Bound the odd-character port window by 36.

research/p104-odd-char-window/NOTICE

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+Odd-character window upper bound.
+Prepared by unpaidmoon with substantial Cursor Grok assistance.
+The port list is the P104 set published in PR #6933.
+New exposition and checks are Apache-2.0.

research/p104-odd-char-window/PROOF.md

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+# Odd-character window of the P120 odd graph
+
+No new κ. The selected value stays `781451/250000000`. This note bounds the
+window of a port set from above. It does not build a word, a decoder, or a
+paid profile.
+
+## 0. Definitions
+
+Let `P120` be the 120 vectors in `F_2^9` of weight 3 or 7 (weight congruent to
+3 mod 4). Two distinct vectors are **adjacent** when their `F_2` dot product
+is 1 and **orthogonal** when it is 0. The **odd graph** has those adjacencies
+as edges.
+
+`P104` is the 104-vector subset stored in `data/ports_p104.json` (70 of weight
+3 and 34 of weight 7). Its complement in `P120` has 16 vectors and exactly two
+cliques of size 8. Each of those cliques, added to `P104`, is a 112-vector set
+`P112`. These are the same sets as in PR #6933: the port file is byte-identical
+to `research/p104-port-window-haemers/data/ports_p104.json` at head
+`a967c521074c2c7450255ae237fda9c439ee1b35`, and the induced odd graph on `P104`
+is 55-regular, as recorded there.
+
+A **1-block pattern** on a vertex set `S` is a matrix, indexed by `S`, with
+nonzero diagonal and with a zero in both off-diagonal positions of every
+orthogonal pair. Entries on adjacent pairs are free, and the matrix need not
+be symmetric. Write `w_1(S; F)` for the minimum rank of a 1-block pattern over
+`F`, and `w(S; F)` for the infimum of `rank(B)/k` over k-block patterns on `S`
+in the sense of #6933. Then `w(S; F) ≤ w_1(S; F)` because the infimum includes
+the case `k = 1`. The identification of this window with the fractional
+Haemers bound is #6933, Theorem 1, and is not used below.
+
+## 1. Degree
+
+**Lemma 1.** Every vertex of the odd graph on `P120` has degree 63.
+
+*Proof.* There are `C(9,3) = 84` weight-3 vectors and `C(9,7) = 36` weight-7
+vectors.
+
+Let `u` have weight 3, and let `T` be the complementary 6-set. A weight-3
+neighbor meets `u` in one coordinate: 3 choices there and `C(6,2) = 15` in
+`T`, so 45 neighbors. A weight-7 vector has a 2-set complement `s`. Its dot
+with `u` is odd if and only if `|u ∩ s|` is even, because the two pieces
+partition the three coordinates of `u`. The even possibilities are `|u ∩ s| = 0`
+(`C(6,2) = 15` sets `s`) and `|u ∩ s| = 2` (`C(3,2) = 3` sets). That is 18
+neighbors. The degree is `45 + 18 = 63`.
+
+Let `u` have weight 7, with 2-set complement `t`. A weight-3 vector `v` has odd
+dot with `u` if and only if `|v ∩ t|` is even. There are `C(7,3) = 35` such
+`v` disjoint from `t`, and 7 that contain `t`. A different weight-7 vector has
+2-set complement `s ≠ t`, and the same parity count used above shows that the
+dot is odd if and only if `s` is disjoint from `t`. There are `C(7,2) = 21`
+such 2-sets. The degree is `42 + 21 = 63`. ∎
+
+`verify.py` recomputes both sides of this count and then enumerates every
+vertex.
+
+## 2. Strongly regular parameters
+
+**Lemma 2.** The odd graph on `P120` is connected and strongly regular with
+parameters `(n, k, λ, μ) = (120, 63, 30, 36)`.
+
+*Proof of the census.* `verify.py` counts common neighbors of every unordered
+pair twice, once with bitsets and once with set intersection, and requires the
+two counts to agree. The degree is constantly 63, adjacent pairs have exactly
+30 common neighbors, and orthogonal pairs have exactly 36. A walk from vertex 0
+reaches all 120 vertices. ∎
+
+The constant parameters are a complete finite enumeration, not a sample. The
+hand argument of Lemma 1 covers only the degree.
+
+## 3. The rank-36 pattern
+
+Let `A` be the adjacency matrix and `J` the all-ones matrix. Lemma 2 is the
+matrix identity
+
+```
+A^2 = k I + λ A + μ (J - I - A) = 27 I - 6 A + 36 J.
+```
+
+**Lemma 3.** `(A - 3I)(A + 9I)(A - 63I) = 0`, the three factors are pairwise
+coprime over `Q`, and
+
+```
+dim ker(A - 63I) = 1,  dim ker(A - 3I) = 84,  dim ker(A + 9I) = 35.
+```
+
+*Proof.* The displayed square rearranges to `A^2 + 6A - 27I = 36 J`. The
