# 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 ```text 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: ` = 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 ` - 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 + ` - 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||)` - 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