Research: repair the mutual-zero neighborhood classification

research/fstar-mutual-zero-neighborhood/PROOF.md

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+<!-- SPDX-License-Identifier: Apache-2.0 -->
+
+# Correct classification of mutual-zero neighborhoods
+
+Let M be a matrix over any field K with rank at most three and nonzero
+diagonal. Define its mutual-zero graph G_0 on the matrix indices by
+
+    {i,j} is an edge iff i!=j and M_ij=M_ji=0.
+
+For a vertex t, let N(t) be its open neighborhood in G_0.
+
+**Theorem.** Every induced graph G_0[N(t)] is a disjoint union of
+complete bipartite components and isolated vertices. In particular,
+every such neighborhood is bipartite and contains no odd cycle.
+
+There is no parallelism assumption on the factors of the center.
+
+## Rank-two classification
+
+First consider any matrix B of rank at most two with nonzero diagonal.
+Factor B_ij=u_i^T v_j with u_i,v_i in K^2, padding a rank-one
+factorization if needed. Every factor vector is nonzero by the
+diagonal condition. Let alpha_i be the projective direction of v_i,
+and let beta_i be the projective direction of the one-dimensional
+kernel of the nonzero functional u_i^T. The diagonal condition says
+
+    alpha_i != beta_i.
+
+A mutual-zero edge between i and j is equivalent to
+
+    alpha_j=beta_i and alpha_i=beta_j.
+
+Group vertices by their ordered type (alpha,beta). A vertex of type
+(a,b) can have neighbors only of type (b,a), and every vertex of the
+first type is adjacent to every vertex of the reciprocal type. Since
+a!=b, there are no edges inside either type. Distinct unordered pairs
+{a,b} give disjoint components. If both reciprocal types occur, they
+give one complete bipartite component; if only one occurs, all its
+vertices are isolated. This proves the classification for B.
+
+The argument is valid in every characteristic. Rank one needs no
+exception beyond the two-dimensional embedding; its mutual-zero graph
+is edgeless. An empty index set is immediate.
+
+## Apply the classification to a neighborhood
+
+Choose a rank factorization M_ij=p_i^T q_j in K^3. The nonzero center
+diagonal ensures p_t is nonzero. Every i in N(t) satisfies
+p_t^T q_i=0, so all q_i on N(t) lie in the two-dimensional plane
+ker(p_t^T). Hence the principal matrix M[N(t),N(t)] has rank at most
+two and still has nonzero diagonal. Its mutual-zero graph is exactly
+G_0[N(t)]. The preceding classification proves the theorem.
+
+An odd wheel would have its odd rim cycle entirely inside the open
+neighborhood of its hub. A bipartite neighborhood cannot contain that
+cycle, even as a non-induced subgraph. Thus the no-odd-wheel conclusion
+of the predecessor's Lemma C4 remains valid, now for every odd rim length.
+
+## The error in the retained C5 case split
+
+The source being corrected is Aaron Wang/navinor's PR2890, head
+`08a44355403604c346f8d907811edce992382cc4`,
+`research/fstar-chi4-zero-digraph-seed73765f29/README.md`, Lemma C5
+(lines 100--118 in the retained bytes). The retained README has Git blob
+`acbea2b0ffc66a87573bcd53bfcd24d3e5a17956` and SHA256
+`87e1de6007d5f4770048153f188561bb2f68720fddaf45938da134f02654e931`.
+
+Its nonparallel branch asserts that the neighborhood has no edges.
+The proposed normalization q_t=e_1, p_t=e_2 would instead force
+M_tt=p_t^T q_t=0, contrary to the standing diagonal hypothesis.
+The parallel branch's same-type clique option also disappears when
+the nonzero diagonal is enforced: the reciprocal type (a,a) is
+forbidden. The uniform classification above avoids this case split.
+
+For clarity, the coordinate changes preserving a factorization are
+dual changes. For any invertible S,
+
+    q_i -> S q_i,       p_i -> S^(-T) p_i
+
+leave every entry p_i^T q_j unchanged. Parallelism between p_t and q_t
+is not invariant under these changes. In fact, since p_t^T q_t=d!=0,
+choose a basis beginning with q_t and completed by a basis of ker(p_t^T).
+The resulting dual coordinates give q_t=e_1 and p_t=d e_1. Every
+admitted center permits this normalization, regardless of whether its
+original factor vectors were parallel.
+
+## Minimal admitted rational counterexample
+
+Take the three-row rational factors
+
+    P = [[1,1,0], [1,0,1], [0,1,1]],
+    Q = [[1,1,-1], [1,-1,1], [-1,1,1]].
+
+Their product is M=P Q^T=2 I_3. The center 0 has nonparallel factors
+p_0=(1,1,0), q_0=(1,1,-1). Its mutual-zero neighborhood consists of
+vertices 1 and 2, which are adjacent. This contradicts the asserted
+empty nonparallel neighborhood. Three vertices are minimal for an
+edge inside an open neighborhood: one center and two distinct neighbors
+are necessary.
+
+All three factor columns are active. Their 2-adic column minima and
+the global normalization exponent gamma are zero. The canonical binary
+anchor masks are A=(3,5,6), B=(7,7,7); every tie has size two. Thus the
+counterexample also lies in the admitted all-even rational anchor domain.
+
+Four copies of each paired factor row give twelve rows and
+M=2 diag(J_4,J_4,J_4). A center's neighborhood is K_(4,4). Giving the
+three groups the respective unit-label masks 1,4,2 gives a full
+twelve-line box witness: each unit belongs to its group's tie, and
+different groups have zero binary dots. This is a counterexample to
+the published C5 branch, not to the twelve-line box-eight target.
+
+## Scope and attribution
+
+The corrected classification and odd-wheel consequence are analytic
+over arbitrary fields. The explicit counterexample and its canonical
+anchor admission are over Q. No predecessor engine or census is needed
+for this correction, and no changed census result is asserted. This
+does not prove F-star, close the twelve-line bound, or change the selected
+multiplication frontier.
+
+Prepared by Sagar Thakrar with substantial OpenAI Codex assistance.
+New work is Apache-2.0. The predecessor C3/C4/C5 statements and F-star
+context retain Aaron Wang/navinor's attribution. The exact retained
+README is the source identified above.

