_pk = V_pk / (2 pi f N A); permeability falls near saturation and the B-H loop's hysteresis is a fixed-size nonlinearity whose relative size grows as the flux swing shrinks.

Granted tier: PROBABLE, set by the code gate on October 4, 2026 · Survived the adversary's attack · Sources: 2 cited, 1 opened and confirmed

• survived a grounded cross-lab adversary (google vs authors ["anthropic"])
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED FORGED: survived a grounded cross-lab adversary AND primary sources confirmed

S13 (100 Hz and 30 Hz bounds on two measured transformers; pub 2011-02-22; captured by the exa leg 2026-10-04, Springer JS-walled to a plain fetch 2026-10-05), S12 (hysteresis at low level, saturation at high level, rapid fall with frequency; fetched 2026-10-05). Observation: the measured distortion bands. Inference: the flux law and the non-monotonic shape between measured points.

• S13Paiva, Pakarinen, Välimäki, Tikander, Real-Time Audio Transformer Emulation for Virtual Tube Amplifiers, EURASIP J. Adv. Signal Process. 2011:347645 (2011-02-22)did not openMeasured Fender NSC041318 and Hammond T1750V output transformers introduce distortion at low frequencies only, below about 100 Hz for the Fender and 30 Hz for the Hammond (INTERVENTION + MODEL; opened by the exa leg 2026-10-04; Springer answers a plain fetch with a JavaScript challenge).
• S12Whitlock, Audio Transformers (handbook chapter, Jensen Transformers; uploaded 2014)opened, quote confirmedMagnetic hysteresis causes increased harmonic distortion for low-frequency signals at relatively low levels and saturation at high levels; distortion decreases rapidly with increasing frequency (ARGUMENT with measured curves).

The page may not attribute warmth or clean-up to the output transformer; its measured contribution is confined to the lowest guitar notes and is one of the places where softer playing raises relative distortion.

The output transformer distorts only the lowest notes, below about a hundred hertz on a measured Fender unit and thirty on a Hammond, and across level the term is not monotonic: hysteresis raises it at low level, saturation at high, so softer playing can raise its share. It survived the adversary on Jensen's handbook curves alone after the Springer page refused a plain fetch, which is why the Paiva measurement stands on one leg's reading. The page may not attribute warmth or clean-up to the transformer; above the low band it is linear filtering, and in the low band it is one of the places where playing softer does the opposite of cleaning up.

A guitar loudspeaker measured at 30 W input, 1.5 m on axis, gave total harmonic distortion relative to the fundamental of -9 dB at 70 Hz (2nd -11, 3rd -13), -25 dB at 100 Hz, -36 dB at 140 Hz, -39 dB at 200 and 400 Hz and -37 dB at 4 kHz; the authors put THD near resonance at roughly 10 to 30 percent and in the high frequencies around 1 percent, and state that much of the salient timbre the loudspeaker imparts comes from its linear filtering (S6, 2008, one driver). Klippel's catalogue puts the low-frequency distortion on force factor Bl(x) and suspension stiffness Kms(x), with cone and surround nonlinearity appearing only after break-up, the break-up irregularities themselves being linear modal behaviour (S19). Fundamental compression is separable into thermal and motor causes: in Klippel's example driver swept 1 to 8 V rms, the fundamental was reduced by 7 dB below resonance, 5 dB at resonance and about 2 dB above it relative to a linear system, the current-based measure isolating the thermal part (S5). On a guitar loudspeaker driven at 20 W average and 60 V peak, voice-coil resistance varied by more than 15 percent, the amplitude response varied 1 to 2 dB within a few seconds from heating, and excursion was asymmetric (+4 mm against -3 mm, a dynamic DC displacement) (S7). Heat releases over seconds, so thermal compression persists into the soft note after a loud passage. Boundary: single drivers in each measurement; the AN12 driver is an example, not a guitar speaker; 30 W was noted as softer than concert level. Population: bench loudspeakers; outcomes: harmonic distortion against frequency, fundamental compression against drive, voice-coil resistance against time; evidence class: INTERVENTION. Room, microphone and playback level after the speaker belong to the room the volume and the record vector.

F = Bl(x) i with Bl falling as the coil leaves the gap and Kms(x) stiffening, both displacement-domain and strongest where excursion is largest (below and at resonance); R_e(T) rising with coil temperature cuts current under voltage drive, a gain change with a thermal time constant; above resonance displacement falls as 1/f^2, so the mid and high band is dominated by linear modal filtering.

Granted tier: FORGED, set by the code gate on October 4, 2026 · Survived the adversary's attack · Sources: 4 cited, 4 opened and confirmed

• survived a grounded cross-lab adversary (google vs authors ["anthropic"])
• citations: refetched 95% (>=90%), resolved 100% (>=80%) -> OK
• dead links: 1 of 19 source(s) would not open, a miss counted in the ratio above, never a misquote
• EARNED FORGED: survived a grounded cross-lab adversary AND primary sources confirmed

S6 (Table 1 THD by frequency at 30 W; 10 to 30 percent near resonance; linear-filtering verdict; 2008), S5 (7, 5, 2 dB compression on a 1 to 8 V sweep), S7 (>15 percent R_e change, 1 to 2 dB in seconds, +4/-3 mm), S19 (which parameter causes which band's distortion). Observation: every number above. Inference: the band sorting by displacement and the thermal persistence into the next phrase. Fetched 2026-10-05.

