Prove the eight-line rank-three anchor-box floor five

research/fstar-eight-line-floor/PROOF.md

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+<!-- SPDX-License-Identifier: Apache-2.0 -->
+
+# Eight rational three-total lines have a five-line box witness
+
+## Statement and the directed box contract
+
+Let `P,Q` be rational matrices with eight rows and three columns, let
+`M=P Q^T`, and assume `M_ii != 0` on every line. Let `A_i,B_i` be the
+canonical binary anchors and suppose every tied set `A_i intersect B_i`
+is nonempty. Then **box(M)>=5** for the directed sub-anchor definition.
+No sharpness claim is made.
+
+The proof uses only the following abstract hypotheses: `M` is a real
+eight-by-eight matrix of rank at most three with nonzero diagonal;
+`A_i,B_i` are binary support sets with nonempty intersection; and
+
+    M_ij=0  implies  A_i dot B_j=0 over F_2.                 (SG)
+
+A witness keeps some lines in an order and assigns labels
+`a_i subset A_i`, `b_i subset B_i`, with `a_i dot b_i=1`. Its off-diagonal
+requirements are only
+
+    i precedes j and M_ij=0  implies  a_i dot b_j=0.
+
+Thus the abstract statement is stronger than the rational canonical case,
+but it does not replace the directed constraint by an unordered condition.
+
+For completeness, canonical normalization supplies SG as follows. Remove
+degenerate product columns, which do not affect M, and put
+
+    m_j=min_i nu_2(P_ij),  n_j=min_i nu_2(Q_ij),
+    gamma=min_j(m_j+n_j),
+    p_ij=2^(-m_j) P_ij,    q_ij=2^(m_j-gamma) Q_ij.
+
+Zero entries have valuation infinity. The normalized entries are rational
+numbers with odd denominator, and `p q^T=2^(-gamma)M`. Their reductions
+modulo two have supports `A_i,B_i`, so every actual zero reduces to the
+zero binary dot product in SG. Nonempty ties are the stated hypothesis.
+This is an algebraic normalization using actual rational valuations, not a
+claim about unbounded inputs to a predecessor's finite valuation sentinel.
+
+## Two ordered blocks give a box witness
+
+Suppose disjoint sets F and L each have an ordering in which every forward
+off-diagonal entry of M is nonzero. Choose a tied coordinate c_i for every
+line in their union. On F use
+
+    a_i=A_i,    b_i=e_(c_i),
+
+and on L use
+
+    a_i=e_(c_i),    b_i=B_i.
+
+Order all of F first, then all of L, respecting the two given orders.
+Every diagonal dot is one and every label is a sub-anchor. Within each
+block there are no forward zero entries to constrain. Between the blocks,
+the forward dot is the original `A_i dot B_j`, so SG supplies every required
+zero. This is a witness on all of F union L.
+
+We shall find a three-line F and a disjoint two-line L. In particular the
+argument permits asymmetric labels: it does not restrict both sides to
+the same tied unit vector.
+
+## Rank three forces a transitive nonzero triple
+
+Call distinct indices `(i,j,k)` a transitive nonzero triple if
+
+    M_ij != 0,    M_ik != 0,    M_jk != 0.
+
+We prove that such a triple exists. Suppose otherwise. Let G be the simple
+undirected graph whose edge `{i,j}` means at least one of `M_ij,M_ji` is
+nonzero. An independent set has a diagonal principal matrix with nonzero
+diagonal; hence G has no independent set of size four.
+
+First suppose G has an independent triple I. Its principal matrix
+`D=M[I,I]` is diagonal and invertible. Since `rank(M)<=3`, its Schur
+complement is zero. For any two indices u,v outside I,
+
+    M_uv = sum_(i in I) M_ui M_ii^(-1) M_iv.
+
+If `u!=v` and `M_uv!=0`, some summand is nonzero. The indices `(u,i,v)`
+then form a transitive nonzero triple, a contradiction. Therefore the
+principal matrix on the five vertices outside I is diagonal with nonzero
+diagonal. Its rank is five, again a contradiction.
+
+It remains that G has no independent triple. For a vertex v, its
+out-neighbors in the directed nonzero graph form an independent set in G:
+an edge between two out-neighbors would give a transitive triple starting
+at v. Its in-neighbors are independent for the same reason. Each set
+therefore has size at most two, and the undirected degree of v is at most
+four.
