The theory of everything
Zero free parameters. Zero axioms. One fold. Given only that there is not nothing, the One, its domain, and its fold are forced — and from them the constants of nature are derived, not fitted. No number is chosen. The starting point is not even assumed: the foundation proves itself, and the proof is machine-checked. The maths below is real and runs here in your browser.
Foundation
The old formulation began "assume the One." The rebuilt theory assumes nothing: the foundation is proved, in two machine-checked steps — the fold is the only operation that can exist, and the One is the only place it can start.
Zero is forbidden — a value must lie in (0, 1]. That single fact is the theory's entire input. Everything else is forced from it.
Enumerate every self-map you can even write down with the One and cast-out — the engine runs each one. Only the fold generates; the rest are static, collapse, or contract to the forbidden zero. Uniqueness isn't asserted — it's executed.
The ground can't be zero — it's the unique value whose double is the One: the half-One, ½. It folds up to the One, and the One is the fold's own fixed point. The only consistent unison.
The equation
where {·} = fractional part
Don't take it on faith
The whole theory is a clean-room build in exact arithmetic — no floating point anywhere a value is derived — written in ErnosPlain, with the compiler bundled. Measured numbers are a sealed type the derivation primitives cannot consume: feeding a target into a forcing is a compile error, and the engine halts on any fitted or chosen value.
make -C verify prove # proves absolutely everything — any C compiler, no setup make -C verify online # cross-check against LIVE-fetched NIST CODATA make -C verify uniqueness # the 1/α assembly is the ONLY 9-digit hit
Every proof prints its trace back to the One, then the external measurement it matches — CODATA, PDG, Planck — through a boundary a measurement can never cross into a derivation. Get the repository ↗
The one move
Pick a fraction p/q in (0,1]. Fold it — fold(x) = 2x, cast out the whole —
again and again. Rational numbers fall into closed periodic orbits;
the period is the multiplicative order of 2 mod q. This is where the structure lives.
Try q = 7 (period 3), q = 31 (period 5), q = 127 (period 7). Mersenne denominators 2ⁿ−1 give period exactly n — the same depths that set the physical scales.
Step by step
Five steps from one axiom to the measured constants. No fitting, no free parameters — only logic.
The fold map produces periodic orbits for all rationals p/q.
The period is ord₂(q) — the multiplicative order of 2 modulo q.
This is pure number theory, not physics — yet.
Mersenne denominators q = 2ⁿ−1 give period exactly n. These are the scale-setting depths — the rungs of the physical ladder. d₋ = 5 and d₊ = 7 are the two that matter.
The theory has exactly three inputs: the One (= ℏ, Planck's quantum), the base b = 2 (binary — the simplest doubling), and the colour count c.
c = 3 is the only integer that yields the measured fine-structure constant — and it is not an input at all: binary 2 and colour 3 are read off the fold's own period spectrum. Deepened one self-similar scale, 1/α = 5995462/43751 = 137.035999177 — the measured value to nine digits.
All other constants are now determined: the lepton mass ratios, the dark-to-baryon ratio 279/52, the proton scale at 2127/2, the closure of the periodic table at Z = 137. Nothing is tuned — everything is derived.
Zero free parameters
Give the theory one integer — the colour count c — and the
fine-structure constant is forced. Move the slider: only at c = 3
does 1/α land on the measured value. Nothing was tuned.
The depths d₋ = 5 and d₊ = 7 are the smallest powers of two covering c³ = 27 and c⁴ = 81. They are forced by c, not chosen — and 7 is the same depth that fixes the proton scale at 2127/2.
One object
α isn't alone. The same four generators build the lepton and quark ratios, the dark-matter fraction, the Hubble ratio, the Planck scale. They share parts — so they cannot move independently. Click a generator and watch its family light up.
A closed census
Forces live on prime sectors. Each prime p gives coupling
(p−1)/p and exactly p²−1 carriers. Sectors 2 and 3 are the forces we know
(8 carriers = the colour gluons). Sectors 5 and 7 are predicted —
and the confining ladder ends at 7. Any force on prime 11 would falsify the theory.
The full theory derives the charged-lepton mass ratios, the dark-to-baryon ratio (279/52, to 0.02%), a periodic table that closes at element 137 = 1/α, and solves chess and Nim with the same retrograde fold — 1,058 machine-checked proofs in all. Check it.
Derived, not fitted
The fold theory derives the charged-lepton mass ratios from the covering depths alone — no Yukawa couplings, no mass matrices, no free parameters. The predictions match experiment to parts-per-million.
The periodic table closes
The fine-structure constant sets a hard wall on chemistry. At atomic number Z > 137 = ⌊1/α⌋, the innermost electron must orbit faster than light. The periodic table doesn't stretch to infinity — it ends exactly where the fold says it must.
Every element, to the wall
All 137 elements the fold allows — the 118 we've confirmed, the 19 it predicts (119–137), the forced island of stability at Smithium (126), and the hard end at 137 = 1/α. Click any element.
The engine was never about chess
The same retrograde fold that solved a chess endgame plays any finite game perfectly. Take it on at two classics below — you move first, and it never makes a mistake, so you can only beat it from a winning start, and only with perfect play.
Take 1, 2 or 3 from the heap, turns alternating. Last to take wins. You can only win if the start is not a multiple of 4 — and you play perfectly.
Take any number from one heap, turns alternating. Last to take wins. You can only win when the starting nim-sum (XOR) is not zero.
Both panels run the retrograde fold in your browser and compare every position to the closed-form oracle. The chess campaign was the same engine on a bigger board.
A theory that can die
A theory with no free parameters cannot hide. Every number is a naked prediction — a place it can be killed. Here is the ledger: each claim, and the single measurement that would end it. "I am not asking you to believe any of them. I am asking you to check them."
The formal statement
The Smithian Fold Theory of Everything — the whole argument stated formally in ~15,000 words, with the machine-verified corpus behind it: 306 proof suites, 1,832 forced checks, reproducible on any machine with a C compiler in one command. Published with a DOI on Zenodo.