• S6Yeh, Bank, Karjalainen, Nonlinear modeling of a guitar loudspeaker cabinet, DAFx-08 (2008)opened, quote confirmedGuitar loudspeaker at 30 W, 1.5 m on axis: THD -9 dB at 70 Hz, -25 dB at 100 Hz, -36 dB at 140 Hz, -39 dB at 200 and 400 Hz, -37 dB at 4 kHz; roughly 10 to 30 percent near resonance, about 1 percent at high frequencies; much of the salient timbre comes from linear filtering (INTERVENTION + MODEL).
• S5Klippel Application Note AN12, Causes for Amplitude Compression (undated)opened, quote confirmedExample driver swept 1 to 8 V rms: fundamental reduced 7 dB below resonance, 5 dB at resonance, about 2 dB above it against a linear system; current-based measure isolates thermal from nonlinear compression (INTERVENTION).
• S7Klippel, Perceptual and Physical Evaluation of Guitar Speakers (undated)opened, quote confirmedGuitar loudspeaker at 20 W average and 60 V peak: excursion +4 mm against -3 mm (dynamic DC displacement), voice-coil resistance varying more than 15 percent, 1 to 2 dB amplitude variation within seconds from heating (INTERVENTION).
• S19Klippel, Loudspeaker Nonlinearities: Causes, Parameters, Symptoms (undated)opened, quote confirmedDominant large-signal nonlinearities are Bl(x), Kms(x), Le(x) and Le(i); low-frequency distortion comes from displacement-driven stiffness and force-factor nonlinearity; break-up irregularities are linear modal behaviour with cone nonlinearity appearing after break-up (MODEL with measurements).

The page may say the speaker is part of the tone; it must say that part is mostly a fixed filter, that its distortion is a low-note phenomenon at high level, and that its heat-driven compression does not reverse when the hands soften.

A guitar speaker's voicing is mostly a fixed linear filter; its distortion is a low-note, high-level term near resonance and its heat-driven compression, a voice-coil resistance change of more than fifteen percent and one to two decibels within seconds on a measured guitar driver, releases over seconds and so persists into the soft note that follows a loud passage. Yeh's cabinet table, Klippel's compression split and the causes catalogue carry it, and it survived the adversary. Together with A4 and A5 it closes the memory argument: the sound at any instant depends on the previous second of playing, and what the hands cannot do by softening is anything the chain remembers. 4.1 states the same speaker physics from the listener's side.

In a comparison of an analog tube chain against a digital model, the measured and perceived difference is governed by the least-matched block inside the declared device-under-test boundary, not by the analog or digital label of the gain stages. For trained listeners judging recorded output, the transducer and its acoustic environment dominate whenever they are not physically shared: the loudspeaker is the largest linear filter in the chain and a major nonlinear generator whose distortion is negligible at small signal and rises steeply with amplitude. A comparison in which one arm drives a real cabinet in a room and the other a headphone or full-range monitor is a comparison of transducers and carries no information about the amplifier model. Where the cabinet is the modeled block, cabinet type sets the result before model architecture does: at 112 dB(A) and one metre, a closed 4x12 modeled by a linear FIR alone reached a 30.2 dB signal-to-error ratio and 20 listeners could not tell it from the recording (45.5 percent correct over 200 ABX responses, 10 trials each), while an open 1x12 under the same protocol was told apart at 78 percent correct (p < .001) at 11.7 dB SER (DAGA 2026; INTERVENTION on 20 listeners, MODEL for the SER). Serious amplifier-model evaluations fix the block: one common cabinet impulse response, one common EQ and loudness matching across all stimuli (Cassidy and De Sena 2023).

The nonlinear loudspeaker parameters Bl(x), Cms(x) and Le(x) vary with cone excursion x, so harmonic and intermodulation products grow faster than linearly with drive; an open-back 1x12 runs at higher excursion for a given SPL than a closed 4x12, so its nonlinear residue is larger and a linear model misses more of it. A microphone at a fixed position samples one point of a radiation pattern set by cone breakup and baffle geometry; two cabinets, or one cabinet at two microphone positions, differ by more decibels across 1 to 5 kHz than any published amplifier model differs from its target.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a step in the reasoning · Sources: 4 cited, 4 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S13 (Klippel: dominant nonlinear loudspeaker parameters, small-signal to large-signal rise; ARGUMENT from a measurement method page). S4 (DAGA 2026, Table 3: cabinet-type dependence of ABX discrimination, 20 listeners, 10 trials per condition, 112 dB(A); INTERVENTION, re-opened by this writer 2026-10-05). S1 (Cassidy and De Sena 2023: common IR and loudness-match control; INTERVENTION, re-opened by this writer). S14 (Pakarinen and Yeh review: amplifier model separated from cabinet model; ARGUMENT).

• S13Klippel, Separated loudspeaker distortion (measurement method page)opened, quote confirmedForce factor Bl(x) and compliance Cms(x) dominate loudspeaker nonlinear distortion; nonlinear distortion is negligible at small signal and rises rapidly with amplitude and depends on stimulus spectrum.
• S4Hybrid Modeling of Guitar Cabinets Using FIR Filters and Neural Networks (DAGA 2026)opened, quote confirmed20 listeners, 10 ABX trials per condition, recordings level-matched to 112 dB(A) at one metre; Table 3: 1x12 linear 78 percent correct at SER 11.7 dB, 1x12 DNN 57.5 percent at 19.8 dB (p = .020), 4x12 linear 45.5 percent at 30.2 dB, 4x12 DNN 50.5 percent at 42.0 dB, neither 4x12 condition above chance.
• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)opened, quote confirmedAll six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)opened, quote confirmedThe modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.

The page may keep the all-analog chain as a description of his rig; it may not say the chain's analog-ness is what a listener hears on a record, because the speaker, cabinet, microphone and room are the larger lever. Any tonal consequence attached to 'the signal path from string to speaker' must name the speaker and room first.

