Proof Chain B: Polynomial → Quantum Mechanics

Proof Chain B: Polynomial → Quantum Mechanics#

N. Joven — 2026 — CC0 1.0

Statement#

From the same two physical properties — energy conservation and stability — the Schrödinger equation and the Born rule follow uniquely at subcritical coupling. The first five propositions are shared with Proof Chain A. The paths diverge at the coupling parameter K.

Shared propositions (P1–P5)#

These are identical to Proof Chain A and are not repeated here. See PROOF_A_gravity.md.

P	Statement	Source
P1	Phase space is S¹	D10 §1
P2	Mediant is unique combining operation	D29
P3	Stern-Brocot tree is configuration space	D10 §2–3
P4	Rational field equation (fixed-point)	D11
P5	d = 3, SL(2,R)	D14

Propositions (quantum branch)#

Q1. The parabola at tongue boundaries [D1, D10 §4]#

Uses: P3 (Stern-Brocot tree), circle map dynamics.

At the boundary of every Arnold tongue, the circle map undergoes a saddle-node bifurcation. Near the bifurcation point, the dynamics reduce to:

\[\dot{x} = \mu - x^2\]

where μ is the distance from the tongue boundary. This is the normal form — the parabola x² + μ = 0 is the fourth primitive.

The basin width scales as:

\[\Delta\theta \propto \sqrt{\varepsilon}\]

where ε = |μ| is the detuning from resonance. This is exact — it is the universal normal form of the saddle-node, not an approximation. The square root is the parabola’s geometry. ∎

Q2. The Born rule [D1, D9]#

Uses: Q1.

The probability of finding the system in a particular locked state is proportional to the basin of attraction of that state. From Q1, the basin width is Δθ ∝ √ε, so the basin area (in the 2D phase-amplitude space) is:

\[P_k \propto (\Delta\theta_k)^2 \propto \varepsilon_k \propto |\psi_k|^2\]

The exponent 2 in |ψ|² is the degree of the parabola at the bifurcation. It is geometry, not a postulate.

Equivalently: the synchronization cost landscape near each attractor is quadratic (C ≈ C₀ + α|ψ − ψ_k|²). Dissipative convergence on this landscape contracts phase-space volume proportional to |ψ_k|². The Born rule is the basin measure of a quadratic cost landscape. ∎

Q3. The subcritical regime K < 1 [D12 §II]#

Uses: P4, P5.

At subcritical coupling (K < 1), the order parameter r < 1. A finite fraction of oscillators remain unlocked — they sit in the gaps of the devil’s staircase with no definite winding number. These are the quantum states.

The population splits:

Oscillators	Fraction	Character
Locked (definite p/q)	r	Classical (metric, P7–P8)
Unlocked (drifting)	1 − r	Quantum (wave, Q4)

The ratio r/(1−r) is set by the self-consistency equation (P4). ∎

Q4. Madelung variables [D12 §II]#

Uses: Q3, P5.

For the unlocked oscillators, define:

Density: ρ(x,t) = density of unlocked oscillators at position x
Phase: S(x,t) = accumulated phase perturbation (action variable)

In the continuum limit on the SL(2,R) manifold (P5), with nearest-neighbor diffusive coupling:

\[\partial_t \rho + \nabla \cdot (\rho\, \nabla S / m) = 0 \qquad \text{(continuity)}\]

\[\partial_t S + \frac{|\nabla S|^2}{2m} + V - \frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}} = 0 \qquad \text{(Hamilton-Jacobi)}\]

where:

Kuramoto quantity	QM quantity
Unlocked oscillator density ρ(x,t)	\|Ψ\|²
Accumulated phase S(x,t)	Action / phase
Tongue-structure effective potential	V(x)
Stern-Brocot RG diffusion D_eff	Quantum potential

The quantum potential ℏ²∇²√ρ / (2m√ρ) arises from the Stern-Brocot renormalization group flow: per-level variance σ²(d) ~ φ^{−4d} sums to a convergent constant D_eff = D₀/(1 − φ^{−4}). ∎

Q5. The Schrödinger equation [D12 §II]#

Uses: Q4.

Define Ψ = √ρ · e^{iS/ℏ}. The Madelung equations (Q4) are algebraically equivalent to:

\[\boxed{i\hbar\,\partial_t \Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi}\]

This is the Schrödinger equation. The transformation is invertible (Madelung, 1927). ∎

Q6. Uniqueness of the quantum branch#

Uses: Q1–Q5.

The Schrödinger equation is the unique output of the subcritical regime because:

The Madelung form is forced by continuity + Hamilton-Jacobi (the only equations consistent with probability conservation and energy conservation for the unlocked population)
The quantum potential is forced by the Stern-Brocot RG (the diffusion constant is set by φ, which is set by the tree structure P3)
The Born rule exponent 2 is forced by saddle-node universality (Q1–Q2)

No postulates of quantum mechanics are assumed. They are derived:

QM postulate	Derivation
States are vectors in Hilbert space	Ψ = √ρ · e^{iS/ℏ} (Q4–Q5)
P = \|ψ\|²	Basin measure of parabolic cost (Q2)
Time evolution is Schrödinger	Madelung ↔ Schrödinger equivalence (Q5)
Observables are operators	Conjugate variables (ρ, S) → (x, p)

∎

The chain#

\[\text{Counting} \xrightarrow{P1} S^1 \xrightarrow{P2} \text{mediant} \xrightarrow{P3} \text{Stern-Brocot} \xrightarrow{P4} \text{field eq.}\]

\[\xrightarrow{K < 1} \text{unlocked oscillators} \xrightarrow{Q3,Q4} \text{Madelung} \xrightarrow{Q5} i\hbar\partial_t\Psi = H\Psi\]

\[\text{Parabola} \xrightarrow{Q1} \sqrt{\varepsilon}\text{ scaling} \xrightarrow{Q2} P = |\psi|^2\]

Seven propositions (five shared, two independent). Same two physical inputs. The quantum branch is the subcritical limit of the same equation whose critical limit gives general relativity.

Dependency graph#

P1–P5 (shared with Proof A)
  ↓
P4 (field eq.) at K < 1
  ↓
Q3 (locked/unlocked split)
  ↓
Q4 (Madelung: ρ, S)
  ↓
Q5 (Schrödinger)

P3 (Stern-Brocot) + Primitive 4 (parabola)
  ↓
Q1 (saddle-node: Δθ ∝ √ε)
  ↓
Q2 (Born rule: P = |ψ|²)

The two limits, one equation#

Parameter	Regime	Locked fraction	Output
K = 1	Critical	r = 1 (all locked)	Einstein (Proof A, P8)
K < 1	Subcritical	r < 1 (partial)	Schrödinger (Q5) + Born (Q2)
K → 0	Decoupled	r = 0 (none locked)	Free particles

The rational field equation (P4) is one equation. General relativity and quantum mechanics are its two continuum limits.

Cross-references#

Proposition	Derivation	Key theorem
P1–P5	D10, D29, D11, D14	(see Proof A)
Q1	D10 §4, D1	Saddle-node normal form
Q2	D1, D9	Basin measure =
Q3	D12 §II	Subcritical: partial locking
Q4	D12 §II	Madelung (1927)
Q5	D12 §II	Schrödinger (1926)
Q6	D1, D9, D12	Uniqueness