Eventual stability of Zhang's l-complexity

Pengcheng Zhang, The 2-complexity of even positive integers, arXiv:2411.19364v1, Integers 24 (2024), A109, Question 3, page 10:

Let l > 1 and define A_l := {n ∈ lZ⁺ : ‖l^m · n‖_l = m + ‖n‖_l for all m ≥ 0}.

Can one say anything about the set A_l? For example, is it true that for every n ∈ lZ⁺, there exists m₀ ≥ 0 such that l^{m₀} · n ∈ A_l?

The every-orbit assertion is proved. The broader request to describe $A_l$ and to bound the first stable index remains open here. Preregistration: #19862.

Statement echo

| Source clause | Formal counterpart | | --- | --- | | $l>1$ | Natural parameter l with 1 < l. | | $n\in l\mathbb Z^+$ | Natural parameter n with 0 < n and l ∣ n. | | Minimal number of copies of $l$, with addition, multiplication and parentheses | Public value evaluates the existing KumarPengTranModularComplexity.Expr; public complexity is the natural infimum of attainable Expr.leaves. | | $A_l$ | Public A states positivity, divisibility and the source equality for every natural exponent, including zero. | | Some $m_0\geq0$ with $l^{m_0}n\in A_l$ | claim quantifies over natural m₀; public result : claim proves it. |

The natural evaluation and minimum are the literal conventions of #19862. The private mem_A_iff identifies them with the frozen evaluator and minimum at modulus zero through injectivity of the natural-to-integer cast. The public claim retains the source's natural definitions. No additional premise or finite cutoff is present.

Every orbit of positive multiples under multiplication by $l$ meets $A_l$. The result gives permanent linear growth from a suitable translate, rather than only an observed constant tail in a finite computation.

The literature screen recorded in #19862 covers the arXiv version, the published question, the ordinary-complexity stabilization method and later abstracts. The search found no dominating settlement of this every-orbit assertion in that scope. The author's thesis and full texts of four later papers are outside the inspected scope. Completeness is ASSUMED-UNVERIFIED.

The generic antitone-chain theorem is already in pinned Mathlib and is used directly. The source-specific obligations are existence of expressions, the exponential leaf bound and extension by one leaf.

For $l>1$, structural induction bounds an expression with $s$ leaves between $l$ and $l^s$. At an addition node both values are at least two, so their sum is at most their product. A positive multiple of $l$ has an expression by repeated addition, and the minimum leaf count is attained. Multiplication by one additional leaf gives $|lu|_l\leq|u|_l+1$.

For a positive multiple $n$, $n\geq l$. Thus $l^{m+1}\leq l^mn\leq l^{|l^mn|_l}$ and $|l^mn|_l\geq m+1$. Consequently the natural defect $d_m=|l^mn|_l-m$ is positive and nonincreasing. WellFoundedLT.antitone_chain_condition gives a constant tail. Its equality and the lower bound allow cancellation of natural subtraction, yielding $|l^r(l^{m_0}n)|_l=r+|l^{m_0}n|_l$ for every $r\geq0$.

A natural $l>1$ and positive multiple $n$ such that every translate $l^{m_0}n$ fails the defining growth equality would contradict result. Finite agreement of computed defects alone does not rule out this falsifier.

Kernel-checked declaration: D5.S3.Arith.Congruence.ZhangLComplexityEventualStability.result, in D5/S3/Arith/Congruence/ZhangLComplexityEventualStability.lean. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

Finite experiment

Published experiment directory. The script is check.py at that revision, SHA-256 9fe54cee5d762ca5b2a00f3e3af293281cc1a4815cb399deb2f6c2031d3d21f8. The verified bytes were executed with python3 /Users/auric/.sshx/bce7d3f527bcf4aa34c6834e/attempt-1/experiment.py, exit 0. The script computes $|54|_2=8$ and $|216|_2=9$; checks Theorem 2(1) for $0\leq m\leq5$, $1\leq r\leq4$ with $2^m6^r\leq2^{15}$; and checks Proposition 1's lower bound using floating logarithms with tolerance $10^{-9}$ on positive multiples up to $2^{15}$ for $l=2$ and $3^9$ for $l=3$. For $n<400$ it tests 199 orbits for $l=2$ and 133 for $l=3$, restricted to those upper bounds: zero increases in the defect and equality of the final two computed values in all tested orbits. These are computed finite checks, not proofs of the corresponding uniform extensions or of an explicit first-stability bound.

Escape audit

The sole public theorem audit target is result. Its four-slot escape audit is unfinished: issue #21973. The existential template does not reconstruct the full universal source telescope; a complete dependent-family reconstruction, bridge, variation, sensitivity and observational-dependence evidence is not supplied. There is no retained Reg registration and no declared_validated claim.

| Item | Status and evidence | | --- | --- | | Every-orbit assertion of Question 3 | Proved, kernel-checked public result. | | Dominating theorem search | Literature reading: no l-complexity settlement found in the scope recorded in #19862; completeness ASSUMED-UNVERIFIED. The generic chain theorem is reused. | | Discrete-defect mechanism | Proved in prose in Route; source-specific steps are the module's private value_bounds, complexity_mul_le, defect_antitone and defect_pos, consumed by result. This applies the established method rather than claiming a new general method. | | Every orbit meets $A_l$ | Proved, kernel-checked result. | | Each orbit has an eventual stable defect | Proved in prose: apply the imported chain condition to the nonincreasing natural defect as in Route. No separate public theorem is asserted. | | Source values, Theorem 2(1) instances, Proposition 1 instances and finite defect checks | Computed, with precisely the scope, command, exit and SHA-256 in Finite experiment. The exact values and Theorem 2(1) instances have only the computed scope stated above; Route proves the general lower-bound argument and result proves eventual stability. | | Fuller description of $A_l$ | Open; no classification is delivered. | | Bound on the first stable index $m_0$ | Open; existence supplies no explicit bound. | | Literature visibility | Literature reading: repo-prior-exposed; supplied search evidence was read. No blinded search or multi-seat consensus is claimed. |

What the settlement shows

The decisive mechanism is a positive integer defect with a nonincreasing orbit, proved in prose in Route and used in the kernel-checked result. The conclusion ranges over every natural $l>1$ and every positive multiple $n$, with no upper bound on either parameter or on the exponent. Positivity and $l>1$ are the proved scope; relaxation and sharpness outside that scope are open here.

The source compares this question with ordinary integer-complexity stabilization under multiplication by three. The proof uses the established discrete-defect method with this source's expression bound. The finite checks reproduce the listed source examples and instances; they do not supply new uniform versions of its other results. No source result is claimed to depend on this answer without a verified dependency. The classification of $A_l$ and effective control of $m_0$ are further candidates.

Literature completeness and first-publication priority are unverified. The source and literature checks are attributed to the preregistration's orchestrator and search seat. The four-slot escape audit remains unfinished under #21973. No model-diversity or exhaustive theorem-absence claim is made.