4.9 KiB
Sine Wave Anti-Music Probe
Purpose
This note defines the simplest silent test signal for Anti-Music Theory and the Mass-Number music / anti-music phase boundary.
The baseline is not an audible tone by default. It is a finite sampled sine wave used as a mathematical carrier for spectral filtering, remainder extraction, and candidate number-set resonance.
AUDIO_RENDER = false
Core Statement
A pure sine wave is the most stable possible musical carrier:
one frequency
one phase
one clean spectral peak
minimal harmonic ambiguity
maximal local predictability
Therefore it is a good first substrate for anti-music perturbation. If Anti-Music cannot destabilize a pure sine carrier in a finite, measurable way, the perturbation is not strong enough or not aligned to the correct metric.
Finite Baseline Signal
Use a finite sampled sine wave:
f_N[n] = sin(2*pi*f0*n/Fs + phi0), 0 <= n < N
where:
N = finite sample count
Fs = sample rate, symbolic or numeric
f0 = carrier frequency
phi0 = initial phase
No infinite waveform is allowed. All tests use finite windows.
Optional Multi-Window Form
For window index j:
f_{N,j}[n] = sin(2*pi*f0*(n+jH)/Fs + phi_j)
where:
H = hop size
j = finite window index
This supports finite-window averaging:
AvgRes(A) = (1/J) * sum_{j=1}^{J} Res(A; R_{N,j})
Anti-Music Perturbation
Given a finite candidate number set:
A = {a_1,...,a_m}
construct a silent perturbation:
P_A[n] = sum_{a in A} w_a sin(2*pi*a*n/N + phi_a)
Apply bounded perturbation:
g_N[n] = f_N[n] + epsilon * P_A[n]
with:
0 <= epsilon <= epsilon_max
Filtered Remainder
Compute:
F_N[k] = FFT(g_N[n])
Remove the known carrier and known noise bands:
F_filtered[k] = H_music[k] F_N[k]
R_N[k] = F_N[k] - F_filtered[k]
where R_N is the anti-music candidate remainder.
Candidate Set Resonance
For the same set A, compute its spectral fingerprint:
S_A[k] = sum_{a in A} w_a exp(i*2*pi*k*a/N)
P_A[k] = |S_A[k]|^2
Score resonance:
Res(A;R_N) = <normalize(P_A), normalize(|R_N|^2)>
High resonance means the candidate set explains the residual energy left after removing the stable sine carrier.
Anti-Music Transition Test
Use:
AMI(A) = AntiMusicScore(A) - MusicScore(A) - RandomnessPenalty(A)
The sine-wave carrier test should classify:
Music basin:
epsilon small, carrier dominates, AMI(A) < 0
Boundary shell:
residual grows, carrier remains recoverable, AMI(A) ~= 0
Anti-music candidate:
structured residual survives filtering, AMI(A) > 0, StructureScore high
Noise quarantine:
residual grows but structure score collapses or variance explodes
Minimal Starting Candidate Sets
Use small finite sets first:
A_music_like = {1,2,3,4,5}
A_sidon_like = {1,2,5,10,17}
A_prime_like = {2,3,5,7,11}
A_anti_candidate = search result maximizing AntiMusicScore(A)
The set {1,2,5,10,17} is useful because it was already used as a finite spectral-void example and has sparse nonuniform spacing.
Arithmetic Audit
After resonance discovery, run:
DifferenceSetReceipt:
|{ |a_i-a_j| : i<j }| = m(m-1)/2
SumSetReceipt:
|{ a_i+a_j : i<=j }| = m(m+1)/2
The sine wave only provides the carrier and residual. It does not prove Sidon.
Minimal Pseudocode
input: N, Fs, f0, epsilon, candidate set A
f = sine(N, Fs, f0)
P = sum_sines_from_A(A, N)
g = f + epsilon * P
F = FFT(g)
R = F - H_music * F
score = resonance(power_fingerprint(A), abs(R)^2)
audit A with difference and sum-set tests
Why This Is the Correct First Probe
A sine wave gives a maximally stable baseline. Anti-Music is then forced to prove it can create structured remainder rather than hiding inside preexisting complexity.
pure sine = stable music basin
anti-music perturbation = controlled destabilizer
filtered remainder = measurable inverse structure
finite average = stability receipt
arithmetic audit = proof boundary
Boundary
Do not claim:
sine perturbation proves anti-music
hearing the tone is required
audio rendering is needed
resonance score proves Sidon
Allowed claim:
A finite sampled sine wave is the cleanest silent carrier for testing whether anti-music number sets create structured, auditable residuals after tonal filtering.
Audit Classification
Receipt: SineWaveAntiMusicProbe
Status: BASELINE_SIGNAL_DRAFT
Gate: U_scope
Reason: defines a finite test carrier and perturbation procedure, but requires actual finite data runs, thresholds, resonance statistics, and arithmetic audits.
Required Receipts
FiniteSignalReceipt
CarrierDefinitionReceipt
PerturbationBoundReceipt
FFTFilterReceipt
RemainderDefinitionReceipt
RemainderResonanceReceipt
FiniteWindowAverageReceipt
RandomnessPenaltyReceipt
DifferenceSetReceipt
SumSetReceipt
ValidatorReceipt