Whatever is least matched inside the comparison sets whether listeners can tell a tube chain from its model, and when the speaker, cabinet, microphone and room are not physically shared they are that block: in the DAGA 2026 ABX the cabinet type, not the processing domain, moved twenty listeners from chance on a closed 4x12 to seventy-eight percent on an open 1x12. The gate clamped every the analog against digital vector axiom after finding two of the vector's source quotations wrong (A7's); this one's own rows were clean and the cabinet record was re-opened by the writer. The adversary called 'least-matched block' a logical leap beyond the sources; it is the reading of the cabinet record and of Cassidy's method of fixing the cabinet, and the page's lesson is unaffected: any sentence about the string-to-speaker path that implies a tonal consequence must name the speaker and room as the larger lever.

For trained listeners judging recorded guitar through a matched output path, the error-to-signal ratio, null depth or signal-to-error ratio of a model orders models by waveform fidelity but does not predict whether a listener can tell the model from the target. The governing quantity is the share of residual that escapes the guitar signal's masking: inharmonic components, components below the fundamental, and residual in attacks and decays. Measured: in the DAGA 2026 ABX (20 listeners, 10 trials per condition) a neural stage raised the 4x12 cabinet model's SER from 30.2 to 42.0 dB and discrimination did not move (45.5 to 50.5 percent correct over 200 responses each, neither above chance), while on the 1x12 an SER rise from 11.7 to 19.8 dB moved discrimination from 78 to 57.5 percent, still above chance (p = .020). Wright et al. (2020) reported test-set ESR of 0.20 and 0.32 percent (about minus 27 and minus 25 dB) yet 14 critical listeners identified the HT-5 Metal models as different from the reference while many could not hear the 5:50 Plus difference (CORRELATIONAL listening on a MODEL). Cassidy and De Sena (16 trained listeners, INTERVENTION) found the same architecture indistinguishable on the high-gain high-pitched excerpt and distinguishable on the lowest-gain excerpt. Changing the perceptual weighting used in training changed rated quality at constant runtime computation (Wright and Välimäki 2019), and the field's review states a single objective metric cannot replace listening tests.

A distorted guitar signal is dense and strongly harmonic, so it masks error locked to its harmonics; the same signal leaves valleys below the fundamental and between harmonics where small error is exposed, and masking is weaker below a masker than above it (sawtooth aliasing thresholds: 3 to 41 dB of attenuation needed above the fundamental, 0 to 41 dB below). Higher gain raises total residual and raises self-masking at once; the two pull in opposite directions, which is why the high-gain excerpt was easiest to model perceptually and the sparse low-gain excerpt hardest. An energy metric sums error wherever it falls; the ear weights error by where it falls relative to the masker.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a step in the reasoning · Sources: 6 cited, 4 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S4 (DAGA 2026 Table 3: SER against percent correct dissociation; INTERVENTION, re-opened by this writer). S3 (Wright et al. 2020: ESR values and listening outcome by amplifier; measured plus CORRELATIONAL). S1 (Cassidy and De Sena 2023: excerpt dependence of ABX; INTERVENTION). S23 (Wright and Välimäki 2019: perceptual loss changed rated quality at constant runtime; INTERVENTION on training). S26 (Vanhatalo et al. 2022 review; ARGUMENT; opened by the anthropic leg, 403 to this writer). S7 (Lehtonen, Pekonen and Välimäki: masking asymmetry around the fundamental; INTERVENTION; opened by the anthropic leg, 403 to this writer).

• S4Hybrid Modeling of Guitar Cabinets Using FIR Filters and Neural Networks (DAGA 2026)opened, quote confirmed20 listeners, 10 ABX trials per condition, recordings level-matched to 112 dB(A) at one metre; Table 3: 1x12 linear 78 percent correct at SER 11.7 dB, 1x12 DNN 57.5 percent at 19.8 dB (p = .020), 4x12 linear 45.5 percent at 30.2 dB, 4x12 DNN 50.5 percent at 42.0 dB, neither 4x12 condition above chance.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)opened, quote confirmedTest-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)opened, quote confirmedAll six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S23Wright and Välimäki, Perceptual Loss Function for Neural Modelling of Audio Systems (2019)opened, quote confirmedChanging the perceptual weighting used during training changed listening-test quality without changing the model's runtime computation.
• S26Vanhatalo et al., A Review of Neural Network-Based Emulation of Guitar Amplifiers (Applied Sciences, 2022)did not openA single objective metric cannot replace subjective listening tests; models with varying time-domain metrics were all judged acceptable in listening; the WaveNet3 HT5M model remained distinguishable from the original. Opened by the anthropic leg; this writer's fetch returned 403.
• S7Lehtonen, Pekonen and Välimäki, Audibility of aliasing distortion in sawtooth signalsdid not openForced-choice thresholds: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation for inaudibility, below the fundamental 0 to 41 dB depending on frequency; masking more prominent above the fundamental than below it. Opened by the anthropic leg; this writer's fetch returned 403.

The page may not quote a null depth, an ESR or a 'percent accurate' figure as proof that a model does or does not sound like the amp; any measurement it cites must be paired with a blind discrimination result on the same material. 'Dig in and it breaks into harmonics' is compatible, but the page may not imply that his high-gain regime is where analog shows, since that is where models are hardest to tell apart.

How much residual a model leaves orders models by fidelity but does not predict whether a listener can hear it; what governs is how much of that residual escapes the guitar signal's own masking, which is why a twelve-decibel improvement in the 4x12 model's signal-to-error ratio moved discrimination not at all and why high-gain excerpts are easier to model perceptually than sparse clean ones. The masking mechanism is the writer's reading, as the adversary said, but the dissociation is measured in three records. For the page it has a sharp edge: the broken-up, high-gain regime is where models are hardest to tell apart, so the page may not imply that his high-gain tone is where analog shows, and it may not quote a null depth as proof either way.