research/fstar-mutual-zero-neighborhood/README.md

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+<!-- SPDX-License-Identifier: Apache-2.0 -->
+
+# Repair the mutual-zero neighborhood classification
+
+PR #2890 Lemma C5's nonparallel-center branch is false as stated. A
+three-row rational example with nonempty, even canonical tied sets
+already contradicts it. The corrected theorem preserves the intended
+no-odd-wheel conclusion:
+
+> For a matrix over any field with rank at most three and nonzero diagonal,
+> every mutual-zero open neighborhood is a disjoint union of complete
+> bipartite components and isolated vertices.
+
+Here a mutual-zero edge means **both** directed matrix entries are zero.
+The theorem concerns the graph induced by a center's open neighborhood,
+not the whole mutual-zero graph or the graph of one-way zeros.
+
+## First working example
+
+```text
+P = [(1,1,0), (1,0,1), (0,1,1)]
+Q = [(1,1,-1), (1,-1,1), (-1,1,1)]
+P Q^T = 2 I3
+```
+
+The center's factors `(1,1,0)` and `(1,1,-1)` are not parallel. Its two
+mutual-zero neighbors are adjacent, contradicting C5's edgeless conclusion.
+Three vertices are minimal for an edge inside an open neighborhood.
+Every factor column is active, the actual valuation minima and gamma are
+zero, and the canonical masks are `A=(3,5,6)`, `B=(7,7,7)`: all ties have
+cardinality two.
+
+Repeating each paired row four times gives `2 diag(J4,J4,J4)`. The center's
+neighborhood is `K4,4`. Unit labels `1,4,2` on the three respective groups
+give a full twelve-line box. This refutes the source lemma, not the
+twelve-line box-eight target or F-star.
+
+## What is corrected
+
+The pinned [PR #2890 source](https://github.com/CrocSwap/integer-mult-bounds/blob/08a44355403604c346f8d907811edce992382cc4/research/fstar-chi4-zero-digraph-seed73765f29/README.md)
+uses `q_t=e1, p_t=e2` in the nonparallel case. That would make the center
+diagonal zero. Coordinate changes preserving a factorization act dually
+on P and Q; parallelism of their corresponding rows is not invariant.
+
+The replacement proof in [PROOF.md](PROOF.md) first classifies every
+rank-two matrix with nonzero diagonal. A vertex has a projective column
+direction and a different row-kernel direction. Mutual-zero edges connect
+exactly reciprocal pairs of these types, giving complete bipartite
+components. A rank-three center's mutual-zero neighbors have principal
+rank at most two, so the classification applies uniformly.
+
+Thus every such neighborhood is bipartite. No odd wheel can occur,
+including as a non-induced subgraph. The predecessor's Lemma C4 survives
+with a valid stronger proof. The source's parallel branch is overbroad:
+its same-type clique option is excluded by nonzero diagonal.
+
+## Verification and source boundary
+
+From the repository root, using CPython 3.12:
+
+```sh
+python3 -B scripts/research_package.py --package research/fstar-mutual-zero-neighborhood --check-only
+python3.12 -B scripts/research_package.py --package research/fstar-mutual-zero-neighborhood
+python3 -B scripts/check_research_commit.py --package research/fstar-mutual-zero-neighborhood --check-source-history
+```
+
+`VERIFY.json` registers normal and isolated/optimized/warnings-as-errors
+replays. The exact controls bind the predecessor statement, check the
+canonical counterexamples, compare literal mutual-zero graphs with the
+reciprocal-type classification, and exercise the nonzero-diagonal,
+one-way-zero and dual-coordinate boundaries. They are finite controls for
+an analytic proof, not a proof by exhaustive rational search.
+
+The predecessor README is frozen unchanged in `inputs/`; its original
+commit, path, blob and SHA256 are in `SOURCES.json`. Read-only inspection
+of the pinned predecessor verifier found no use of the false nonparallel
+case as a search prune: its sequential construction uses actual nullspaces
+and dot products. That engine was **not rerun**, and no changed census
+result or comprehensive downstream audit is claimed.
+
+New work is Apache-2.0, by Sagar Thakrar with substantial OpenAI Codex
+assistance. The original C3/C4/C5 statements, source package and F-star
+context retain Aaron Wang/navinor's attribution. Separate same-session
+agents contributed proof, controls and review; these are not independent
+human acceptance or formal kernel checking. No supplier admission,
+selected-frontier change or κ improvement is claimed.
+
+## Open next questions
+
+1. Which downstream arguments, if any, use the rejected nonparallel-empty
+   branch? Check the actual caller before changing it; a citation to the
+   predecessor engine is not evidence that its search used this lemma.
+2. What additional information controls canonical anchor boxes when local
+   bipartite mutual-zero neighborhoods do not settle global coloring?
+   Keep ordinary directed coloring separate from tied-coordinate-compatible
+   labelings and actual rational realizability.
+
+Coordination: [canonical claim and handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1317#discussioncomment-19009680),
+following the [released predecessor handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-18966804).