Metadata
bibkey: manabe2024restrictedamalgamation
authors: Hikaru Manabe
year: 2024
title: An Amalgamation Nim with Restriction
doi: 10.48550/arXiv.2411.14488
url: https://arxiv.org/abs/2411.14488v1
claim: Conjecture 1 relates both parities of the Grundy value to the same adjacent pair of XOR values in restricted three-pile amalgamation Nim.
strata_touched:
- D5/S0/Certificates/Games/ManabeRestrictedAmalgamationNim
license: citation-only
triage: anchor
Verified locator
DOI: 10.48550/arXiv.2411.14488
Source: https://arxiv.org/abs/2411.14488v1
Game and options
Definition 4, page 2:
Two players take turns to do one of the following (i) and (ii). (i) A player chooses a pile and removes any number of stones from the pile. (ii) A player can replace two piles with a single pile containing the total number of stones in the original two piles when the numbers of stones in two piles are equal to or more than 2.
Definition 5, pages 2–3:
For x, y, z ∈ Z≥0, we define
The six sets printed as (1)–(6) are the three strict single-pile reductions and the three possible pair mergers. Each merger includes both placements of the sum and zero. The equality signs joining these sets denote their union. The formal move definition includes exactly those six sets, with zero allowed as a pile size. The Grundy number is the mex of the option values. The recursion measure is the stone total plus the number of nonempty piles; every option has smaller measure, making the recursion guard inactive.
Conjecture 1
Section 3, page 14:
For n ∈ Z≥0, we have the following (i) and (ii). (i) If the Grundy number of a position (x, y, z) is 2n, then x ⊕ y ⊕ z = 2n or x ⊕ y ⊕ z = 2n + 1. (ii) If the Grundy number of a position (x, y, z) is 2n + 1, then x ⊕ y ⊕ z = 2n or x ⊕ y ⊕ z = 2n + 1.
The formal claim preserves both implications for every natural pile size and every natural n. Nat.xor denotes bitwise XOR.
Scope and literature
The exact formula derived in the repository is 2h + (p XOR δ): h is the XOR of the half-pile quotients, p the XOR of their remainders, and δ indicates the existence of a positive disjoint quotient pair without all three quotients being positive and pairwise disjoint. This implies Conjecture 1. The entire Definition 6 P-set equivalence has only numerical evidence here. Four or more piles and unrestricted amalgamation Nim remain open.
The source and literature screen are recorded at https://github.com/the-omega-institute/trureturing/issues/19861. The arXiv history lists v1; the CGTC V abstract still states a conjecture. Later Manabe articles were keyword-screened, without a systematic audit of implicit general theorems. Citation-service coverage and unread references leave completeness ASSUMED-UNVERIFIED. No publication-priority claim is made.
The independent finite experiment is in the-omega-institute/trureturing-experiments, commit ffbf16ab, docs/reports/manabe-2024-restricted-amalgamation-nim-conjecture-1/check.py, SHA-256 9fe9d0293073584a03f7249591ae254519e12932685709745b6a2d2f5dab30c4. The recorded run exits 0 on 68,921 positions with each pile at most 40; it reports no mismatch for the conjecture, exact formula or Theorem 2. The output SHA-256 is ed2dc754154a5a61b65044dab6e538b5856d79833e34d863538b2cf5dfce8d12.