Collatz Final Gate v3.7c: Asymptotic Dirichlet Spectral Gap and Entropic Heatflow Proof

Summary (v3.7)

We recast Collatz dynamics on the 2-adic space $M_C \simeq \mathbb{Z}_2$ through a reversible, self-adjoint Markov contraction $K$ and the entropic Laplacian $\Delta_C^{(\infty)} := I-K$. A unitary dilation gives the compression identity $K=J^\dagger U_T J$ and the energy equality

$$\mathcal{E}[f] = \langle f,(I-K)f \rangle = \frac{1}{2}\|Jf - U_T Jf\|_{L^2}^2,$$

identifying entropy dissipation with parity-branch coherence loss.

Scope

Exponential relaxation/absorption is proved unconditionally for the annealed model. Transfer to deterministic (quenched) Collatz trajectories holds conditionally under explicit refresh regularity (uniform ERF, bad-block budget, or ERF-average) or the existence of a finite 'Gate B' residue certificate.

Non-claim

This work does not claim an unconditional proof of the Collatz conjecture.

What’s new in v3.7

1. Appendix F: The 'Gate B' Residue Certificate

This is the main update. We formalize the final number-theoretic step required to activate the v3.6 framework.

• Gate B (Def F.1): Isolates a finite, verifiable 'residue certificate' $(m,L,S,\eta)$.

• Activation (Lemma F.3): Proves that the existence of Gate B implies a uniform good-block structure ($\rho \ge 1/(2L)$ and $b(N)=O(1)$), which satisfies the (previously conditional) sublinear deficit requirement of Thm 8.18.

• Result (Thm F.4): The existence of Gate B + the annealed gap $\lambda_*^{(D)}$ $\implies$ unconditional exponential absorption for all quenched trajectories $n > N_0$.

2. Verifiable Routes to Proving Gate B (App F)

Appendix F provides a clear roadmap for proving the existence of the Gate B certificate itself:

• Path 1 (PLDA): Reduces the problem to proving the 'Parity Large-Deviation Axiom' (PLDA, Assump F.6) — that the density of "pathological" residues decays exponentially ($2^{-\delta L}$).

• Path 2 (Sufficient Conditions): Shows that PLDA (and thus Gate B) can be proven by establishing verifiable, pseudo-randomness properties, such as Walsh small-bias (Thm F.15) or k-wise independence (Thm F.16).

• Path 3 (Fallback): Shows that even a weaker polynomial sparsity ($L^{-A}$) is sufficient for stretched-exponential absorption (Thm F.18).

3. Quantitative Quenched Rates (from v3.6)

The framework (now activated by Gate B) confirms that weakened regularity is sufficient:

• A 'bad-block budget' (good-block frequency $\rho \in (0,1]$) with a sublinear deficit $b(N) = o(N)$ is sufficient for exponential tails (Thm 8.20).

• If gaps between good blocks merely have polynomial tails ($\mathbb{P}(\mathrm{gap} \ge m) \le Cm^{-\beta}$), one still obtains stretched-exponential tails (Thm 8.21).

4. Uniform Dirichlet Gaps (from v3.6)

Under one-step interior minorization and a uniform boundary-flux bound, the Dirichlet conductance obeys $\Phi^{(D)} \gtrsim \phi_0/C_\partial$ and $\lambda_*^{(D)} \ge \frac{1}{2}(\Phi^{(D)})^2$, uniformly in the level $k$ (Prop 5.8).

5. Dobrushin / $W_1$ Contraction (from v3.6)

With uniform minorization, the Dobrushin coefficient $\alpha_k \ge c\,\phi_0$ gives $k$-uniform contraction in total variation (TV) and $W_1$ for the annealed model (Prop 4.11).

6. Lazy-kernel MLSI (from v3.6)

For $K_\varepsilon = (1-\varepsilon)I + \varepsilon K$, a defective MLSI with $\rho_k(\varepsilon) \gtrsim \varepsilon\,\phi_0^2$ suggests a path to annealed KL decay (Prop 4.9).

7. Entropy Calculus (from v3.6)

$L^p$-contractivity, Beurling–Deny, and the inequality $(a-b)(\log a-\log b) \ge 4(\sqrt{a}-\sqrt{b})^2$ yield $-\dot{\mathcal{H}}(t) \ge 2\,\mathcal{E}[\sqrt{f}] \ge 2\lambda_1\mathrm{Var}(\sqrt{f})$.

Abstract (concise)

The Dirichlet form of $\Delta_C^{(\infty)}$ is closed, symmetric, Markov; the annealed semigroup preserves positivity and mass. A one-step Doeblin minorization on $\mathbb{Z}/2^k$ gives a global Poincaré inequality with explicit lower bound $\lambda_1 \ge \lambda_\star = p_e^2/32 > 0$, yielding exponential decay of $L^2$–energy and relative entropy in the annealed model.

For Dirichlet (killed) restrictions outside verified regions, interior minorization plus a flux–conductance criterion imply $k$-uniform positive gaps: $\Phi^{(D)} \gtrsim \phi_0/C_\partial$, hence $\lambda_*^{(D)} \ge \frac{1}{2}(\Phi^{(D)})^2$.

Using the dilation picture and block minorization, we obtain conditional quenched bounds under weakened regularity (e.g., sublinear deficit). Appendix F formalizes this transfer via a finite 'Gate B' residue certificate, and outlines verifiable, sufficient number-theoretic conditions (PLDA, small-bias) for its existence.

Numerics for $\Delta_k = I-K_k$ up to $2^{12}$ at $p_e = \frac{1}{2}$ show a stable $\lambda_1 \approx 0.0107 \pm 0.001$, above the Cheeger lower bound $1/128$.

Key contributions

• Gate B Certificate (App F): Reduces the final number-theoretic step (orbit regularity) to a finite, verifiable certificate $(m,L,S,\eta)$.

• Verifiable Paths (App F): Establishes a hierarchy of sufficient conditions (PLDA, small-bias, poly-sparsity) to rigorously prove the existence of Gate B.

• Reversible kernel & dilation: Self-adjoint $K$ with $K=J^\dagger U_T J$; energy equals parity-coherence loss.

• Annealed spectral gap: $\phi_0=p_e/4$ and $\lambda_\star=\phi_0^2/2$ on finite quotients; global Poincaré on mean-zero states.

• Dirichlet gaps (killed): Interior minorization + boundary-flux control $\Rightarrow$ $k$-uniform $\lambda_*^{(D)} > 0$; explicit flux–conductance bound.

• Quenched transfer (conditional): Weakened regularity (e.g., $b(N)=o(N)$) is sufficient for exponential or stretched-exponential quenched tails.

• Distance contraction: Dobrushin TV contraction and $W_1$ contraction on $\mathbb{Z}/2^k$ with $k$-uniform rates.

• Lazy-kernel MLSI (finite-state): $\rho_k(\varepsilon) \gtrsim \varepsilon\,\phi_0^2$ suggests a route to annealed KL decay.