Abstract
We lift discrete/continuous convolution and correlation to symbolic numerals (m, a) by computing the classical operation on magnitudes and attaching an alignment tag via a rapidity-weighted mean of contributors. This preserves linearity and the convolution/correlation theorems on the magnitude channel while keeping stability metadata audit-ready. Under collapse phi(m,a) = m, all results reduce exactly to the classical formulas.
Definitions (magnitude first, alignment summary)
Discrete-time convolution
For x[n] = (m_x[n], a_x[n]) and a scalar kernel k[n] (classical; tagged (k[n], +1)),
y[n] = (m_y[n], a_y[n])
m_y[n] = Σ_r k[r] * m_x[n - r]
u_x[n] = atanh( clamp(a_x[n], -1+eps, +1-eps) ), eps = 1e-6
w_r(n) = | k[r] | * | m_x[n - r] |^gamma # gamma >= 0 (default 1)
u_y[n] = ( Σ_r w_r(n) * u_x[n - r] ) / ( Σ_r w_r(n) ) if Σ w_r(n) > 0
a_y[n] = tanh( u_y[n] )
Continuous-time convolution
For x(t) = (m_x(t), a_x(t)) and scalar kernel k(t),
m_y(t) = ∫ k(τ) * m_x(t - τ) dτ
w(τ; t) = | k(τ) | * | m_x(t - τ) |^gamma
u_y(t) = ( ∫ w(τ; t) * u_x(t - τ) dτ ) / ( ∫ w(τ; t) dτ )
a_y(t) = tanh( u_y(t) )
Why absolute-weights? Using |k| prevents cancellation from oscillatory kernels from corrupting the stability tag; it mirrors physical “load” on the output.
Options (manifest):
align.weights = "|k|*|m|^gamma" # default
align.gamma = 1 # 0 for kernel-only weighting
align.fallback = "mean_u" | "+1" # when Σ weights = 0
Correlation (cross and auto)
Discrete cross-correlation
For x[n], y[n],
R_xy[ℓ] = ( M_xy[ℓ] , a_xy[ℓ] )
M_xy[ℓ] = Σ_n m_x[n] * m_y[n + ℓ]
u_pair[n,ℓ] = u_x[n] + u_y[n + ℓ] # product term (M2 on per-term product)
w_n(ℓ) = | m_x[n] * m_y[n + ℓ] |
u_xy[ℓ] = ( Σ_n w_n(ℓ) * u_pair[n,ℓ] ) / ( Σ_n w_n(ℓ) )
a_xy[ℓ] = tanh( u_xy[ℓ] )
Autocorrelation R_xx[ℓ] follows by setting y = x. Continuous-time forms use integrals.
Link to spectra (magnitude channel): by Wiener–Khinchin, the PSD is the Fourier transform of the autocorrelation of m. Alignment diagnostics can use the rapidity spectrum from 2.25.
Properties (magnitude channel) and collapse
- Linearity: convolution/correlation are linear in
m. - Shift/scale: standard theorems hold for
m. - Convolution theorem:
F{m_x * k} = F{m_x} · F{k}. - Collapse: if all
a = +1, thena_y = +1and the outputs are exactly classical.
Alignment tags are summaries; they do not alter any magnitude identities.
Filter stability and guards
- BIBO stability (magnitude): discrete
Σ_r |k[r]| < ∞or continuous∫ |k(τ)| dτ < ∞. - Normalization: moving-average/low-pass kernels often satisfy
Σ k[r] = 1(or∫ k = 1). - Signs/oscillation: kernels with negative lobes are allowed; weights use
|k|by default to ensure nonnegative averaging. - Boundary handling (discrete):
conv.mode = "full" | "same" | "valid" conv.pad = "zero" (default, pad magnitude 0 with (0,+1) tag) | "reflect" | "wrap" - Causality: for real-time filtering use
k[r]=0forr<0; otherwise declare noncausal.
Worked examples
A) 3-point moving average (discrete).k = [1/3, 1/3, 1/3]. Around index n0, suppose
x[n0-1] = (1, 0.3) → u ≈ 0.3095
x[n0] = (4, 0.8) → u ≈ 1.0986
x[n0+1] = (7, -0.2) → u ≈ -0.2027
Magnitude:
m_y[n0] = (1 + 4 + 7)/3 = 4
Weights (gamma=1): (1/3)*|1|, (1/3)*|4|, (1/3)*|7| = (0.333, 1.333, 2.333), sum = 4.0.
Alignment:
u_y = (0.333*0.3095 + 1.333*1.0986 + 2.333*(-0.2027)) / 4.0 ≈ 0.2739
a_y = tanh(0.2739) ≈ 0.267
Result: y[n0] ≈ ( 4.000 , 0.267 )
B) Exponential moving average (causal, discrete).y[n] = α m_x[n] + (1-α) m_y[n-1], α = 0.2. Take
x[n] = (5, 0.1) → u_x ≈ 0.1003
y[n-1] = (4, 0.7) → u_prev ≈ 0.8673
Magnitude:
m_y[n] = 0.2*5 + 0.8*4 = 4.2
Weights: 1.0 and 3.2.
Alignment:
u_y = (1.0*0.1003 + 3.2*0.8673)/4.2 ≈ 0.684
a_y = tanh(0.684) ≈ 0.594
Result: y[n] ≈ ( 4.2 , 0.594 )
C) Cross-correlation at lag 0 (length-2).
x = [(1,0.9), (2,0.3)] → u ≈ (1.4722, 0.3095)
y = [(2,0.2), (1,0.8)] → u ≈ (0.2027, 1.0986)
Magnitude:
M_xy[0] = 1*2 + 2*1 = 4
Alignment (two terms, equal weights 2 and 2):
u_pair1 = 1.4722 + 0.2027 = 1.6749
u_pair2 = 0.3095 + 1.0986 = 1.4081
u_xy[0] = (2*1.6749 + 2*1.4081)/4 ≈ 1.5415
a_xy[0] = tanh(1.5415) ≈ 0.912
Result: R_xy[0] ≈ ( 4 , 0.912 )
Options and extensions (manifest)
conv.type = "discrete" | "continuous"
align.weights = "|k|*|m|^gamma" # gamma = 1 default
align.gamma = 1
align.parameter_dep = false # true to include kernel shape at each eval
conv.mode = "full" | "same" | "valid"
conv.pad = "zero" | "reflect" | "wrap"
causal = true | false
kernel.normalize = true | false # Σ k = 1 (or ∫ k = 1)
stability.check = "L1" # BIBO: Σ|k| < ∞ or ∫|k| < ∞
diagnostics = "rapidity_spectrum" # compute A_u(ω) as in 2.25 (optional)
zero_policy = "canonical_zero" # (0,+1) for exact zeros
Alternative weight modes (advanced):
"kernel_only":w = |k|— useful to expose the filter’s structural footprint independent of signal scale."signed_pairs"(correlation): weights= |m_x m_y|but form per-term rapidities asu_x + u_y; keep denominator positive (use absolute weights).
Takeaway
Convolution and correlation in SSM keep the classical magnitude math and add a principled, reproducible alignment summary via rapidity-weighted means. This preserves linear systems theory on magnitudes (linearity, BIBO, convolution/correlation theorems) while making stability drift traceable through filtering and pattern matching. Under collapse (a = +1), everything is classical.
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1; convolution/correlation alignments use rapidity-weighted means with declared weights. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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