For trained listeners judging level-matched, cabinet-matched recordings through headphones, published discrimination of a state-of-the-art model from its tube target is small and above chance, neither zero nor large. Cassidy and De Sena (2023; 16 Tonmeister students in the ABX, 72 trials each, 12 per model, 192 pooled responses per model; INTERVENTION) found every one of six neural models of a Peavey 6505 above the 95 percent critical level, the rock-trained LSTM (about 57 percent correct by the openai leg's reading of Figure 8) below the 99 percent level. Düvel et al. (2020; N = 177 online listeners passed through a headphone-or-loudspeaker screening, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses on an 8-point source-plus-confidence scale; CORRELATIONAL) found d' = 0.34 (range 0.0 to 0.74 across stimuli), 547 of 1,062 responses correct (51.5 percent), and a weak relation to musical sophistication. Wright et al. (2020; 14 critical listeners) found the HT-5 Metal models identifiable and the 5:50 Plus models not for many listeners. On preference, Cassidy and De Sena's 21 listeners rated the real amplifier and all six models inside the minus 10 to plus 10 'indifferent' band in a bass-and-drums mix, and a rock-trained LSTM was rated significantly more realistic than the real amplifier on the lowest-gain excerpt. The population is trained listeners and the task is recorded listening; the result does not transfer to a player in the loop without a separate test. The largest 2026 dataset (NAM A2, 1,184 participants, 105,842 MUSHRA ratings over 37 tones) is rated similarity and carries no identification count.

Discrimination is bounded above by the total variation distance between the two output distributions (ideal observer: P correct = (1 + TV)/2) and a human falls below that bound; a modern model's residual is a few tenths of a percent of signal energy, mostly harmonically locked and masked, so the information available to a listener is small and the measured rate sits a few points above chance. Preference is a different estimand that depends on what a listener likes rather than on what differs, so the same data can show a detectable difference and no preference.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Survived the adversary's attack · Sources: 4 cited, 4 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S1 (Cassidy and De Sena 2023; INTERVENTION; ABX, MUSHRA-style and preference; re-opened by this writer 2026-10-05). S2 (Düvel et al. 2020; CORRELATIONAL online discrimination; N = 177, 1,062 responses, hits 0.526, false alarms 0.403, d' = 0.34; re-opened by this writer). S3 (Wright et al. 2020; CORRELATIONAL listening on 14 subjects). S19 (NAM A2 MUSHRA raw data; CORRELATIONAL platform data; similarity ratings only).

• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)opened, quote confirmedAll six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S2Düvel, Kopiez, Wolf and Weihe, Confusingly Similar (Music and Science, 2020)opened, quote confirmedOnline discrimination study, N = 177, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses, hits 0.526 and false alarms 0.403 (547 of 1,062 correct, 51.5 percent), d' = 0.34, range 0.0 to 0.74, weak relation to musical sophistication.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)opened, quote confirmedTest-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S19NAM A2 MUSHRA raw data repository (tone-3000, 2026)opened, quote confirmed105,842 similarity ratings from 1,184 participants over 37 tones, collected April 18 to May 29, 2026, MUSHRA-style with hidden reference and anchor; rated similarity, not identification trials.

The page may not say 'you can hear the difference' and may not say 'no one can'; the true sentence is that trained listeners pick the tube amplifier out of a matched recording a little more often than a coin would and do not prefer it. The all-analog chain survives on the page as his choice, not as a claim about what a listener hears on a record.

Trained listeners judging level-matched, cabinet-matched recordings pick the tube amplifier out from a state-of-the-art model a little more often than a coin would and do not prefer it: Cassidy and De Sena's sixteen Tonmeister students were above the ninety-five percent criterion for every model and below the ninety-nine for the best, Düvel's one hundred seventy-seven online listeners landed at a d-prime of a third, and in a bass-and-drums mix every stimulus sat inside the indifferent band. The writer re-opened both primary records; the adversary found the claim supported; gemini's 'drops to random chance' ruling rested on similarity ratings with no identification count and is recorded as the against leg. The page may not say 'you can hear the difference' and may not say 'no one can'; the all-analog chain survives as his choice, not as a claim about what a listener hears on a record.

Because the best published identification rates sit near 57 to 59 percent correct, a two-alternative forced-choice test needs on the order of 150 pooled trials per condition to detect that rate at 80 percent power and one-sided alpha 0.05; about 60 trials for 65 percent, about 33 for 70 percent, about 620 for 55 percent (binomial power calculation by the anthropic leg and this writer; MODEL class, not a published number). A 10-of-20 result bounds the true rate only to a Wilson 95 percent interval of about 30 to 70 percent; 100 of 200 bounds it to about 43 to 57 percent. Trials from one listener on one excerpt are not independent population observations, so participant-level analysis is required. The informal protocol of asking 'do you hear a difference' carries a response bias independent of the stimuli: in a published critique of an amplifier listening test, listeners reported a difference 61 to 66 percent of the time regardless of whether the amplifiers differed, and an unbalanced same-different mix converts that bias into apparent correctness (ARGUMENT class). An equivalence claim must be stated as a bound (for example, the upper 95 percent limit on identification is below 60 percent under these conditions), never as a failure to reject chance; preference is a separate dependent variable with its own interval.

The standard error of a proportion near 0.5 is 0.5/sqrt(n); detecting a 7-point departure from chance requires that error to be about a third of the departure, hence hundreds of trials. A yes-biased difference count is a criterion shift in signal-detection terms, not sensitivity; d' with a reported criterion separates them. BS.1116, the double-blind triple-stimulus hidden-reference method, is the standard for impairments small enough to need this control, and the EBU recommends it over MUSHRA once impairments are small.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed PROBABLE) · Refuted by the adversary on a number larger than the sources support · Sources: 4 cited, 3 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed PROBABLE -> granted SIGNAL (gate lowers, never raises)

S1 (Cassidy and De Sena: 192 responses per model against 95 and 99 percent binomial criteria; INTERVENTION). S2 (Düvel: 1,062 responses with hits and false alarms tabulated and d' reported; CORRELATIONAL). S18 (Stereophile blind listening letters: the 61 to 66 percent bias and the 52.3 against 50.9 percent chance comparison; ARGUMENT; opened by the anthropic leg, 403 to this writer). S24 (EBU Tech 3339: BS.1116 for small impairments; ARGUMENT, a standards document).

• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)opened, quote confirmedAll six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S2Düvel, Kopiez, Wolf and Weihe, Confusingly Similar (Music and Science, 2020)opened, quote confirmedOnline discrimination study, N = 177, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses, hits 0.526 and false alarms 0.403 (547 of 1,062 correct, 51.5 percent), d' = 0.34, range 0.0 to 0.74, weak relation to musical sophistication.
• S18Stereophile, Blind Listening letters, part 4 (critique of an amplifier listening test)did not openParticipants reported a difference 61 to 66 percent of the time regardless of whether the amplifiers differed; an unbalanced same-different mix converts that bias into apparent correctness; a separate test's 52.3 percent correct did not differ from the 50.9 percent chance level. Opened by the anthropic leg; this writer's fetch returned 403.
• S24EBU Tech 3339, Evaluations of multichannel audio codecs (BS.1116 versus MUSHRA guidance)opened, quote confirmedBS.1116 double-blind triple-stimulus hidden-reference is recommended for small impairments, MUSHRA for intermediate ones; listeners who downgrade the hidden reference are excluded.

The page may not cite any single blind test, published or informal, as settling the analog-versus-digital question; if the investigation page reports a test it carries trial count, denominator and interval. A video or forum 'nobody could tell' is not a row.

At the effect size the published records show, an informal blind test is silence: a forced-choice null needs on the order of a hundred and fifty pooled trials per condition to have power, ten-of-twenty bounds nothing, and the common 'did you hear a difference' protocol carries a yes-bias that listeners showed regardless of whether the amplifiers differed. The adversary objected to 'exactly 150' as an overstatement; the axiom states it as this writer's binomial arithmetic, a model result, and the figure moves with the assumed rate. For the investigation page the rule is procedural: any test it reports carries trial count, denominator and interval, and a forum 'nobody could tell' is not a row.

Every other model error (gain, equalization, nonlinear shape, noise, sag) also exists in analog form and can in principle be matched; foldover of harmonics past the Nyquist frequency exists only in a sampled nonlinear process. Folded components are inharmonic and descend as the note ascends, which defeats the masking the guitar's own harmonics provide. Measured thresholds on sawtooth stimuli in forced-choice listening: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation to be inaudible, those below the fundamental 0 to 41 dB depending on frequency, low-frequency audibility limited only by the hearing threshold (INTERVENTION on trained listeners). In Wright et al. (2020) a neural amplifier model's aliased components sat about 40 dB below the first harmonic at low and mid frequencies and the authors heard no clear aliasing on guitar and bass, an observation and not a threshold experiment. The 2025 engineering state: Esqueda and Murai's antialiased recurrent model shows most aliasing components below about minus 60 dB with no perceivable aliasing in informal listening up to fundamentals of at least 5 kHz; Sato and Smith trade waveform accuracy against alias suppression and leave the listening test to future work (MODEL class, measured spectra). Antiderivative antialiasing reduces low-frequency aliasing by up to about 50 dB in a four-point version. A cabinet filter applied after folding cannot remove an alias already inside the band.

A nonlinearity applied to a band-limited signal generates harmonics at h times f0; in discrete time any component above fs/2 reflects to |h f0 minus k fs| inside the Nyquist interval, where it is no longer a multiple of f0. Because masking is weaker below a masker than above it, and because a rising note drives aliases downward into the sparse region under the fundamental, the audibility specification is a frequency-dependent noise-to-mask ratio, not a THD figure. Oversampling raises fs/2 so fewer harmonics fold; ADAA smooths the nonlinearity's effective bandwidth; both cost computation that real-time hardware must pay.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Survived the adversary's attack · Sources: 7 cited, 5 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S7 (aliasing audibility thresholds; INTERVENTION; opened by the anthropic leg). S3 (Wright et al. 2020: aliasing about 40 dB under the first harmonic, no clear audibility on program material; measured). S6 (Esqueda and Murai, DAFx 2025: minus 60 dB and informal listening to 5 kHz; measured; re-opened by this writer 2026-10-05). S5 (Sato and Smith, DAFx 2025: accuracy against alias suppression; measured; re-opened by this writer). S25 (Parker et al. ADAA: up to about 50 dB low-frequency reduction; MODEL). S22 (Shannon 1949: band-limited sampling; the theorem). S14 (Pakarinen and Yeh review: aliasing inherent in sampled nonlinear modeling; ARGUMENT).

• S7Lehtonen, Pekonen and Välimäki, Audibility of aliasing distortion in sawtooth signalsdid not openForced-choice thresholds: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation for inaudibility, below the fundamental 0 to 41 dB depending on frequency; masking more prominent above the fundamental than below it. Opened by the anthropic leg; this writer's fetch returned 403.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)opened, quote confirmedTest-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S6Esqueda and Murai, Antialiased Black-Box Modeling of Audio Distortion Circuits Using Real Linear Recurrent Units (DAFx 2025)opened, quote confirmedAntialiased model spectrum with most aliasing components below approximately minus 60 dB; informal listening with a sine sweep found no perceivable aliasing up to fundamentals of at least 5 kHz.
• S5Sato and Smith, Aliasing Reduction in Neural Amp Modeling by Smoothing Activations (DAFx 2025)opened, quote confirmedA measured trade-off between waveform accuracy and alias suppression in neural amp models; perceptual validation left to future listening tests.
• S25Parker, Zavalishin and Le Bivic, Antiderivative Antialiasing for Memoryless Nonlinearitiesdid not openThe antiderivative method attenuates low-frequency aliased components, up to about 50 dB in a four-point version, at low oversampling factors. Opened by the anthropic leg.
• S22Shannon, Communication in the Presence of Noise (1949)opened, quote confirmedBand-limited sampling and ideal reconstruction; the distinction between predictable distortion and unpredictable noise.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)opened, quote confirmedThe modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.

