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The AND-Gate Does Not Gate: A Soundness Gap in the Win-Condition Construction

IRF Technical Note No. 15 — an erratum to Section 6 of Raumschach and Computational Complexity

International Raumschach Federation  ·  IRF Technical Notes, No. 15  ·  2026

Section 6 of the consolidated paper (Raumschach and Computational Complexity) describes an AND-gate/win-condition mechanism in which a Rook V, sitting on a checking piece’s line, retreats if and only if every one of several Boolean input tokens along its rank line has branched off. Continued work toward full assembly of a nontrivial formula tested this mechanism, for the first time, against an actual legal-move generator with more than one input token in genuine sequence — a configuration no prior test, including Section 6’s own verification, appears to have exercised. The result is a genuine soundness gap: the mechanism delivers checkmate as soon as the single token nearest V branches off, regardless of the state of any token further down the line, which is the logical opposite of the claimed AND. This note documents the finding, traces its cause to an assumption about the checking piece’s ray that does not hold under partial retreat, identifies which of this project’s other constructions are and are not affected, and presents a verified resolution: V’s own gate is sound for exactly one adjacent token, and arbitrary Boolean formulas can be composed correctly by computing all sub-formula logic off V’s square entirely and wiring the result into that single token’s presence or absence, at the cost of the single-token “triple duty” composition this project had been pursuing.

I. Summary

The consolidated paper (Raumschach and Computational Complexity, Section 6) describes the AND-gate/win-condition mechanism as follows: a Rook V sits at the point where its own rank line intersects a triagonal line running to the King, with Boolean input tokens further along that same rank line, each blocking V’s retreat when False and branching off the line when True; V’s retreat is legal “if and only if every token has branched off.” Once legal, the paper states, V’s retreat opens the triagonal line for a checking Unicorn, which delivers check and closes the King’s last flight square simultaneously.

Continued computational work, building toward full assembly of a nontrivial formula, checked this mechanism against an actual legal-move generator for the first time with more than one input token in genuine sequence — a case the original verification (Note No. 9 and the consolidated paper’s own Section 6 tests) does not appear to have exercised, since all prior tests checked only whether V could reach one specific, fully-retreated escape square under a single pre-chosen assignment. That check found a genuine soundness gap: the mechanism, as specified, does not implement AND for more than one input token. It delivers checkmate as soon as any single token nearest to V has branched off, regardless of the state of any token further down the line. This is the opposite of the claimed behavior, and closer to OR than AND.

II. The Underlying Cause

The gate’s soundness depends on an assumption that turns out not to hold: that V must retreat all the way to its designated escape square before the checking Unicorn’s ray is uncovered. In fact, the checking Unicorn’s ray passes through V’s home square specifically — the one point where V’s rank line crosses the King’s triagonal line — not through the escape square. The instant V leaves that one home square, for any reason and to any destination, the Unicorn’s ray is clear and discovered check is delivered. This was confirmed directly: moving V a single square away from its home position, with a checking Unicorn placed anywhere further out on the same triagonal line with a clear path, produces immediate checkmate — before the Unicorn itself has made any move, and regardless of how far V actually got.

This collapses the intended multi-token AND. Consider two input tokens, T1 nearest to V and T2 next, both immediately adjacent with no gap between them (the strongest possible placement, tested directly rather than assumed). If x1 is True, T1 branches off and vacates the square adjacent to V. V now has a legal move — one square, onto T1’s old position — regardless of whether x2 is True or False, because a Rook’s legality to make some move in a direction depends only on whether the nearest obstacle is present, not on any obstacle further along the line. That one-square move is enough: it vacates V’s home square, and checkmate follows immediately, whether or not T2 (representing x2) has branched off at all.

This was verified directly and precisely: with x1 = True and x2 = False, V has exactly one legal move (onto the square T1 vacated); making that move produces a position where the King is in check with zero legal replies — checkmate — even though the conjunction x1 ∧ x2 is False. A control case (x1 = False, x2 = True) confirms the nearest token alone is sufficient to block V completely, regardless of tokens further down: V has zero legal moves. Between the two cases, the pattern is unambiguous: the outcome tracks only the state of the token nearest to V. Every other token in the chain is logically inert.

