Submediant

Submediant#

N. Joven — 2026 — ORCID 0009-0008-0679-0812 — CC0 1.0

Source: harmonics

You know this pattern.

Take two numbers. Add them. Use the last two to make the next. 1, 1, 2, 3, 5, 8, 13, 21 …

The ratio between consecutive terms settles to \(\phi = (1 + \sqrt{5})/2\). This is the golden ratio — the number that appears in sunflower spirals, nautilus shells, and the branching of trees.

It also appears in the cosmic microwave background.

The polynomial behind the Fibonacci sequence, \(x^2 - x - 1 = 0\), has two roots. Three properties of these roots correspond to three physical measurements:

What the roots do

What we observe

Residual

Their product is \(\pm 1\)

Born rule: \(P = \lvert\psi\rvert^2\)

exact

\(\phi^2 = \phi + 1\) (self-similarity)

CMB spectral tilt \(n_s \approx 0.965\)

0.2%

\(\phi - \psi = \sqrt{5}\) (gap width)

Inflation lasted \(\sim 61\) e-folds

CMB-S4, ~2028

The same polynomial also determines that space has three dimensions (because fractions have two parts: \(\dim \mathrm{SL}(2) = 2^2 - 1 = 3\)) and that the MOND acceleration scale is \(a_0 = 1.25 \times 10^{-10}\) m/s\(^2\) (observed: \(1.2 \times 10^{-10}\), residual: 4%).

The frequency distribution \(g(\omega)\) — the one remaining free input — determines itself: the distribution that produces dynamics reproducing that distribution is unique. The framework introduces no free parameters and no free functions — every quantity is determined by the recurrence and the boundary conditions.

This site walks through the derivation chain, starting from the polynomial and arriving at known physics:

Counting \(\to\) fractions \(\to\) the golden ratio \(\to\) mode-locking \(\to\) the devil’s staircase \(\to\) a self-consistency equation \(\to\) general relativity and quantum mechanics

Each step is a composition of the previous ones. The derivation chain is 30 steps. The engine solves the equation numerically and produces all predictions from a single run.

Where to start#

  • Curious about the math? Start with The Alphabet — four primitives that generate all the structure

  • Curious about the physics? Start with Predictions — what the framework says and how it compares to measurement

  • Want the punchline? K = 1: Einstein — how one equation produces general relativity, uniquely

  • Want to run it? rfe — python -m rfe --observables

  • From scratch? First Principles — sin(ωt) to Einstein in 10 steps

  • Where are we? Our Address — the universe’s computational clock on the Stern-Brocot tree

Source#

  • harmonics — the derivation chain (Derivations 1–30)

  • rfe — the engine (one equation, all observables)

  • proslambenomenos — one frequency, zero free parameters

  • 201 — gravity as synchronization in a frictional medium

  • intersections — stick-slip dynamics and dark matter