The page may say a digital model can carry one artifact an analog chain never has, aliasing, and that in current designs it is small and measured rather than heard in the published record; it may not say digital 'sounds' like anything on this ground. For the investigation page the probe is a sine sweep and high notes at high gain, not a riff.

Aliasing is the one error a digital model can have and a tube amp cannot: harmonics past half the sample rate fold back as inharmonic tones that descend as the note rises and land under the fundamental where masking is weakest. Measured thresholds on sawtooth stimuli set how far down it must be, and the two DAFx 2025 papers the writer re-opened put current designs near minus sixty decibels with perceptual validation still pending; the adversary found this supported. The page may say a model can carry an artifact an analog chain never has and that it is small and measured, not heard, in the published record; the honest probe for the investigation page is a sine sweep and high notes at high gain, not a riff.

A digital chain adds conversion, buffering and filter group delay that an analog chain does not, and this delay reaches the player through a mechanism independent of timbre, so no null test removes it and no interactive comparison that leaves loop delay unequal is a timbre test. The measured record, none of it on guitar: with a percussive digital instrument under sub-millisecond latency control, 10 ms with ±3 ms jitter and 20 ms were rated worse on six quality measures than zero added latency while stable 10 ms was not, and metronome synchronisation error did not change (Jack, Stockman and McPherson 2016; replicated in 2018 with 11 amateur musicians and 10 professional percussionists over 12 randomized pairs, professionals more aware of the change; INTERVENTION, within-person). Conscious detection thresholds are far higher: a PEST staircase study (n = 37) found a mean JND of 49 ms at zero base latency, 27 ms at 64 ms base and 77 ms at 512 ms, lower for high musical sophistication (INTERVENTION). Acceptable monitoring latency for performers ranged from 42 ms to possibly under 1.4 ms by instrument and by wedge against in-ear monitoring (Lester and Boley 2007; CORRELATIONAL). On a keypress-to-sound task musicians' delay thresholds averaged about 102 ms against about 180 ms for non-musicians (van Vugt and Tillmann 2014; CORRELATIONAL). The gemini leg's mechanism for sub-5 ms detection, comb filtering between bone-conducted string sound and a delayed monitor, is a hypothesis with no opened measurement; the anthropic leg's prediction that a guitarist's threshold sits nearer 10 to 20 ms because the acoustic string sound is a zero-delay reference is likewise a stated hypothesis.

A causal real-time system cannot emit a response to a pick attack before the attack has been sampled; total delay sums conversion, block buffering, processing and the acoustic path (about 2.9 ms per metre of air). A listener evaluating a recording can have a constant offset removed; a player cannot subtract delay from an ongoing sensorimotor loop, and the quality rating falls at a delay the same player neither consciously detects nor shows in timing error, which is why the two dependent variables must be collected separately. The threshold moves with the operating point (27 ms at a 64 ms base against 49 ms at zero), so a figure at one base does not transfer to another.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Survived the adversary's attack · Sources: 5 cited, 4 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S8 (Jack, Stockman and McPherson 2016: the 10 ms, jittered 10 ms and 20 ms conditions and the quality-against-timing dissociation; INTERVENTION; reachable). S9 (Jack et al. 2018: 11 amateurs and 10 professional percussionists, 12 pairs; INTERVENTION; opened by the openai leg, 403 to this writer). S10 (Schmid et al. 2024: PEST, n = 37, 49, 27 and 77 ms; INTERVENTION; reachable). S11 (Lester and Boley 2007: 42 ms to under 1.4 ms by instrument and monitor; CORRELATIONAL; reachable). S12 (van Vugt and Tillmann 2014: about 102 against 180 ms; CORRELATIONAL; reachable).

• S8Jack, Stockman and McPherson, Effect of latency on performer interaction and subjective quality assessment of a digital musical instrument (Audio Mostly 2016)opened, quote confirmedWith sub-millisecond latency control on a percussive digital instrument, 10 ms with jitter and 20 ms were rated worse on six quality measures while stable 10 ms was not, with no change in metronome synchronisation error; the acceptable threshold placed between 10 and 20 ms.
• S9Jack, Mehrabi, Stockman and McPherson, Action-sound Latency and the Perceived Quality of Digital Musical Instruments (Music Perception, 2018)did not open11 amateur musicians and 10 professional percussionists, 12 randomized pairs, conditions 0, 10, 20 ms and 10 ms with ±3 ms jitter; 20 ms and the jitter condition rated worse, 10 ms not different from zero, professionals more aware of latency. Opened by the openai leg; this writer's fetch returned 403.
• S10Schmid, Ambros, Bogon and Wimmer, Measuring the Just Noticeable Difference for Audio Latency (2024)opened, quote confirmedPEST staircase, n = 37: mean JND 49 ms at zero base latency, 27 ms at 64 ms base, 77 ms at 512 ms; lower JND for high musical sophistication.
• S11Lester and Boley, The Effects of Latency on Live Sound Monitoring (AES 123rd Convention, 2007)opened, quote confirmedAcceptable monitoring latency ranged from 42 ms to possibly under 1.4 ms depending on instrument and on wedge versus in-ear monitoring.
• S12van Vugt and Tillmann, Thresholds of Auditory-Motor Coupling Measured with a Simple Task in Musicians and Non-Musicians (PLOS One, 2014)opened, quote confirmedKeypress-to-sound delay detection thresholds: musicians about 102 ms, non-musicians about 180 ms.

