manabe 2024 restricted amalgamation nim conjecture 1

Hikaru Manabe, An Amalgamation Nim with Restriction, arXiv:2411.14488v1, Section 3, Conjecture 1, 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 game is Definitions 4–5: strictly reduce any one of three natural pile sizes, or merge two piles each at least two, allowing both placements of the sum and zero. Preregistration: #19861.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | Three nonnegative pile sizes | Natural arguments of move, grundy and claim | Zero is included | | Definition 5 sets (1)–(3) | First three alternatives of move | All strict reductions | | Definition 5 sets (4)–(6) | Last three alternatives of move | Both piles at least two; both placements | | Grundy value is mex of option values | grundy, with the existing AbukuEnforceNimberDomination.mex | Private move_decreases and grundy_eq remove the well-founded guard on options | | Both even and odd implications, for every natural n | Two conjuncts of claim, proved by result | Neither implication is dropped | | Termination measure in the binding conventions | size: stone total plus number of nonempty piles | An implementation measure, not an extra game hypothesis |

The four public definitions preserve the binding expressions in #19861. The measure and recursion guard add no hypothesis to the universal result.

The public theorem D5.S0.Certificates.Games.ManabeRestrictedAmalgamationNim.result settles both parts of Conjecture 1 for every position. Its mathematical proof establishes an exact value, not a finite approximation. The imported D5.S0.Certificates.Games.AbukuEnforceNimberDomination.mex supplies the minimum-excluded definition; that owner's game theorem is not on this module's dependency path.

Four-slot escape registration of the sole public theorem result is unfinished: #20798. No Reg registration is delivered. The quantified conjunction of guarded disjunctions needs a lawful enrolled readout, exact source reconstruction, source selection, intervention/sensitivity and observational-dependence evidence. This gap does not weaken the Lean theorem.

Write $a=x/2$, $b=y/2$, $c=z/2$ using natural quotients, and put $h=a\mathbin{\mathrm{xor}}b\mathbin{\mathrm{xor}}c$ and $p=(x\bmod2)\mathbin{\mathrm{xor}}(y\bmod2)\mathbin{\mathrm{xor}}(z\bmod2)$. Let $B$ mean that some pair of $a,b,c$ is positive and bitwise disjoint; let $E$ mean that all three are positive and pairwise bitwise disjoint. Put $\delta=1$ exactly when $B\land\neg E$, and otherwise $\delta=0$. The private exact_formula proves $g(x,y,z)=2h+(p\mathbin{\mathrm{xor}}\delta)$.

The private merger_low_bit shows that a legal merger preserving the high XOR reverses the corrected low bit. exclusion rules out the proposed value at every option; lower_blocks and last_value provide every smaller value. Strong induction on size and mex then give grundy_formula, exact_formula and result.

A position violating either implication of claim would refute the settlement. The stronger formula also predicts every Grundy value under the stated six-option relation. The finite computation below tests these predictions independently; the universal result is kernel-checked Lean.

The source module and Blueprint Scribe have the same owning path under D5/S0/Certificates/Games/ManabeRestrictedAmalgamationNim. result proves the expanded claim. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$.

Finite experiment

Experiment entry at commit ffbf16ab17fc78fc53ff70b7bd8850aade42a8d4, script check.py. Command: python3 /Users/auric/.sshx/7d2cc64fd2186d0cad557293/attempt-1/experiment.py, exit 0. Script SHA-256: 9fe9d0293073584a03f7249591ae254519e12932685709745b6a2d2f5dab30c4. For all 68,921 positions with $0\le x,y,z\le40$, there are zero mismatches for Conjecture 1, the exact formula, and equality of the zero set with Definition 6's P-set. Computed examples are $g(0,2,4)=7$ with XOR $6$, $g(2,2,0)=0$, and $g(2,4,6)=1$ with XOR $0$.

What the settlement shows

• Settlement — proved, kernel-checked: result proves both clauses of Conjecture 1 under the exact binding conventions, at all natural pile sizes.
• Stronger result — proved, kernel-checked: private exact_formula proves $g=2h+(p\mathbin{\mathrm{xor}}\delta)$ universally. Private xor_half identifies $h=(x\mathbin{\mathrm{xor}}y\mathbin{\mathrm{xor}}z)/2$; low_value_lt bounds the corrected low bit by two.
• Mechanism — proved, kernel-checked: merger_low_bit states that every legal merger retaining the high XOR reverses the corrected low bit. merger_exclusion and even_merge_target use that fact.
• Adjacent zero-set characterization — proved in prose from the exact formula: both summands are nonnegative and the low bit is below two, so $g=0$ is equivalent to $h=0$ and $p=\delta$. No additional public theorem asserts this characterization.
• Definition 6 comparison — computed: equality with the paper's complete P-set is checked only on $0\le x,y,z\le40$ by the experiment above. A separate universal Lean equivalence with Definition 6 is open.
• Sharpness — computed: $g(0,2,4)=7\ne6$ and $g(2,4,6)=1\ne0$ show that replacing the adjacent-pair conclusion by exact equality with ordinary XOR fails. These values are in the stated finite run.
• Other source results — literature reading: Theorem 2 states the P-position set independently of Conjecture 1. The settlement supplies the conjectured adjacent-pair Grundy property; the source's full Definition 6 equivalence is not separately proved by this module.
• Open frontier — open: four or more piles and unrestricted amalgamation Nim. The proof's three-pile correction is not asserted for those games.
• Novelty — literature reading; completeness ASSUMED-UNVERIFIED: the source history and the literature screen in #19861 and the search evidence cover the named sources only. No exhaustive publication-priority claim is made.

The arXiv history lists v1. The CGTC V abstract retains a conjecture. Seven later-author papers were keyword-screened, not audited for implicit general theorems: 2501.15786, 2504.18606, 2510.03088, 2601.18715, 2603.10414, 2605.14321 and 2609.05358. Citation-service coverage and unread references leave literature completeness ASSUMED-UNVERIFIED; no zero-citation claim is made. The unfinished escape audit is #20798.