Abstract
We lift classical generating functions and integral transforms to symbolic numerals (m, a) by acting on the magnitude and carrying a clear alignment summary. The magnitude channel uses the standard Z-transform, Laplace transform, and moment generating function (MGF). For alignment, we recommend a rapidity-based weighted average that is reproducible and collapse-safe. Under collapse phi(m,a) = m, these constructions reduce exactly to their classical forms.
Alignment summaries for transforms (policy)
Transforms map many samples to one function; we therefore report the transform on the magnitude and attach a single alignment tag summarizing contributors.
Rapidity mean (default).
u_n = atanh( clamp(a_n, -1+eps, +1-eps) ), eps = 1e-6
a_T = tanh( (sum w_n * u_n) / (sum w_n) )
Suggested weights (declare in manifest):
w_n (discrete) = |m_n|^gamma with gamma = 1 (default)
w(t) (continuous) = |m(t)|^gamma with gamma = 1 (default)
Options: include transform-specific weights (e.g., |z|^{-n} for Z, e^{-Re(s) t} for Laplace) if you want the summary to reflect the evaluation point; otherwise keep a_T parameter-independent for stability audits.
Diagnostic channel (optional).
For deeper analysis, also compute a rapidity transform of u:
Z_u(z) = sum u_n * z^{-n} (discrete)
L_u(s) = ∫_0^inf u(t) * e^{-s t} dt (continuous)
This is for plots/diagnostics; the canonical transform value remains on the magnitude.
Z-transform (discrete-time)
For a sequence x[n] = (m[n], a[n]), define
Z_s{x}(z) = ( M(z) , a_Z )
M(z) = sum_{n=-inf}^{+inf} m[n] * z^{-n}
a_Z = rapidity_mean( {a[n]}, weights = w_n )
- ROC (region of convergence): identical to the classical ROC of
M(z). - Linearity (magnitude):
Z{alpha m1 + beta m2} = alpha Z{m1} + beta Z{m2}withalpha, betaclassical. - Shifts/scales: time shifts and scaling carry over on
M(z)exactly; recomputea_Zusing the chosen weights if you want the summary to reflect the new window.
Collapse: if all a[n] = +1, then a_Z = +1 and Z_s reduces to the classical Z-transform.
Laplace transform (continuous-time)
For x(t) = (m(t), a(t)) (causal case for concreteness),
L_s{x}(s) = ( M(s) , a_L )
M(s) = ∫_0^inf m(t) * e^{-s t} dt with Re(s) > sigma_0
a_L = rapidity_mean( {a(t)}, weights = w(t) )
- ROC: identical to the classical Laplace ROC for
m(t). - Linearity and shift theorems: hold on
M(s); alignment summary is metadata.
Collapse: a(t) = +1 for all t ⇒ a_L = +1 and the transform is classical.
Moment generating function (MGF) and probability GFs
For a symbolic random variable X = (m, a) with classical law on m:
MGF_s(t) = ( M_X(t) , a_M )
M_X(t) = E[ exp( t * m ) ]
a_M = rapidity_mean( {a}, weights = exp( t * m ) * p(m) )
- Other generating functions (PGF for integer-valued
m, cumulant GF, etc.) lift analogously: compute classically onmand attach an alignment summary using the natural weights from the definition.
Collapse: if a = +1 almost surely, the alignment tag is +1 and the GF is classical.
Worked examples
A) Z-transform of a geometric sequence.m[n] = r^n * u[n] with u[n] the unit step; a[n] = a0 constant.
M(z) = sum_{n=0}^inf (r^n z^{-n}) = 1 / (1 - r z^{-1}), ROC: |z| > |r|
a_Z (gamma=1, constant a0) = a0
Example: r = 1/2, a0 = 0.8 ⇒ Z_s{x}(z) = ( 1 / (1 - 0.5 z^{-1}) , 0.8 )
B) Z-transform with alternating stability.m[n] = (1/2)^n u[n], a[n] = (-1)^n * 0.6. With w_n = |m[n]|:
u_n = atanh( (-1)^n * 0.6 ) = (-1)^n * atanh(0.6) ≈ (-1)^n * 0.6931
a_Z = tanh( sum w_n u_n / sum w_n ) = tanh( 0 ) = 0
Magnitude transform: M(z) = 1 / (1 - 0.5 z^{-1}); alignment summary reveals net neutral stability due to alternating signs.
C) Laplace of a decaying exponential with drifting alignment.m(t) = e^{-t}, a(t) = 0.3 + 0.7 e^{-t} on t >= 0.
Magnitude: M(s) = 1 / (s + 1) (ROC Re(s) > -1).
Alignment summary (gamma=1, weight w(t)=|m(t)|):
u(t) = atanh( 0.3 + 0.7 e^{-t} )
a_L = tanh( ∫_0^inf u(t) e^{-t} dt / ∫_0^inf e^{-t} dt )
= tanh( ∫_0^inf u(t) e^{-t} dt ) # denominator = 1
Numerically this lands between atanh(0.3) and atanh(1.0) and reports a moderate Pearo.
D) MGF of a two-point distribution.m in {1, 2} with P(m=1)=0.6, P(m=2)=0.4; alignments a(1)=0.5, a(2)=0.9.
M_X(t) = 0.6 * e^t + 0.4 * e^{2t}
u(1) = atanh(0.5)=0.5493, u(2)=atanh(0.9)=1.4722
a_M = tanh( (0.6 e^t * u(1) + 0.4 e^{2t} * u(2)) / (0.6 e^t + 0.4 e^{2t}) )
As t grows, the heavier-weighted m=2 term dominates, so a_M trends toward 0.9.
Properties and notes
- Linearity and convolution theorems hold on the magnitude channel exactly as in the classical theory (e.g., time-domain convolution ↔ product of Z/Laplace magnitudes). If you require a stability analogue, use the diagnostic rapidity transforms
Z_u,L_uto study coupling; the canonical numeric result remains on magnitude. - Stability windows. For streaming or nonstationary data, compute
a_Tover sliding windows using the same rapidity-mean rule; publish the window size and hop in the manifest. - Parameter-dependent summaries (optional). You may define
a_T(z)ora_T(s)by including transform kernels inw_norw(t); default is parameter-independenta_T.
Manifest fields (recommended)
transform = "Z" | "Laplace" | "MGF" | "PGF" | "CGF"
alignment_summary.policy = "rapidity_mean"
alignment_summary.weights = "abs_m^gamma" # gamma = 1 by default
alignment_summary.parameter_dependent = false # true to include |z|^{-n} or e^{-Re(s) t}
diagnostic_alignment_channel = "off" | "Z_u" | "L_u"
eps = 1e-6 # clamp for atanh
zero_canonicalize = true
Also declare ROC (Z/Laplace), domains for parameters, and any windowing choices for streaming data.
Takeaway
Generating functions and transforms lift as-is on the magnitude channel; a principled rapidity mean supplies a single, audit-ready alignment tag. This keeps transforms compatible with classical toolchains while exposing stability information in a compact, reproducible way. Collapse (a = +1) returns the classical Z/Laplace/MGF exactly.
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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; alignment summaries use rapidity means unless declared otherwise. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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