trureturing — Discovering truth

bibkey: larssonmanabe2026additive
authors: Urban Larsson and Hikaru Manabe
year: 2026
title: Additive Subtraction Games
doi: 10.48550/arXiv.2603.10414
url: https://arxiv.org/html/2603.10414v2#Thmthm20
claim: Question 20(iii) separately asks whether the collision density of an arbitrary bracket expression is always rational.
strata_touched:
  - D5/S3/Arith/Density/RationalBracketCollisionDensity
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2603.10414

Primary version: https://arxiv.org/html/2603.10414v2#Thmthm20

Section 1 defines bracket expressions. Section 7, Question 20(iii), asks the separate rationality question. The primary v2 HTML has 462659 bytes and SHA-256 ba317318db8f71382c62bdea468c538f9abfe84ab3c84e31340af1d363b21cd3.

Source statements

Section 1 defines a bracket expression on finitely many terms as $w_n={\sum }_{i=1}^t{c}_i\lfloor n{\alpha }_i\rfloor$, with integer coefficients and positive rational moduli. Sequence indices include zero.

Question 20 defines $W=\{w_n:n\ge 0\}$ and, for a positive integer $k$, $C(k)=\{x\in W:x+k\in W\}$. Item (iii) asks, for an arbitrary bracket expression, about $\rho (k)={\lim }_{N\to \infty }|C(k)\cap [0,N]|/N$ and explicitly asks: “Is it always rational?”

Mathematical answer and scope

The repository-derived theorem proves that this full limit exists and is a rational real number for every finite number of terms, every integer coefficient vector, every positive rational modulus vector, and every positive integer shift. The count is over distinct nonnegative image values, including both endpoints, with denominator N. Negative and zero drift and the empty expression remain included.

Clearing the finitely many denominators gives an affine recurrence with a positive index period and integer drift. Its actual image has an eventually periodic nonnegative tail. Collision membership inherits this period. The existing eventual-cycle average theorem then gives a rational mean of a finite indicator cycle; the inclusive endpoint and denominator are normalized to the stated limit. The source supplies the question and definitions, not this proof.

The other characterisation, closed-form and game-theoretic requests in Question 20 are outside this result. No extension to irrational moduli is asserted.

Literature qualification

The bounded source and literature screen is specified in https://github.com/the-omega-institute/trureturing/issues/23511. The v1 and v2 question and rational-modulus definition were read. Two observed citing preprints, arXiv:2605.14321 and arXiv:2609.05358, were screened for an answer, without a matching answer located. MathDB’s first 20 displayed results for the source title, authors, source ID, and collision/bracket/density queries did not identify an exact target record. These readings were supplied by the source qualification; their full searches have not been independently repeated here.

Citation coverage is incomplete: the source’s OpenAlex count omitted two observed citations, and further citation-service requests returned HTTP 429. General web and arXiv discovery also had documented capability gaps. Exhaustive absence of a prior answer and global priority remain ASSUMED-UNVERIFIED. A matching published answer excludes new-resolution credit. No global novelty or priority claim is made.