+scalar polynomial factors as `x^2 + 6x - 27 = (x - 3)(x + 9)`, since
+`(-6)^2 + 4·27 = 144`. Right-multiplication by `A - 63I` kills `J`, because
+`AJ = 63 J` by regularity, so
+
+```
+(A - 3I)(A + 9I)(A - 63I) = 0.
+```
+
+The linear factors are distinct over `Q`. If a matrix satisfies `fg(A) = 0`
+with `gcd(f, g) = 1`, polynomials `a, b` with `af + bg = 1` split the space
+into `ker f(A)` and `ker g(A)`. Applying that twice, `Q^{120}` is the direct
+sum of the three kernels. The all-ones vector lies in `ker(A - 63I)`. If
+`(A - 63I)x = 0`, every coordinate of `x` equals the average of its neighbors.
+A coordinate of maximum value forces the same value on its neighbors, and
+connectedness forces `x` to be constant. Thus that kernel is one-dimensional.
+
+Let `f = dim ker(A - 3I)` and `g = dim ker(A + 9I)`. Then `f + g = 119`.
+The diagonal of `A` is zero, so `tr(A) = 0`, and that trace is
+`3f - 9g + 63`. Equivalently `f - 3g = -21`.
+Substitution gives `4g = 140`, so `g = 35` and `f = 84`. ∎
+
+**Theorem 1.** Let `H = 3I - A`. Over `Q`, `H` has rank 36, diagonal 3, and
+zeros on every orthogonal pair. For every field `F` of characteristic not 3,
+the reduction of `H` is a 1-block pattern of rank at most 36. For every
+nonempty `S ⊆ P120`,
+
+```
+w(S; F) ≤ w_1(S; F) ≤ 36.
+```
+
+*Proof.* `H = -(A - 3I)`, so Lemma 3 gives rank 36 over `Q`. The diagonal is 3
+and the off-diagonal support is the edge set. In characteristic not 3 the
+diagonal remains nonzero, and rank cannot rise under reduction. A principal
+submatrix is a 1-block pattern on its index set and has rank at most 36. ∎
+
+`verify.py` checks that the reduction has rank exactly 36 modulo 5, 7, 11 and
+10007, which matches the rational rank, and that a single planted orthogonal
+entry is rejected. Modulo 3 the diagonal of `H` is 0, so this matrix is not a
+pattern in characteristic 3. That control is the reason for the next section.
+
+In characteristic 2 the same reduction is a legal pattern of rank at most 36.
+It does not improve the rank-9 Gram pattern already recorded for `P104`,
+`P112` and `P120` in #6933. The new ceiling is the odd-character one.
+
+## 4. Characteristic 3
+
+Index features of a vector `v` by the nine coordinates and the `C(9,2)`
+coordinate products, evaluated in `{0,1}` and read in `F_3`. This is a
+45-column matrix `Φ`. For each vertex `v`, the linear system
+
+```
+Φ_u · y = 1 if u = v,   Φ_u · y = 0 if u is orthogonal to v,
+```
+
+with adjacent rows unconstrained and with free variables set to 0, has a
+solution `y_v`. `verify.py` solves all 120 systems and substitutes the
+solutions back into the equations. The matrix `M_uv = Φ_u · y_v` is therefore
+a 1-block pattern over `F_3`, and `M = Φ Y^T` has rank at most 45. Gaussian
+elimination gives rank exactly 38.
+
+**Theorem 2.** Over `F_3`, `w(S; F_3) ≤ w_1(S; F_3) ≤ 38` for every nonempty
+`S ⊆ P120`.
+
+The nine linear coordinates alone do not solve these systems: all eight
+sampled columns in `verify.py` are inconsistent. The quadratic coordinates are
+part of this factorisation, not an unused pad. The same eight samples are not
+a proof that no rank-36 pattern exists in characteristic 3.
+
+## 5. The interval, conditional on #6933
+
+#6933, Theorem 4, states that in every characteristic other than 2,
+
+```
+w(P104; F) ≥ 13,   w(P112; F) ≥ 14,   w(P120; F) ≥ 15,
+```
+
+because `n/8` equals those three integers and the E8 root-line form has rank
+at most 8. That argument is not replayed here. Accepting it, Theorems 1 and 2
+give
+
+| field | P104 | P112 | P120 |
+|---|---|---|---|
+| char ≠ 2, 3 | [13, 36] | [14, 36] | [15, 36] |
+| char 3 | [13, 38] | [14, 38] | [15, 38] |
+
+#6933 already places the characteristic-2 window of `P104` in `[53/6, 9]` and
+the characteristic-2 windows of `P112` and `P120` at exactly 9. Theorem 1 does
+not narrow those characteristic-2 intervals.
+
+The lower bounds are not claimed for arbitrary subsets. #6933 exhibits a
+14-port subset with a 2-block pattern of rank 17, ratio `17/2 < 9`.
+
+## 6. What this does not say
+
+The window is the block-mask quantity of #6933. That note already records that
+a physical decoder for `k > 1` with exactly this mask is not derived. A rank-36
+pattern is not a multiplication algorithm, not a paid profile, and not a
+change to the selected frontier. No human has reviewed this proof.

