Beyond Fourier: When the Transform Transforms Itself

By John Gavel

The Realization

I was working through the mathematics of temporal flow physics — a framework where flows are more fundamental than time, and time emerges from asymmetric flow states — when something clicked. I realized we had a form of recursion that looked eerily similar to Fourier transforms, but operating on something deeper: geometric situations themselves.

The key observation was this expression:

$$ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} e^{i \Phi_k} \neq 0 $$

In standard signal processing, when you average phases over many oscillations, you typically get zero — everything cancels out unless there’s coherent structure. But here, even as \( n \) approaches infinity, the phase average persists. Something was maintaining coherence across infinite recursion.

This wasn’t just Fourier analysis in disguise. It was something fundamentally different.

What Fourier Analysis Assumes

Standard Fourier theory makes several implicit assumptions:

  • The environment is fixed: signals are decomposed in flat, unchanging space
  • The basis is predetermined: sines and cosines are complete and orthogonal
  • The transform is linear and static: decomposition does not alter the basis
  • Periodicity is assumed: full cycles, infinite extension, closure under translation

These assumptions work brilliantly for continuous media — vibrations, waves, and fields. But they break down when the environment itself is discrete, causal, and recursive.

The Recursive Transform

In temporal flow physics, we remove the closure assumption. Instead of asking “what frequencies exist in this signal?”, we ask:

What patterns can survive recursive interaction with a discrete causal environment?

Define a recursive phase accumulator:

$$ \Phi_{n+1} = \Phi_n + \Delta \Phi(\text{environment}, \text{adjacency}) $$

Where:

  • \( \Delta \Phi \) depends on how many neighbors actively participate
  • Recursion replaces periodicity
  • Survival replaces orthogonality

A “mode” now exists not because it is an eigenfunction of an operator, but because it is recursively coherent:

$$ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} e^{i \Phi_k} \neq 0 $$

This is not a Fourier eigenmode. It is a recursively stable partial mode — a pattern that survives geometric dissolution.

How Particle Ratios Emerge

Consider a 12-neighbor environment — the kissing number in three dimensions. At each recursive step, phase distributes across adjacency paths.

Two stable partitions emerge:

  • Tetrahedral sector: 4 paths out of 12
  • Octahedral sector: 8 paths out of 12

Define a projection operator:

$$ P_{\text{sector}} = \frac{\text{active paths}}{\text{total paths}} $$

The recursive phase contribution becomes:

$$ \Delta \Phi_{\text{tet}} = \frac{4}{12} \Delta \Phi_0 = \frac{1}{3} \Delta \Phi_0 $$ $$ \Delta \Phi_{\text{oct}} = \frac{8}{12} \Delta \Phi_0 = \frac{2}{3} \Delta \Phi_0 $$

After \( N \) recursive steps:

$$ \Phi_N = N \cdot \frac{\text{paths}}{12} \cdot \Delta \Phi_0 $$

The ratio survives recursion because it is topological, not dynamical.

The fractions \( 1/3 \) and \( 2/3 \) are not arbitrary quantum numbers. They are the stable Fourier weights that survive recursive deformation in a 12-neighbor causal geometry.

The Meta-Transform

Standard Fourier transform:

function → spectrum (fixed basis)

Recursive transform:

environment → admissible spectra (evolving basis)

Formally:

$$ \text{Basis}_k(\text{env}_{n+1}) = \mathcal{R}(\text{env}_n) \circ \text{Basis}_k(\text{env}_n) $$

The spectrum evolves with the environment.

This is the “Fourier transform of the transform” — the measuring basis undergoes the same recursive dynamics as what it measures.

Visualizing the Evolution

  • Step 0: isotropic phase impulse
  • Step 1: discrete adjacency → 12 channels
  • Step 2: recursive reinforcement suppresses instability
  • Step 3: partial phase closure locks 4/12 and 8/12
  • Step 4: ratios become invariant → charge, coupling, mass

This is geometric dissolution: symmetry erodes until only recursively stable remnants remain.

The Speed of Light

The speed of light itself emerges from recursive geometry.

Raw propagation speed:

$$ v_{\text{raw}} = \frac{L}{\Delta t} $$

Information compression:

$$ \rho_I = \frac{\ln(12)}{\ln(4096)} \approx 0.2987 $$

Minimal tension scalar:

$$ Q = \frac{4}{9} $$

Geometric symmetry factor:

$$ \frac{5}{4} $$

Result:

$$ c = \frac{L}{\Delta t} \cdot \sqrt{12} \cdot \frac{\ln(12)}{\ln(4096)} \cdot \frac{4}{9} \cdot \frac{5}{4} $$

The speed of light is not an input constant — it is the output of recursive geometric constraints.

The Core Insight

Ratios like \( 1/3 \), \( 2/3 \), and \( 4:8 \) are fixed points of a recursively deformed Fourier transform acting on a 12-neighbor causal substrate.

They are survivability coefficients — the weights that remain coherent when geometry measures itself.

Perhaps particles are not things in spacetime.

Perhaps they are the harmonics that remain when spacetime recursively folds through itself.

This is part of an ongoing exploration of temporal flow physics. The framework is still being developed, and engagement is welcome.