Abstract
Classical Fourier analysis decomposes signals into sums of sinusoids. In Shunyaya Symbolic Mathematics (SSM), signals are sequences or fields of symbolic numerals (m, a), so both amplitude and alignment contribute to frequency content. Under collapse phi(m,a) = m, all results reduce to the classical theory.
Symbolic Fourier transform (two channels)
For a (sufficiently integrable) symbolic time series x(t) = (m(t), a(t)), define
F_s(omega) = ( M(omega), A(omega) )
M(omega) = ∫ m(t) * exp(-i*omega*t) dt
A(omega) = ∫ a(t) * exp(-i*omega*t) dt
Interpretation
• M(omega): classical Fourier transform of the magnitude channel.
• A(omega): Fourier transform of the alignment signal.
Collapse: if a(t) = +1 for all t, then A(omega) is a DC spike at omega = 0, and M(omega) reduces to the classical transform.
Rapidity-spectrum (default; geometry-aware)
To respect boundedness and M2 geometry, analyze the rapidity
u(t) = atanh( clamp(a(t), -1+eps, +1-eps) ), eps = 1e-6
and take
A_u(omega) = ∫ u(t) * exp(-i*omega*t) dt
Benefits
• Linearizes alignment composition (additive in u).
• Avoids edge distortion when a(t) approaches ±1.
Manifest field (recommendation): spectrum.alignment_channel = "rapidity" (default) or "alignment".
Discrete-time DFT (implementation)
For samples x[n] = (m[n], a[n]), n = 0..N-1, let
u[n] = atanh( clamp(a[n], -1+eps, +1-eps) )
With window w[n]:
M[k] = sum_{n=0}^{N-1} w[n] * m[n] * exp(-i*2*pi*k*n/N)
A_u[k] = sum_{n=0}^{N-1} w[n] * u[n] * exp(-i*2*pi*k*n/N)
Window, zero-padding, detrend, overlap, and averaging follow classical practice and must be declared in the manifest.
Symbolic power spectra
Define two-channel power (continuous or discrete, per convention):
P_s(omega) = ( |M(omega)|^2 , |A_u(omega)|^2 )
Interpretation
• |M(omega)|^2: classical signal power at frequency omega.
• |A_u(omega)|^2: “drift power,” the strength of alignment dynamics at that frequency.
Parseval (channelwise, under standard conventions):∫ |m(t)|^2 dt = (1/(2*pi)) ∫ |M(omega)|^2 d omega, and similarly for u(t) and A_u(omega).
Numeric illustration (qualitative).
Let m(t) = cos(2*pi*f0*t), a(t) = 0.5*cos(2*pi*f0*t).
Then M(omega) has impulses at ±f0 (amplitudes per the chosen convention).u(t) ≈ atanh(0.5)*cos(2*pi*f0*t) near 0, so A_u(omega) has impulses at ±f0 scaled by atanh(0.5) ≈ 0.549.
The ratio highlights that only part of the oscillatory content is centre-aligned.
Cross-spectra and coherence (m–u coupling)
Define cross-spectrum and magnitude–rapidity coherence:
S_m_u(omega) = M(omega) * conj( A_u(omega) )
Coh_m_u(omega) = |S_m_u(omega)|^2 / ( |M(omega)|^2 * |A_u(omega)|^2 )
Guards: define Coh_m_u(omega) = 0 when the denominator is zero.
Interpretation
• Coh_m_u near 1 indicates tight coupling between amplitude and alignment oscillations at that frequency.
• Useful for diagnosing modes that grow while stability simultaneously decays.
Autocorrelation and Wiener–Khinchin (symbolic form)
Channelwise autocorrelations:
R_m(tau) = E[ m(t) * m(t+tau) ]
R_u(tau) = E[ u(t) * u(t+tau) ]
Their Fourier transforms give the corresponding spectral densities.
A mixed correlation R_m_u(tau) = E[ m(t) * u(t+tau) ] yields S_m_u(omega) by Fourier transform.
Symbolic eigen-decomposition (operators on H_beta)
For a bounded beta-linear operator T on H_beta, a symbolic eigenpair satisfies
T(x) = M2_scale( lambda, x )
with lambda = (m_lambda, a_lambda) a symbolic scalar and M2_scale denoting scalar multiplication via the M2 rule (componentwise on entries).
Interpretation
• m_lambda: classical growth/rotation factor for the mode.
• a_lambda: stability of the mode (Pearo vs Nearo).
Collapse: if a_lambda = +1, lambda reduces to the classical eigenvalue.
Informal spectral theorem
For self-adjoint symbolic operators (with respect to <•,•>_beta), there exists an orthonormal basis of symbolic eigenvectors and a diagonal representation with symbolic eigenvalues lambda = (m, a). Under collapse, this recovers the classical spectral theorem.
Example (symbolic harmonic decomposition)
x(t) = ( cos(omega*t), +1 ) oplus ( sin(omega*t), a0 )
Magnitude spectrum: lines at ±omega from cos(omega*t).
Rapidity spectrum: lines at ±omega from atanh(a0)*sin(omega*t).
Symbolic analysis reveals dual spectra: magnitude oscillations and stability oscillations at the same frequency, with amplitudes governed by atanh(a0).
Time–frequency analysis (nonstationary signals)
Define the symbolic STFT:
STFT_m(t0, f) = sum_n w[n - t0] * m[n] * exp(-i*2*pi*f*n)
STFT_u(t0, f) = sum_n w[n - t0] * u[n] * exp(-i*2*pi*f*n)
Report both magnitude and rapidity spectrograms to visualize when and where drift power emerges.
Implementation notes (manifest)
spectrum.alignment_channel = "rapidity" # or "alignment"
spectrum.window = <type>
spectrum.nfft = <N>
spectrum.overlap = <percent>
eps = 1e-6 # clamp for a before atanh
detrend = <method>
cross_spectra = on/off
coherence = on/off
normalization = <DFT convention> # declare scaling for Parseval
Takeaway
Symbolic Fourier and spectral analysis attach alignment to every frequency:
• Each frequency has a dual signature (amplitude and stability).
• Power splits into size power and drift power; coherence quantifies their coupling.
• Eigenvalues carry stability tags.
Collapse recovers classical Fourier and spectral results exactly. Using u(t) by default keeps alignment analysis bounded, geometry-aware, and M2-consistent.
Navigation
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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 with rapidity u = atanh(a) where needed. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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