Research-Stack/0-Core-Formalism/otom/docs/audit/SpectralVoidPatternRecognition.md

Spectral Void Pattern Recognition

Purpose

This note refines the torsional-corridor model by treating voids and bandwidth drops as frequency-domain signatures of the active-cell selector.

The useful claim is:

bandwidth drops / spectral voids can reveal structured destructive interference in the material throat.

The unsafe claim is:

spectral voids by themselves prove the Sidon property.

Voids can be used as a diagnostic and candidate generator. A separate pair-sum audit or nonseparable encoding proof is still required.

Core Model

For a finite active index set A = {a_1,...,a_m}, define an idealized spectral field:

F_A(omega) = sum_{a in A} w_a cos(a omega + phi_a)

or complex form:

S_A(omega) = sum_{a in A} w_a exp(i a omega)
Power_A(omega) = |S_A(omega)|^2

Expanding the power spectrum:

Power_A(omega) = sum_a w_a^2 + 2 sum_{a<b} w_a w_b cos((a-b) omega + phi_ab)

Important correction:

The direct Fourier power spectrum exposes the difference set A-A, not the sum set A+A.

The Sidon/B2 property is about uniqueness of unordered sums:

a+b = c+d -> {a,b} = {c,d}

while a Golomb ruler is about uniqueness of positive differences:

a-b = c-d -> (a,b) = (c,d)

For finite integer sets, distinct differences imply Sidon, but the diagnostic being visualized by cos(a omega) is primarily a difference-spectrum diagnostic.

Void / Bandwidth Drop Definition

A spectral void is a frequency interval where transmission or spectral power drops below threshold:

Void_epsilon = { omega : Gamma(omega) <= epsilon_G }

or for the idealized field:

Void_theta = { omega : F_A(omega) <= -Theta }

In a physical phononic system, the measured version is:

Void_measured = { omega : TransmissionLoss(omega) >= TL_min }

Physical Interpretation

Voids can come from:

Bragg scattering from periodic structure
local resonance stop bands
coupled Bragg/local resonance interference
viscoelastic damping
chiral/rotational coupling
boundary/truncation resonances
self-healing repair failure windows

Therefore a void pattern is evidence of structured wave control, not automatically arithmetic uniqueness.

Spectral Receipt

To use voids as an audit receipt, require:

1. measured or simulated transmission spectrum Gamma(omega)
2. extracted void intervals V_k
3. model-predicted void intervals from active index set A
4. fit/error score between predicted and measured voids
5. independent pair-sum or difference-set audit of A

Proposed fit score:

VoidFit(A) = 1 - measure(V_measured symmetric_difference V_predicted) / measure(V_measured union V_predicted)

For discrete samples:

VoidFit(A) = 1 - |V_measured Δ V_predicted| / |V_measured ∪ V_predicted|

Difference-Spectrum Diagnostic

Because power expansion exposes A-A, a clean diagnostic is:

D_A = { |a_i - a_j| : i < j }

Golomb/difference uniqueness:

|D_A| = m(m-1)/2

If this holds, then A is a Golomb ruler and therefore also Sidon under ordinary integer addition.

This is stronger than merely seeing repeated voids.

Sum-Set Diagnostic

For the Sidon condition directly:

Sigma_A = { a_i + a_j : i <= j }

Sidon/B2 uniqueness:

|Sigma_A| = m(m+1)/2

This must be checked explicitly if the claim is Sidon rather than Golomb.

Updated Gate

The spectral void gate selects candidate active indices:

chi_void(i,t) = 1 iff
  omega_i lies outside measured voids
  and mode_overlap_i >= eta_min
  and Gamma_i(omega,t) >= Gamma_min

or, for destructive-threshold activation:

chi_void(i,t) = 1 iff F_A(omega_i,t) <= -Theta_N

The selected set then requires:

DifferenceSetReceipt or SumSetReceipt
NonseparableEncodingReceipt
CompactDensityReceipt

Literature Anchors

Phononic crystals and metamaterials support the physical basis for voids/bandwidth drops through Bragg scattering, local resonance, destructive interference, chiral coupling, and bandgap engineering. These support the spectral receipt, not direct proof of Sidon uniqueness.

Audit Classification

Receipt: SpectralVoidPatternRecognition
Status: DIAGNOSTIC_RECEIPT_DRAFT
Gate: U_scope
Reason: spectral voids can diagnose structured destructive interference and candidate active sets, but they require explicit difference-set/sum-set audits before being treated as Sidon evidence.

Required Receipts

TransmissionSpectrumReceipt
VoidExtractionReceipt
VoidFitReceipt
DifferenceSetReceipt
SumSetReceipt
NonseparableEncodingReceipt
CompactDensityReceipt