III. Why Earlier Verification Did Not Catch This

Every test since Section 6’s own verification — through this project’s subsequent composition, tempo-matching, and game-tree work — checked V’s legality by asking whether V could reach one specific, fully-retreated destination square, with input tokens placed (or omitted) directly according to a pre-chosen Boolean assignment. That check happens to give the right-looking answer whenever the geometry never gives V a partial, non-terminal legal move to take instead — which was true of every configuration tested, generally because tokens were spaced with gaps, or because only a single token was in play, or because composition tests wired truth values in directly rather than building a genuine multi-token chain with adjacent placement. The gap only surfaces when (a) more than one token sits in true sequence on V’s line, (b) the nearest one is capable of branching off independently of the others, and (c) the win-condition check asks the question the mechanism actually answers — “does V have any legal move at all” — rather than the proxy question — “can V reach this one specific square.” No test prior to this one asked (c) in the presence of (a) and (b) simultaneously.

IV. What Is and Is Not Affected

Not affected: the corridor, level-crossing via, and OR-gate constructions (Sections 5 and 7), which do not depend on V’s home-square/triagonal-line trick. The general finding that Raumschach’s third dimension removes flat chess’s wire-crossing obstacle (Section 3) is a structural, board-geometric argument independent of this gadget and is also unaffected.

Affected: the AND-gate/win-condition construction of Section 6, specifically its claim to correctly gate an arbitrary number of Boolean inputs. The construction is sound for exactly one input token (there is nothing for a second token to be inert relative to). It is not sound, as specified, for two or more.

Also affected, pending re-examination: the three-input scaling claimed in Section 6 (“verified to scale to three inputs with no change”) and the OR-gate’s own soundness under composition, since the OR-gate’s escape mechanism was, in this project’s later composition work, wired directly into the same rank line as an AND-gate input using the identical vacate-the-line logic now shown to be unsound for sequential multi-token gating. Any composed structure built on this project’s test_assembly_composite.py, test_double_composite.py, or test_single_token_composition.py inherits this gap for the same reason: each wired a computed Boolean value into a single proxy token’s presence or absence on V’s line, which is exactly the one-token-at-a-time case this note shows is sound — but none of them tested two or more independently movable tokens on the same line in true sequence, which is where the gap actually lives.

V. Resolution: A Sound Construction, With a Real Cost

Update, added after further work: a sound resolution exists, though it is more limited than the original claim. Neither of the two directions named above (cascaded pairwise gates, decoupling the checking ray from V’s far square) was needed; the fix is simpler and comes at the cost of scope. It was confirmed directly against the generator:

The cost: this is a genuine repair of soundness, not of the original ambition. It is no longer true that a single physical piece can serve double or triple duty — as a sub-formula’s own mobile token, as a rank-line blocker, and as a piece sitting on the checking ray, all at once — the way Section 6 and this project’s own test_single_token_composition.py and test_double_composite.py assumed. Composition now requires an explicit, separate wiring step between a verified sub-formula and the one token that actually matters for mate. Whether a design exists that recovers single-token, no-wiring composition without reintroducing this note’s original flaw remains open; what is no longer open is whether some sound construction exists for arbitrary formulas of this shape — it does, and has been checked.

VI. Conclusion

The AND-gate/win-condition mechanism in Section 6 of the consolidated paper, as originally specified, does not gate more than one Boolean input correctly: mate is delivered as soon as the token nearest V branches off, independent of every other token in the chain, which is closer to OR than to the claimed AND. This was verified directly against the legal-move generator with tokens placed immediately adjacent to V — the strongest arrangement available — ruling out placement as the cause. The gap was not visible in any test prior to this one because every earlier test, including this project’s own composition and game-tree work, checked a proxy condition (can V reach one specific square) that coincides with the real win condition (does V have any legal move at all) only in the single-input case. A sound repair was found and verified for arbitrary formulas built from OR and AND: keep V’s own gate to exactly one adjacent token, and compute all sub-formula logic off V’s square entirely, wiring in the result. This restores correctness at the cost of the single-token, triple-duty composition this project had been pursuing; recovering that stronger property, if possible at all, remains open.

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