The page's unbroken-path phrase may be rewritten as the one defensible physical difference between his rig and a modeler, zero added delay, bounded by the fact that the published quality effect appears at 10 to 20 ms in percussion instruments and that no guitar threshold exists. The investigation page records each modeler's measured loop delay, not its buffer size.

Added loop delay reaches a player through a channel timbre masking cannot touch: with a percussive digital instrument, ten milliseconds with jitter and twenty milliseconds were rated worse on quality while timing error did not move, and conscious detection sits far higher at about forty-nine milliseconds. None of this is on guitar, and both the sub-five-millisecond comb-filtering mechanism gemini asserted and the ten-to-twenty-millisecond guitar prediction anthropic offered are hypotheses with no opened measurement. For the page this is the one defensible physical difference between his rig and a modeler, zero added delay, with the bound that no guitar threshold has been measured and that the investigation page records every modeler's measured loop delay, not its buffer size.

An all-analog chain is continuous-time processing without numerical conversion; it is not one galvanically continuous circuit, since coupling capacitors pass AC and block DC between stages, the output transformer transfers energy between separate windings, and every active stage draws its output energy from a power supply, so no current and no electrons travel from pickup to speaker. The two consequences of bidirectional termination that do reach the ear are measurable and large. At the input, pickup inductance and total shunt capacitance form a second-order low-pass with a resonant peak at fr = 1/(2π√(LC)) whose height is set by the load resistance: a Strat-type pickup loaded by an extra 2,200 pF peaks near 2.4 kHz and the same pickup with 47 pF near 8 kHz, and a lower load impedance lowers the peak (MODEL plus measurement). At the output, the power stage's source impedance interacts with the speaker's non-flat Z(f) so that a high source impedance equalizes the driver along its impedance curve; pentode stages have the highest output impedance, triode-strapped the lowest, and less global negative feedback raises it. The output transformer itself adds distortion only below about 100 Hz for a measured Fender unit and 30 Hz for a Hammond (MODEL fitted to measurement). Any buffered analog pedal, active pickup or FET input stage terminates the pickup's view at that buffer, so a buffered analog pedalboard is a cascade of one-way blocks exactly as a digital chain is; a digital chain whose input presents the same complex load and whose power stage or model reproduces the output interaction matches both parameters, while one with a flat 1 megohm converter input and a high-damping full-range power amplifier matches neither and is discriminable for that reason alone.

A passive magnetic pickup is a voltage source in series with an inductance of several henries and a resistance of several kilohms, shunted by cable and input capacitance and by the input resistance; the circuit is a damped resonator whose peak frequency and height are set by the downstream load, which is what bidirectional means here. A tube output stage without heavy feedback has a source impedance comparable to the speaker's nominal impedance, so the speaker's impedance peak at resonance and its inductive rise at high frequency receive more voltage there, an equalization that follows Z(f). Both effects are parameters of the terminations, not of the processing domain; a model that measures and reproduces them removes them from the residual.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed FORGED) · Refuted by the adversary on a source said less than the claim · Sources: 5 cited, 2 opened and confirmed

• TRIPWIRE: this axiom's own source(s) S15 (misquote), S16 (number_not_on_page) were wrong -> SIGNAL (the tripwire hits the axiom whose use of a source was wrong)
• CLAMPED: claimed FORGED -> granted SIGNAL (gate lowers, never raises)

S15 (USPTO 7548850: the resonance equation and the 2.4 kHz against 8 kHz example; MODEL plus measurement; reachable). S16 (damping factor: frequency-dependent source impedance and the Z(f)-following response; ARGUMENT; reachable). S17 (Paiva, Pakarinen and Välimäki 2011: transformer distortion below about 100 Hz and 30 Hz; MODEL fitted to measurement; reachable). S14 (Pakarinen and Yeh review: coupling capacitors, separate stages and transformer behavior as modeled blocks; ARGUMENT; reachable). S21 (Aiken reactive load emulator: the LRC network reproducing a cabinet's impedance curve; ARGUMENT; reachable).

• S15USPTO patent 7548850 (pickup and cable loading)misquotePickup inductance and cable capacitance resonate at fr = 1/(2π√(LC)); a Strat-type pickup with an extra 2,200 pF peaks near 2.4 kHz and with 47 pF near 8 kHz; lower cable capacitance lowers the peak amplitude; load resistance sets the Q.
• S16Damping factor (reference article)opened, the number was not on the pageAmplifier output resistance varies with frequency in tube and solid-state amplifiers; the frequency-dependent loudspeaker response change caused by output resistance exceeds the damping effect itself; pentode stages have the highest output impedance, triode-strapped the lowest, less global feedback raises it.
• S17Paiva, Pakarinen and Välimäki, Real-Time Audio Transformer Emulation for Virtual Tube Amplifiers (EURASIP JASP, 2011)did not openMeasured Fender and Hammond output transformers introduce distortion only below about 100 Hz and 30 Hz respectively; the model is bidirectional, feeding speaker current back to the power amplifier.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)opened, quote confirmedThe modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.
• S21Aiken Amplification, Designing a Reactive Speaker Load Emulatoropened, quote confirmedThe LRC network that reproduces a guitar cabinet's impedance curve for an amplifier not driving a speaker; a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction.

The page's sentence 'the signal path from string to speaker is an unbroken electrical circuit' is rewritten: the chain is continuous-time with no conversion, and what that buys is zero added delay plus two termination interactions the page may name, the pickup load and the amp-speaker coupling; any claim that continuity itself is audible is dropped. The Physics of Music (February 2026) is bounded by name to vinyl-against-digital playback: its even-against-odd harmonic distinction does not govern amplifier modeling, where harmonic structure follows the implemented nonlinearity, not its representation.

