Gaussianity of random Zeckendorf game lengths via mixing

The unordered Zeckendorf game starts from N copies of F_1; at each step it combines consecutive Fibonacci terms or splits a repeated term until the Zeckendorf decomposition is reached. Let L_N be the number of moves.

Conjecture 1.7, quoted from arXiv:2210.11038v1:

“In the limit N\to\infty, the distribution of the number of moves in a random Zeckendorf game on input N converges to a Gaussian, with expectation and variance approximately 0.215N.”

The paper studies both the uniform measure mu_N on complete games and the path measure P_N obtained by choosing uniformly among legal moves at each state. Because "approximately" and the measure are not fixed inside the conjecture sentence, the first declaration-ready target should be the paper's exact mixing Question 6.5 for the locally uniform path measure.

Let d_N be the longest game length and let X_{N,i} be the Bernoulli indicator that move i is a split, padded by zero after termination. With alpha_N(k) the strong-mixing coefficient between the sigma-algebras before m and after m+k, the paper asks whether there exist delta, C > 0 such that for all N, the sum over k >= 0 of (k+1)^2 alpha_N(k)^(delta/(4+delta)) is less than C.

It states that an affirmative answer immediately yields the Gaussianity conjecture via the cited mixing central limit theorem. Question 6.6 proposes the stronger pointwise bound |P_N(A ∩ B) - P_N(A) P_N(B)| <= C/k^(2+epsilon).

The paper states the difficulty:

“The scope of this result is admittedly restricted...”

“...it is unclear how to extend this to the entirety of \Omega_N.”

The paper obtains Gaussianity only on certain natural partition components. Its full-space obstacle is dependence between separated move indicators; the open section explicitly turns that into Questions 6.5 and 6.6.

• Game states are finite raw Fibonacci multiplicities and game moves are value-preserving local rewrites ending at a unique Zeckendorf normal form.
• The weighted-probability family already supports exact finite product probabilities, event capture counts, inclusion-exclusion, and moments. It is suitable for finite Omega_N certificates once paths are encoded.
• The missing theoretical bridge is a locality/regeneration theorem: a sufficiently wide canonical digit barrier should make early and late split indicators nearly independent. Frozen carry normalization is a plausible source of such a barrier.

• The frozen rewrite relation is not the paper's game relation and carries no random path measure.
• No enumeration of complete paths, local branching weights, stopping time, sigma-algebra, mixing coefficient, or weak convergence API exists.
• Finite exact moment identities do not imply uniform-in-N mixing or a central limit theorem.
• Conjecture 1.7's intended measure and the mathematical meaning of "approximately 0.215N" require an owner choice before formalization.

1. Define the finite game DAG and both measures exactly. For P_N, use the recursion G_S(z) = z * average over successors of G_T(z); for mu_N, weight successors by their number of terminal suffix paths.
2. Prove the invariant that the number of combine moves is fixed, so game length differs from a constant by the split count, matching the paper's triangular array.
3. Search for regenerative barriers in the bin representation: an interval of empty or canonical bins may prevent carry/split influence from crossing except through a rare sequence of moves.
4. Bound the probability that influence crosses a gap of width k; a polynomial exponent greater than 2 would establish Question 6.6, while exponential decay would be stronger.
5. Use existing finite probability and moment declarations for exact small-N checks; import a mixing central limit theorem only after the uniform bound is formal.

The Gaussian limit and the existential mixing constants have no single finite falsifier. A claimed explicit bound with fixed epsilon, C, N0 is falsified by exact N, m, k, A, B violating it.

A structural regeneration lemma is finitely falsified by two legal histories agreeing on the proposed barrier but producing different reachability or conditional distributions beyond it. Such history pairs should be emitted as minimal counterexamples.

For every N <= 35, use dynamic programming on the exact game DAG to compute under both mu_N and P_N:

1. the full rational probability generating function of L_N;
2. exact mean, variance, skewness, and fourth standardized moment;
3. distributions of each split indicator and selected two-block joint events;
4. lower bounds on alpha_N(k) obtained by exhaustive event search for small sigma-algebras;
5. normalized histograms and Kolmogorov-Smirnov distance, explicitly separated by measure.

The main Evidence output should be whether a candidate digit barrier predicts the observed dependence decay. Gaussian-looking histograms alone are insufficient.

window. Exact finite distributions and candidate regeneration lemmas are reachable; a uniform mixing theorem across all game paths is a genuine new probabilistic layer.

• Conjecture 1.7 is intended for P_N, for mu_N, or for both; the sentence itself is ambiguous.
• The coefficient 0.215 has a limiting exact value rather than being only a finite-simulation fit.
• A canonical-bin barrier yields decay strong enough for Question 6.5 or 6.6.
• Whether the conjecture was resolved after arXiv v1 is unverified; novelty of the regeneration route is unassessed.