+
+The nonneighbors of v form a clique in G, since two nonadjacent ones
+together with v would form an independent triple. G has no clique of size
+four: choose one nonzero direction of each edge in such a clique, obtaining
+a tournament on four vertices; some vertex has two in-neighbors or two
+out-neighbors, and the edge between those two completes a transitive
+triple. Thus v has at most three nonneighbors. On eight vertices its
+undirected degree is at least `8-1-3=4`.
+
+The degree bounds are equalities. The in-neighbor and out-neighbor sets
+must each have size two and be disjoint, or their union would have size
+less than four. Consequently
+
+    M_ij M_ji=0 whenever i!=j.                            (1)
+
+Choose a real rank factorization `M_ij=u_i^T v_j` with vectors in R^3,
+padding unused dimensions by zero if needed, and form the three-by-three
+matrices `T_i=v_i u_i^T`. The trace identity and (1) give
+
+    tr(T_i T_j)=M_ij M_ji=0  for i!=j,
+    tr(T_i^2)=M_ii^2>0.
+
+The symmetric three-by-three real matrices have dimension six. Therefore
+the eight symmetric parts `(T_i+T_i^T)/2` have a nontrivial real linear
+dependence. For its coefficients c_i, not all zero, the matrix
+`T=sum_i c_i T_i` is skew-symmetric, yet
+
+    tr(T^2)=sum_i c_i^2 M_ii^2>0.
+
+A real skew-symmetric matrix instead satisfies
+
+    tr(T^2)=-sum_(a,b) T_ab^2<=0.
+
+This contradiction finishes the proof that a transitive nonzero triple
+exists. Positivity over the reals is used explicitly; the argument does
+not assert the same matrix lemma over an arbitrary field.
+
+## Completion of the five-line witness
+
+Use a transitive nonzero triple as F in its defining order. Among the
+remaining five vertices there is an edge of G, since otherwise their
+principal matrix would be diagonal of rank five. Orient that edge in a
+nonzero direction and use its two vertices as L. The two-block construction
+keeps `3+2=5` lines and proves the theorem.
+
+## The six-line consequence of the same construction
+
+With six lines, the nonzero graph G has two disjoint edges. Otherwise its
+edges pairwise intersect, so it is a star, a triangle with isolated vertices,
+or empty; each possibility has an independent set of four and therefore a
+diagonal rank-four principal matrix. Orient the two disjoint edges in nonzero
+directions and use them as the two blocks. The same construction gives
+`box(M)>=4`. Thus the transfer argument also gives a short proof of PR3393's
+six-line result, without its anchor-type case analysis.
+
+## Scope
+
+The proof is analytic and has no finite-census premise. It proves the scoped
+box bound, not a complete multiplication theorem, a new selected frontier,
+or a physical supplier/controller invoice. It does not prove `box>=6` at nine lines or an upper bound of five at eight lines.
+
+Prepared by Sagar Thakrar with substantial OpenAI Codex assistance.
+New work is Apache-2.0. The directed box definition, canonical normalization,
+and SG implication retain the Aaron Wang/navinor predecessor attribution.

research/fstar-eight-line-floor/README.md

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+# Eight-line anchor-box floor
+
+For rational `P,Q` with eight rows and three columns, `M=P Q^T`, nonzero
+diagonal, and canonical `tau_i>=1`, **`box(M)>=5`**. This answers the next
+threshold question in
+[PR3393](https://github.com/CrocSwap/integer-mult-bounds/pull/3393).
+Sharpness at five is not established.
+
+[PROOF.md](PROOF.md) proves a stronger matrix statement: every real
+eight-by-eight matrix of rank at most three with nonzero diagonal has distinct
+indices `(i,j,k)` with all three forward entries `M_ij,M_ik,M_jk` nonzero.
+An independent triple would give a rank-basis identity forcing the other five
+vertices to be independent, a contradiction. Otherwise neighborhood counts
+eliminate bidirectional nonzero pairs. Eight associated rank-one matrices
+would then span a positive eight-dimensional space for the trace form on
+three-by-three real matrices, whose positive index is six.
+
+An ordered triple and a disjoint nonzero pair become a five-line box witness:
+use full source anchors and tied unit target labels in the first block, then
+tied unit source labels and full target anchors in the second. The directed
+zero constraints between the blocks are precisely the original SG implication.
