Research-Stack/4-Infrastructure/shim/signal_equation_invariant_roots_summary.md

Signal Equation Invariant Roots

These are invariant roots for accessible local signal equations. They are route/control priors and hardware design handles, not external physics proof or compression proof without exact byte receipts.

Root count: 33 Receipt hash: 10ec6bf94808b4517c6e866889d8c9cca02969fbcb43c574b0abd27c3cac3a33

Unifying Root

SignalRoute = (coordinate, invariant_root, admissible_transform, receipt_barrier)

Every accessible signal equation reduces to a coordinate map plus an invariant root. The invariant root says what can survive rescaling, basis changes, phase shifts, lane permutation, or compression-route projection. Promotion still requires a receipt barrier.

Roots

SIGROOT001_spectral_overlap

• Equation: <s1,s2> = sum_i s1_i s2_i
• Invariant root: inner-product pairing on aligned spectral coordinates
• Admissible transforms: common bin permutation; orthonormal basis change when both signatures transform together
• Compression use: route similarity, duplicate-island pruning, nearest repair template
• FPGA use: DSP dot-product lane with accumulator and saturation guard

SIGROOT002_piecewise_merge

• Equation: merge_i = min(1, left_i + right_i)
• Invariant root: bounded semilattice occupancy over [0,1]^n
• Admissible transforms: coordinatewise monotone maps that preserve zero, one, and order
• Compression use: safe feature union without unbounded sidecar growth
• FPGA use: saturating add primitive

SIGROOT003_resonance_degeneracy

• Equation: deg(left,right) = |support(left) intersect support(right)|
• Invariant root: support-intersection cardinality
• Admissible transforms: positive amplitude scaling and common support-preserving permutation
• Compression use: overlap score for tokenbook/feature collisions
• FPGA use: bitmask AND plus popcount

SIGROOT004_wavefront_value

• Equation: value = (A - gamma*d) * osc(omega*d) for d <= v*t, else 0
• Invariant root: retarded wavefront cone plus phase class modulo cycle
• Admissible transforms: translations and metric-preserving coordinate changes
• Compression use: event influence radius for local route activation
• FPGA use: distance gate, phase LUT, envelope subtractor

SIGROOT005_signal_band_policy

• Equation: band(x) = threshold_partition(x)
• Invariant root: ordered threshold cell
• Admissible transforms: monotone rescaling with transformed thresholds
• Compression use: route budget scheduler
• FPGA use: comparator ladder

SIGROOT006_acoustic_gradient

• Equation: Z_acoustic ~ |grad f|
• Invariant root: metric norm of field gradient
• Admissible transforms: coordinate changes with explicit metric tensor
• Compression use: manifold steepest-descent route proposal
• FPGA use: finite-difference gradient and norm pipeline

SIGROOT007_fitness_entropy_compensation

• Equation: f + alpha*H = f_max
• Invariant root: affine fitness-entropy conserved total
• Admissible transforms: unit changes that transform alpha coherently
• Compression use: semantic/fitness score must pay entropy cost
• FPGA use: linear score lane with conserved budget comparator

SIGROOT008_gibbs_free_energy

• Equation: G = H - T*S
• Invariant root: Legendre-transformed available-energy potential
• Admissible transforms: thermodynamic coordinate changes preserving conjugate pair T,S
• Compression use: available byte-gain after entropy/side-info cost
• FPGA use: cost potential lane for thermal/energy-aware routing

SIGROOT009_affine_erasure_permutation

• Equation: pi(i) = a + s*i mod n
• Invariant root: cycle structure determined by gcd(s,n)
• Admissible transforms: offset translation and invertible modular scaling
• Compression use: repair stream interleaving with deterministic owner
• FPGA use: modular address generator

