trureturing — Discovering truth

slug: kasel-2026-displacement-ladder
bibkey: kasel2026erdos197
doi: 10.48550/arXiv.2609.02939
url: https://arxiv.org/abs/2609.02939v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/Permutation/KaselDisplacementLadder.result

Problem

Kasel, arXiv:2609.02939, Appendix B.4, states: “The conjectured growth L(m) = m − 2”. The exact target was preregistered in #14885.

Let $b(v)=\lceil {\log }_{2}v\rceil$ for $v\ge 2$, implemented by the total natural-number function Nat.clog 2 v. Set

$$S_A(m)=\{v\in [1,4^m]:b(v)\ge 2,b(v)\text{ even}\},D=\{3,4,9,10,11,12,13,14,15,16\}.$$

A scheme consists of natural-valued stages $s$ and fibre positions $r$. It orders values by stage, then by fibre position. Validity requires injectivity within each stage and excludes both increasing and decreasing three-term arithmetic progressions in that concatenation. Normalization requires $s(v)\ge \lfloor b(v)/2\rfloor$ on $S_A(m)$. Write $\delta (v)=s(v)-\lfloor b(v)/2\rfloor$ and let $L(m)$ be the minimum, over valid normalized schemes, of ${\max }_{v\in D}\delta (v)$. The proved statement is $L(m)=m-2$ for every natural $m\ge 2$.

Motivation

The displacement ladder measures the finite-horizon structural obstruction behind Erdős Problem 197. Its exact growth quantifies the obstruction already supplied by the repository’s frozen Erdos-Graham order-gadget theorem.

Gap

The literature readings recorded for preregistration found only arXiv v1; the author’s TeX still labels the growth conjectural. Searches on Kasel, displacement ladder and L(m) found no resolution. The formal-conjectures file ErdosProblems/197.lean states only the parent problem. The repository’s earlier order-gadget theorem supplies the lower-bound obstruction. The remaining upper construction and its matching lower-bound transfer are now proved in Lean. These readings delimit the searched material; they do not establish an exhaustive originality claim.

Route

For the upper bound, put 3 and 4 at stage one and all other values at stage $m$. Order evens before odds, using reversal of the lowest $2m+1$ bits after XOR with target 4 for evens and target 3 for odds. The bounded numeric rank is injective and puts each target first. A recursive parity argument excludes a middle term between its endpoints in rank order. The dyadic gap excludes an AP starting at 3 with a later even middle term. These facts prove validity, while block bounds prove normalization and distinguished displacement at most $m-2$.

For the lower bound, $m=2$ follows from normalization at 3. For $m\ge 3$, assume both 15 and 16 have stage below $m$, and set $M=2\cdot 4^{m-1}$. Every value in $(M,2M]$ lies in block $2m$ and has stage at least $m$. Counting predecessors gives a rank preserving and reflecting the valid concatenation order. Its AP exclusion and forced guard comparisons form the forbidden order gadget at $M\ge 32$. Hence 15 or 16 has stage at least $m$, giving distinguished displacement at least $m-2$. Combining the two bounds proves the exact claim.

Falsifier

A normalized valid scheme at some horizon $4^m$, $m\ge 2$, with all distinguished displacements below $m-2$ would contradict the lower bound. Failure of existence of a normalized valid scheme with all distinguished displacements at most $m-2$ would contradict the upper bound. For an integer cap $w\ge 0$, the equivalent first-failure exponent is $m=w+3$.

Evidence

KaselDisplacementLadderDefs fixes the definitions and dyadic membership lemmas. KaselDisplacementLadderLower.lower_bound proves the universal distinguished lower bound using the frozen ErdosGrahamOrderGadget.result. KaselDisplacementLadderUpper.upper_bound constructs an attaining normalized valid scheme. KaselDisplacementLadder.result : claim combines both bounds. These declarations lie under D5.S3.Combinatorics.Permutation; their corresponding Blueprint Scribes describe the definitions and proofs, and the result node declares a Proved resolution of this problem slug.

Triage

• [proved] For integer $w\ge 0$ and natural $m\ge 2$, a normalized valid scheme at horizon $4^m$ with every distinguished $\delta \le w$ exists if and only if $m\le w+2$. This follows from the matching bounds.
• [proved] $L(m)\to \infty$, which re-derives Theorem 44’s conclusion for $S_A$, already known through the order gadget.
• [open] Erdős #197 itself.

ASSUMED-UNVERIFIED

The literature readings above are carried from preregistration; no fresh external search was performed for this delivery because network access is unavailable. The first-failure equivalence and divergence are consequences of the exact proved bounds, rather than separate named Lean declarations. The Lean result resolves the displacement-ladder claim with the definitions stated above; it does not resolve the parent Erdős problem.