+This permits asymmetric labels and needs no anchor-type census. A short
+corollary also gives the six-line floor four from two disjoint nonzero edges.
+
+## Exact evidence
+
+The working rational fixture duplicates Aaron Wang's PR2978 core with block
+sizes `3,3,2`:
+
+```
+P0 = [(-1,-3,0), (1,-1,-2), (3,3,3)]
+Q0 = [(1,-3,2),  (3,-1,-2), (3,-1,2)]
+M  = diag(8 J3, 8 J3, 12 J2).
+```
+
+It has rank three, `gamma=0`, and every tied count is two. Its box is **six**:
+two size-three blocks give six kept lines, while every seven-set meets all
+three blocks and forces a triangular rank-three label minor inside a common
+two-dimensional target space. This is a positive control, not a sharpness
+example for the lower bound five.
+
+`verify.py` constructs the ordered blocks using exact arithmetic and checks
+their labels both literally and with the unchanged target-first Gaussian
+oracle exported from corrected PR3393 head
+`8e1e574fa68465dc0ec69d07852266b9597f9597`. The export is authenticated before
+execution. Controls exercise the directed-cycle boundary, rank-four exclusion,
+SG/nonempty-tie/diagonal admission, the rank-basis identity, and the distinction
+between symmetric and skew trace directions. [RESULT.json](RESULT.json)
+records finite checks; the general theorem rests on the written proof.
+
+From the repository root, with Python 3.12 (tested with 3.12.15):
+
+```sh
+python3 -B scripts/research_package.py --package research/fstar-eight-line-floor --check-only
+python3.12 -B scripts/research_package.py --package research/fstar-eight-line-floor
+```
+
+Normal and isolated optimized replays are registered. There are no third-party
+Python dependencies. Exact source history and hashes are in
+[SOURCES.json](SOURCES.json); the selected record and task catalogue are
+historical provenance. Development searches are not proof premises or claims
+of an exhaustive rational census.
+
+## Scope
+
+The abstract result needs real rank at most three, nonzero diagonal, nonempty
+tied sets and SG. Canonical rational three-total factors satisfy SG through
+their actual 2-adic normalization. The trace argument uses real positivity;
+no arbitrary-field extension is asserted. The proof is analytic, not formally
+machine verified or independently reviewed by a human.
+
+This is a scoped box theorem. The full F-star conjecture, complete multiplication
+theorem, physical controller/supplier admission and selected kappa are unchanged.
+The explicitly changed `independent-moment` scope, baseline, stop rule and next
+question are recorded in [TASK.json](TASK.json). The canonical investigation is
+[Discussion1317](https://github.com/CrocSwap/integer-mult-bounds/discussions/1317#discussioncomment-19008770).
+
+## Open next questions
+
+Does `v=9` force six kept lines? A sufficient next target is two disjoint
+transitive nonzero triples. Start from the `3,3,3` three-block duplication,
+which has box six. One arbitrarily chosen triple can leave a six-line matrix
+with no transitive triple, so the current argument does not answer this
+question. Choosing a better triple or using further anchor information needs
+a new argument; the matrix condition is sufficient, not necessary.
+
+## Attribution
+
+New work by **Sagar Thakrar with substantial OpenAI Codex assistance**, including
+separate proof, exact-control and review agents. Separate AI runs are not
+independent human review. New work is Apache-2.0.
+
+Aaron Wang/navinor supplied the directed sub-anchor box, SG and valuation
+lineage through [PR2914](https://github.com/CrocSwap/integer-mult-bounds/pull/2914)
+and [PR3031](https://github.com/CrocSwap/integer-mult-bounds/pull/3031), and the
+three-line core in [PR2978](https://github.com/CrocSwap/integer-mult-bounds/pull/2978).
+DreamingOfClouds supplied the F-star/certificate origins and earlier fixtures in
+[PR1590](https://github.com/CrocSwap/integer-mult-bounds/pull/1590) and
+[PR2111](https://github.com/CrocSwap/integer-mult-bounds/pull/2111).
+Arham Tahir's [PR3105](https://github.com/CrocSwap/integer-mult-bounds/pull/3105)
+gave the earlier joint triple classification. Those contributors retain their
+source-specific assistance disclosures. The frozen independent oracle and the
+six-line predecessor are Sagar Thakrar's
+[PR3393](https://github.com/CrocSwap/integer-mult-bounds/pull/3393).