slug: takemura-fixed-point-apd-conjecture-2
bibkey: takemura2025apd
doi: 10.48550/arXiv.2512.18169
url: https://arxiv.org/abs/2512.18169v1
triage: theorem
motivation_gids:
  - D5/S3/ArithSums/TakemuraFixedPointAlternatingPowerDifference.resultValue

Takemura’s fixed-point first appearance value

Problem

For n >= 2, let fix(sigma) be the number of fixed points of a permutation sigma of Fin n, and define

APD_m(fix) = sum_{sigma in S_n} sign(sigma) * fix(sigma)^m.

Takemura’s Conjecture 2 in arXiv:2512.18169v1 asks to prove the value at degree n - 1:

theorem resultValue (n : Nat) (hn : 2 <= n) :
    apd n fix (n - 1) = (n.factorial : Int)

The source reports numerical verification through n = 10 and leaves the statement unproved.

Motivation

The conjecture gives the exact first nonzero alternating power difference, not only its degree. Its factorial value reflects the count of injections from an (n - 1)-element index type into an n-element set.

Gap

The source explicitly labels the formula Conjecture 2. The bounded literature check registered in issue #8837 found only arXiv:2512.18169v1 and a later paper by the same author that restates the APD definition but does not prove this conjecture. The checked Semantic Scholar citation list was empty. Repository, pinned Mathlib, and external Lean ecosystem searches found no theorem with the complete APD value statement. This is a bounded search result, not an exhaustive worldwide absence claim.

Route

Expand fix(sigma)^(n-1) as a sum over (n - 1)-tuples and exchange the tuple and permutation sums. For a tuple image S, reindex permutations fixing S pointwise by permutations of its complement. Their signs sum to zero when the complement has at least two elements and to one otherwise. At degree n - 1, the surviving tuples are exactly the injections from Fin (n - 1) to Fin n. Their descending-factorial count simplifies to n!.

Falsifier

A natural n >= 2 for which the signed sum over all permutations differs from n! would refute the statement. A computation using unsigned permutations, a subset rather than all of S_n, or a different fixed-point convention addresses a different claim.

Evidence

The source statement and definitions are recorded in Library/ArithSums/takemura2025apd.md. The frozen theorem is D5/S3/ArithSums/TakemuraFixedPointAlternatingPowerDifference.resultValue, with declaration statement ID sha256:53c7bb8dc65bcf39d1fd31f07237f879f761da939051c515e3b00ed67ce72ac6. Its axiom closure is exactly propext, Classical.choice, and Quot.sound. The module freeze has no prerequisite frozen project nodes; the proof uses only pinned Mathlib facts.

Triage

First tier: an explicitly numbered conjecture in a 2025 arXiv paper, preregistered in issue #8837. Resolution: proved.

After local helper expansion, the proof consists of pinned-library instantiation, reindexing, finite-sum normalization, injection counting, and factorial normalization. It therefore has no escape witness. The external conjecture is admitted under open-problem-resolution, not escape-witness. The uniform symbolic theorem is not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies.

ASSUMED-UNVERIFIED

Google Scholar and MathSciNet were not searched. The literature finding does not claim exhaustive global novelty, priority, or absence of an independent proof. Bounded numerical checks through n = 10 do not prove the universal statement.