'An unbroken electrical circuit' is a statement about topology whose audible residue is two termination parameters a modeler can match or miss: the pickup's load resonance and the output stage's source impedance against the speaker's impedance curve; coupling capacitors and transformers already break DC continuity in every tube amplifier, and a buffered analog pedalboard is a cascade of one-way blocks exactly as a digital chain is. The gate found two of this axiom's source quotations wrong (the pickup patent misquoted, the damping-factor page not carrying the stated numbers), which clamped the whole vector; the physics is the same as 2.1 and 2.4, which carry it on clean rows, so the page should cite those. The sentence the page keeps is that the chain is continuous-time with no conversion; the claim that continuity itself is audible is dropped.

As of the legs' searches in October 2026, no published study puts a guitarist blind, in the loop, on a tube chain and its model under shared cabinet, room, level and delay with a discrimination statistic; the nearest record is Buffa and Lebrun (2017), blind comparisons and interviews with four guitarists among several amplifier simulations, qualitative, with configuration latencies from under 1 ms on dedicated hardware to much longer on computer paths. The gap is a gap, not a null. A player closes loops a recording does not: acoustic regeneration from speaker to strings (sustain and controlled feedback, set by SPL, distance and cabinet radiation), the input load loop, the output load loop (source impedance against a reactive speaker; a capture made through a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction, which is why reactive load boxes exist), and thermal and supply nonstationarity over a take. An interactive comparison discriminates on whichever of these the arms do not share before any model error is reached; a tube amp into its own cabinet at 100 dB against a model into headphones differs on three loops and says nothing about the model. An unblinded report ('it breathes', 'it blooms', 'it feels disconnected') is a claim about a mechanism (compression time constant, supply sag, speaker excursion at that SPL, pickup resonance Q, loop delay), and its lawful output is the variable to measure, never a discriminability result. The largest 2026 dataset (NAM A2 MUSHRA) is rated similarity of recordings and does not fill the gap.

With acoustic feedback the chain is y = G(x_hand + H y), so y = G/(1 minus G H) x_hand; near a strong feedback condition small differences in gain, phase or delay between arms produce large differences in closed-loop behavior, and matching a prerecorded performance (open loop) is weaker than matching the response to a family of player actions (closed loop). The loudspeaker's motional impedance returns a voltage to the output stage that a resistor does not, so the amplifier's distortion and frequency response under a real speaker differ from those under a dummy load; a model captured through a resistor models a different operating point.

Granted tier: SIGNAL, set by the code gate on October 4, 2026 (the writer claimed PROBABLE) · Refuted by the adversary on a claim no opened source could verify · Sources: 5 cited, 4 opened and confirmed

• VECTOR KILL: 2 misquote flag(s) in this vector (>= 2) -> SIGNAL for every axiom in it
• CLAMPED: claimed PROBABLE -> granted SIGNAL (gate lowers, never raises)

S20 (Buffa and Lebrun 2017: four guitarists, blind comparisons among simulations, measured latencies by configuration; ARGUMENT plus a small qualitative INTERVENTION; reachable at the QMUL repository). S21 (Aiken reactive load emulator: resistive against reactive load; ARGUMENT). S13 (Klippel: loudspeaker nonlinearity rises with amplitude, the SPL dependence of the regeneration loop; ARGUMENT from measurement method). S14 (Pakarinen and Yeh review: the acoustic feedback path from speaker to string as a modeled block; ARGUMENT). S19 (NAM A2 raw data: rated similarity of recordings, not interactive play; platform data).

• S20Buffa and Lebrun, Real time tube guitar amplifier simulation using WebAudio (2017)did not openBlind comparisons and interviews with four guitarists among several amplifier simulations, qualitative; measured latencies from under 1 ms on dedicated modeling hardware to much longer on computer paths.
• S21Aiken Amplification, Designing a Reactive Speaker Load Emulatoropened, quote confirmedThe LRC network that reproduces a guitar cabinet's impedance curve for an amplifier not driving a speaker; a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction.
• S13Klippel, Separated loudspeaker distortion (measurement method page)opened, quote confirmedForce factor Bl(x) and compliance Cms(x) dominate loudspeaker nonlinear distortion; nonlinear distortion is negligible at small signal and rises rapidly with amplitude and depends on stimulus spectrum.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)opened, quote confirmedThe modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.
• S19NAM A2 MUSHRA raw data repository (tone-3000, 2026)opened, quote confirmed105,842 similarity ratings from 1,184 participants over 37 tones, collected April 18 to May 29, 2026, MUSHRA-style with hidden reference and anchor; rated similarity, not identification trials.

The page may say he plays an all-analog chain and that no blind test of players in the loop exists either way; it may not say players can feel the difference and may not say they cannot. The investigation page's open item is the three-arm protocol: offline matched-input discrimination, blinded native-chain playing with shared cabinet and SPL and the delay difference reported in milliseconds, and a delay-equalized repeat to isolate latency from the rest.

As of the legs' searches no published study puts a guitarist blind, in the loop, on a tube chain and its model under shared cabinet, room, level and delay with a discrimination statistic; Buffa and Lebrun's four guitarists are qualitative and the largest 2026 dataset rates recordings, not play. The adversary rightly noted an absence cannot be verified from unopened sources; it is recorded as a gap, not a null. The page may say he plays an all-analog chain and that no blind test of players in the loop exists either way; it may not say players can feel the difference and may not say they cannot, and an unblinded 'it breathes' is a variable to measure (compression time constant, sag, speaker excursion, pickup Q, loop delay), never a result.