SIGROOT010_genomic_weight

• Equation: W = (rho + v + tau + sigma + q) / ((1+kappa^2)*(1+epsilon))
• Invariant root: dimensionless normalized field-strength ratio
• Admissible transforms: common scale-normalization of numerator terms
• Compression use: adaptive erasure threshold
• FPGA use: fixed-point ratio approximation

SIGROOT011_pbacs_phi_accumulator

• Equation: phi_{t+1} = phi_t + c mod 2^32
• Invariant root: circle rotation orbit class
• Admissible transforms: phase offset; modular conjugacy preserving increment
• Compression use: deterministic phase owner for route symbols
• FPGA use: free-running modular accumulator

SIGROOT012_pbacs_error_feedback

• Equation: e_next = v + e - b*theta
• Invariant root: bounded quantization residual
• Admissible transforms: threshold-preserving fixed-point rescale
• Compression use: exact residual lane for symbol decisions
• FPGA use: sigma-delta style feedback cell

SIGROOT013_mutual_information_gain

• Equation: MI = baseline_bpb - actual_bpb
• Invariant root: byte-per-symbol improvement under one ratio schema
• Admissible transforms: comparisons that keep baseline and actual schema identical
• Compression use: route evidence coordinate
• FPGA use: counter difference after codec run

SIGROOT014_weighted_mi_prediction

• Equation: MI_pred = sum_i w_i MI_i S_i / sum_i w_i S_i
• Invariant root: barycentric coordinate in similarity-weighted evidence simplex
• Admissible transforms: common positive scaling of all weights
• Compression use: nearest-prior route prediction
• FPGA use: weighted accumulator plus reciprocal approximation

SIGROOT015_surprise_metric

• Equation: S = log(1 + |delta_MI|)
• Invariant root: monotone function of absolute prediction residual
• Admissible transforms: monotone reparameterization of residual magnitude
• Compression use: route anomaly detector
• FPGA use: absolute-delta threshold; log optional

SIGROOT016_structure_yield

• Equation: rho = MI / (cost + eps)
• Invariant root: information-per-cost efficiency ratio
• Admissible transforms: unit changes preserving numerator/denominator interpretation
• Compression use: candidate route priority
• FPGA use: score-per-cycle allocator

SIGROOT017_weighted_feature_distance

• Equation: d(z1,z2) = sqrt(sum_i w_i*((z1_i-z2_i)/s_i)^2)
• Invariant root: diagonal metric distance after scale normalization
• Admissible transforms: coordinate rescaling absorbed into s_i and w_i
• Compression use: route family clustering
• FPGA use: scaled L2 distance pipeline

SIGROOT018_energy_gradient_waveform

• Equation: wave_E = (|grad E|, omega_gradE, phi_gradE)
• Invariant root: gradient magnitude and phase trajectory
• Admissible transforms: metric-aware coordinate changes
• Compression use: energy/cost-aware transform scheduling
• FPGA use: gradient magnitude plus phase accumulator

SIGROOT019_shape_energy_coupling

• Equation: C_SE = alpha <grad h, grad E>
• Invariant root: metric inner product of shape and energy gradients
• Admissible transforms: coordinate changes preserving the metric pairing
• Compression use: align geometry witness only when it reduces route cost
• FPGA use: dual-gradient dot-product lane

SIGROOT020_spectral_field_score

• Equation: score = mM + pP + <s,F>
• Invariant root: bilinear pairing between local state and field
• Admissible transforms: paired basis changes that preserve the bilinear form
• Compression use: local route-field compatibility score
• FPGA use: three-term MAC lane

SIGROOT021_parabolic_j_score

• Equation: J(k) = 32 - 0.5*(k-22)^2
• Invariant root: distance from resonant vertex k=22
• Admissible transforms: translation to vertex coordinate u=k-22
• Compression use: resonance-ranked candidate pruning
• FPGA use: subtract-square-threshold circuit

