Maximum order complexity along polynomial Zeckendorf subsequences

Let s_Z(n) be the sum of the digits in the Zeckendorf representation, and set S_Z = (s_Z(n) mod 2) for n >= 0 and S_{Z,P} = (s_Z(P(n)) mod 2) for n >= 0. For a binary sequence S, M(S,N) is the least feedback-register length needed to generate its first N values, as defined in the paper.

Conjecture 3, quoted from arXiv:2106.09959v1:

“The sequence \mathcal S_Z along polynomial subsequences, denoted by \mathcal S_{Z,P} for a polynomial P of degree d\geq2, verifies M(\mathcal S_{Z,P},N)\asymp N^{1/(2d)}, i.e. there are c,C>0 such as for all N large enough we have cN^{1/(2d)}\leq M(\mathcal S_{Z,P},N)\leq CN^{1/(2d)}.”

The paper has already proved the matching lower bound for monic P in Z[X] of degree d >= 2 with P(N_0) contained in N_0; the new mathematical content is the upper bound. A safe first formal target is the concrete case P(X) = X^2:

∃ C > 0, ∃ N0, ∀ N ≥ N0,
  maximumOrderComplexity (fun n => zeckendorfDigitSum (n^2) % 2) N ≤ C * N^(1/4)

The real-valued bound must be encoded with explicit ceilings and floors.

The paper states the difficulty:

“The maximum order complexity of \mathcal S_\varphi is algorithmically more difficult to handle.”

“With our program and our machine, it is not possible to compute the maximum order complexity of a sequence any further than 10^9 terms.”

The authors say the square plot supports the conjecture but the cube plot does not clarify it; proving the conjecture would show their lower bound is sharp.

• WDigits already gives the exact canonical digit set, so s_Z is a finite cardinality/sum on a frozen object rather than a new numeral system.
• Carry and Normalize give a local, terminating way to study how polynomial increments change digits; any upper bound must control how far those carries propagate.
• The frozen word-complexity layer supplies definitions and techniques for repeated factors, but maximum order complexity is a different invariant. The needed bridge is the collision criterion: every repeated length-M block in the first N positions must have the same successor bit.
• The exponent 1/(2d) suggests a two-sided noninterference window for polynomial values; this matches the paper's explanation that Lucas expansions around a center affect both sides.

• No zeckendorfDigitSum mod 2 sequence is declared.
• No maximum-order-complexity, feedback-register, or DAWG API exists.
• Frozen factor complexity does not imply the required successor determinism.
• There is no quantitative carry-propagation theorem for P(n+h) - P(n) at polynomial scale.

1. Define s_Z directly from wdigits; prove compatibility with the raw-digit normalizer.
2. Replace the existential feedback-polynomial definition by the equivalent repeated-block criterion used by maximum order complexity: two equal length-M blocks before N must have equal next bit.
3. For P = X^2, decompose (n+h)^2 - n^2 = 2nh + h^2 into separated Fibonacci/Lucas blocks. Use normalization to isolate a bounded carry zone around each block.
4. Show that an M-bit history with M = O(N^(1/4)) determines the relevant boundary state of the carry automaton, hence determines the next parity bit.
5. Generalize the separation lemma to degree d; the already-proved paper lower bound can remain an external target until formalized.

The asymptotic claim has no single finite falsifier because c, C, and N0 are existential. A proposed effective upper bound M <= C ceil(N^(1/(2d))) is falsified by an exact N above its stated threshold where the inequality fails.

The key successor-determinism lemma is sharply falsifiable: find i < j with identical length-M blocks of S_{Z,P} but different following bits. Every computational certificate should emit this pair, not only a complexity value.

Build two independent exact implementations of M(S,N): naive repeated-block checking for small N, and a suffix-array or DAWG method for larger N. For P = X^2 and P = X^3:

1. compute s_Z(P(n)) mod 2 from exact greedy WDigits;
2. cross-check the two complexity implementations through at least N = 10^6;
3. sample at Fibonacci-scale N and record M/N^(1/(2d)), local log-log slopes, and witness pairs causing every step increase;
4. separately measure the maximum changed digit index under P(n) -> P(n+1) to test the proposed carry-window lemma.

The result must not be reported as proof of an asymptotic exponent.

window. The lower bound and numerical shape are known, and the repository can formalize exact finite complexity and carry behavior; the uniform upper bound needs a new quantitative noninterference theorem.

• Conjecture 3 inherits the earlier monic/integer/nonnegative scope for P; the quoted conjecture itself abbreviates that context.
• The repeated-block characterization will be the most useful formal definition for the chosen edge cases of M(S,N).
• Carry propagation admits a uniform state bound of the conjectured scale.
• Whether the conjecture was resolved after arXiv v1 is unverified; novelty of any proposed carry lemma is unassessed.