Proof Chain A: Polynomial → General Relativity

Proof Chain A: Polynomial → General Relativity#

N. Joven — 2026 — CC0 1.0

Statement#

From two physical properties of coupled oscillators — energy conservation and stability — the Einstein field equations follow uniquely. Each proposition uses only the previous ones. No proposition uses the continuum, coordinates, or any physics beyond oscillator coupling.

Definitions#

D1. An oscillator is a process with an integer cycle count.

D2. Two oscillators couple when they share energy without external input.

D3. A winding number p/q means: q cycles of one oscillator correspond to p cycles of the other.

Propositions#

P1. The circle [D10 §1]#

Uses: integers, fixed-point condition.

A period-q orbit with winding number p satisfies f^q(x) = x + p and f^q(x) = x simultaneously. Therefore p ≡ 0 in the phase space. Since p is an arbitrary integer, the phase space is R/Z = S¹. ∎

P2. The mediant [D29]#

Uses: P1 (circle), energy conservation (D2), Arnold tongue stability.

Two coupled oscillators at winding numbers a/b and c/d lock to a frequency between them (energy conservation: no external source). Among all rationals in (a/b, c/d), the one with smallest denominator has the widest Arnold tongue — width w ~ (K/2)^q — and is reached first as coupling increases.

Theorem (Stern-Brocot, 1858). For adjacent fractions (|ad − bc| = 1), the unique rational in (a/b, c/d) with smallest denominator is the mediant (a+c)/(b+d). ∎

P3. The Stern-Brocot tree [D10 §2–3]#

Uses: P1, P2.

Iterating the mediant from the endpoints 0/1 and 1/0 enumerates every positive rational exactly once, ordered by denominator. The tree is the unique configuration space: every winding number appears at its natural complexity level. No other enumeration respects the Arnold tongue stability ordering (P2). ∎

P4. The rational field equation [D11]#

Uses: P3, fixed-point condition.

The population N(p/q) at each node of the Stern-Brocot tree satisfies:

\[N(p/q) = N_{\text{total}} \times g(p/q) \times w(p/q,\; K_0 F[N])\]

where g is the frequency distribution, w is the tongue width, and F[N] is the global order parameter. This is the fixed-point equation x = f(x) applied to the population: the distribution determines the coupling which determines the distribution. ∎

P5. Three dimensions [D14]#

Uses: P2, P3.

The mediant (a+c)/(b+d) acts on column vectors by addition: \(\binom{a}{b} + \binom{c}{d} = \binom{a+c}{b+d}\). The group generated by these operations on pairs of coprime integers is SL(2,Z). Its continuum completion is SL(2,R).

Self-consistent adjacency (the spatial manifold must be the group itself, so that every point can serve as mediator) forces the spatial manifold to have dimension dim SL(2) = 2² − 1 = 3.

SL(2,C) ≅ Spin(3,1) gives Lorentz symmetry by complexification. ∎

P6. SL(2,R) is unique [D15]#

Uses: P5.

Four conditions characterize SL(2,R) among all connected real Lie groups:

Condition	Source
Arithmetic skeleton from the mediant	P2 → SL(2,Z)
Projective action on frequency ratios	Winding numbers are ratios
Dynamical trichotomy from Iwasawa KAN	Elliptic/parabolic/hyperbolic
Farey-hyperbolic geometry	Stern-Brocot tree tiles H²

The Bianchi classification of 3-dimensional Lie algebras eliminates every alternative. ∎

P7. The continuum limit at K = 1 [D12 §I]#

Uses: P4, P5, P6.

At critical coupling K = 1, all oscillators are locked: the order parameter r = 1. The rational field equation on the Stern-Brocot tree, in the continuum limit on the SL(2,R) manifold, produces the ADM evolution equations:

\[\partial_t \gamma_{ij} = -2N \mathcal{K}_{ij} + D_i N_j + D_j N_i\]

\[\partial_t \mathcal{K}_{ij} = -D_i D_j N + N\left(R_{ij} + \mathcal{K}\mathcal{K}_{ij} - 2\mathcal{K}_{ik}\mathcal{K}^k{}_j\right) + \text{matter}\]

The dictionary:

Kuramoto	ADM
Coherence r(x,t)	Lapse N
Phase gradient ∂ᵢθ	Shift / momentum
Correlation ⟨∂ᵢθ ∂ⱼθ⟩	Spatial metric γᵢⱼ
Phase curvature ⟨(∂ᵢ∂ⱼθ)²⟩	Extrinsic curvature Kᵢⱼ

The first equation is exact under locked-state conditions. The second follows from differentiating and substituting the Kuramoto dynamics. ∎

P8. Uniqueness: Einstein [D13, Lovelock 1971]#

Uses: P7, P5, P6.

The continuum limit satisfies four conditions:

Metric theory on a 3+1 manifold (P5: dim = 3, P7: the metric is the correlation tensor)
Self-consistency origin (P4: fixed-point equation)
Second-order in metric derivatives (P7: Kuramoto is first-order in θ, so correlations are second-order in γ)
General covariance (P6: SL(2,R) acts transitively)

Lovelock’s theorem (1971): In 4 dimensions, the unique divergence-free, symmetric, second-order tensor built from the metric and its first two derivatives is:

\[G_{\mu\nu} + \Lambda g_{\mu\nu}\]

Therefore:

\[\boxed{G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu}}\]

The Einstein field equations are the unique output. Not the intended output — the only one that satisfies the four conditions. ∎

The chain#

\[\text{Counting} \xrightarrow{P1} S^1 \xrightarrow{P2} \text{mediant} \xrightarrow{P3} \text{Stern-Brocot} \xrightarrow{P4} \text{field eq.} \xrightarrow{P5,P6} d{=}3,\;\text{SL}(2) \xrightarrow{P7,P8} G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}\]

Eight propositions. Two physical inputs (energy conservation, stability). One output (Einstein). Every step is a theorem, not a choice.

Dependency graph#

P1 (circle)
 ↓
P2 (mediant) ← energy conservation + stability
 ↓
P3 (Stern-Brocot tree)
 ↓           ↘
P4 (field eq.)  P5 (d=3, SL(2))
 ↓               ↓
 ↓             P6 (uniqueness of SL(2,R))
 ↓               ↓
P7 (ADM from Kuramoto at K=1)
 ↓
P8 (Lovelock → Einstein)

Cross-references#

Proposition	Derivation	Key theorem
P1	D10 §1	Circle from integers + fixed-point
P2	D29	Stern-Brocot (1858)
P3	D10 §2–3	Stern-Brocot tree enumeration
P4	D11	Rational field equation
P5	D14	dim SL(2) = 3
P6	D15	Bianchi classification
P7	D12 §I	ADM from Kuramoto
P8	D13	Lovelock (1971)