SIGROOT022_cmyk_frequency_lattice

• Equation: f_ch(h) = base_ch + delta*h
• Invariant root: channel-local affine frequency lattice coordinate h
• Admissible transforms: affine frequency calibration preserving delta steps
• Compression use: symbol carrier with exact inverse
• FPGA use: base-plus-shift frequency synthesizer

SIGROOT023_rydberg_gap

• Equation: nu_bar = R*(1/n1^2 - 1/n2^2)
• Invariant root: reciprocal-square quantum gap
• Admissible transforms: unit conversion between wavenumber, wavelength, frequency, and energy
• Compression use: stable physical spectral basis index
• FPGA use: small table of canonical spectral lines

SIGROOT024_lorentzian_resonance

• Equation: L(delta) = 1/(1+delta^2)
• Invariant root: squared detuning from spectral center
• Admissible transforms: sign flip of detuning; normalized wavelength units
• Compression use: nearest spectral-basis assignment
• FPGA use: detuning-square LUT

SIGROOT025_kmer_base4_index

• Equation: idx = 16*b1 + 4*b2 + b3
• Invariant root: base-4 coordinate of codon symbol
• Admissible transforms: base relabeling with explicit inverse map
• Compression use: fixed codon/tokenbook coordinate
• FPGA use: two-bit shift-and-or indexer

SIGROOT026_dct2_basis

• Equation: basis_{j,k} = cos(pi/n*(j+1/2)*k)
• Invariant root: orthogonal cosine projection coefficient
• Admissible transforms: orthogonal transforms preserving coefficient energy
• Compression use: spectral coefficient compaction
• FPGA use: fixed cosine basis or LUT butterfly

SIGROOT027_qpsk_phase_class

• Equation: phase in Z_4
• Invariant root: phase class modulo pi/2
• Admissible transforms: global phase rotation with receiver correction
• Compression use: 2-bit symbol carrier
• FPGA use: quadrant decoder

SIGROOT028_qam16_constellation

• Equation: symbol = (a in A4, phase in Z4)
• Invariant root: finite amplitude-phase lattice point
• Admissible transforms: affine constellation calibration with preserved decision cells
• Compression use: 4-bit symbol carrier / QAM transfer metaphor
• FPGA use: amplitude slicer plus quadrant decoder

SIGROOT029_dmt_subcarrier_quotient

• Equation: phase_base = phase_out - offset_i mod cycle
• Invariant root: phase quotient after subtracting subcarrier offset
• Admissible transforms: subcarrier permutation with receipted offset table
• Compression use: parallel lane carrier with exact demodulation
• FPGA use: per-lane phase subtractor

SIGROOT030_hann_window_fft_energy

• Equation: E_bin = avg_{k in bin} |FFT(window*x)_k|
• Invariant root: windowed spectral-energy distribution
• Admissible transforms: time shift up to phase; amplitude normalization when max-normalized
• Compression use: audio/signal route feature vector
• FPGA use: window multiply, FFT, magnitude, bin accumulator

SIGROOT031_transient_features

• Equation: transient = (max dx+, max dx-, zero_crossings/n, peak/rms)
• Invariant root: edge/impulse morphology of the signal chunk
• Admissible transforms: time-local scaling with normalized crest and ZCR preserved
• Compression use: decide raw vs spectral vs hybrid route
• FPGA use: delta extrema, sign-change counter, RMS/peak lane

SIGROOT032_predictability_autocorrelation

• Equation: pred = 0.5*(corr(x_t, x_{t-1}) + 1)
• Invariant root: normalized temporal correlation
• Admissible transforms: affine amplitude scaling removed by mean/variance normalization
• Compression use: predictor suitability signal
• FPGA use: sliding dot product and norm lane

SIGROOT033_cosine_similarity

• Equation: cos(theta)=<a,b>/(||a|| ||b||)
• Invariant root: projective direction on spectral feature sphere
• Admissible transforms: positive scaling of either vector
• Compression use: chunk reuse / skip decision
• FPGA use: dot product and reciprocal norm threshold