research/p104-odd-char-window/README.md

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+# Odd-character window of the P120 odd graph (no new κ)
+
+**No new κ.** The selected record stays `781451/250000000`. This package
+upper-bounds the port-window of `P120` and of every nonempty subset, including
+`P104` and both copies of `P112`. It does not build a word or a paid profile.
+
+The proof is in [PROOF.md](PROOF.md). The finite census, the rank checks and
+the failure controls are in [verify.py](verify.py).
+
+## Result
+
+Let the odd graph have an edge when two weight-3-or-7 vectors in `F_2^9` have
+odd dot product. `P120` is all 120 of those vectors. `P104` is the 104-vector
+subset in `data/ports_p104.json`, byte-identical to the port list of PR #6933
+at `a967c521074c2c7450255ae237fda9c439ee1b35`.
+
+1. The odd graph on `P120` is connected and strongly regular with parameters
+   `(120, 63, 30, 36)`. The degree count is written out. The parameters `λ = 30`
+   and `μ = 36` are the exhaustive pair census.
+2. `H = 3I - A` has rank 36 over `Q`, nonzero diagonal in every characteristic
+   other than 3, and zeros on non-edges. So `w(S; F) ≤ 36` for every nonempty
+   `S ⊆ P120` and every field of characteristic not 3.
+3. In characteristic 3 that matrix has zero diagonal. A degree-at-most-2
+   factorisation produces a 1-block pattern of rank 38, so `w(S; F_3) ≤ 38`.
+
+Accepting the lower bounds in #6933, Theorem 4, without replaying them:
+
+| field | P104 | P112 | P120 |
+|---|---|---|---|
+| char ≠ 2, 3 | [13, 36] | [14, 36] | [15, 36] |
+| char 3 | [13, 38] | [14, 38] | [15, 38] |
+
+Characteristic 2 is unchanged: #6933 already has `w(P104; F_2) ∈ [53/6, 9]` and
+`w = 9` on `P112` and `P120`. The reduction of `H` is a legal pattern there of
+rank at most 36, which does not beat the Gram pattern of rank 9.
+
+## Scope
+
+Original card: `complex-rank-shape-decoder`. Scope status: **changed**. The
+card asks whether a decoder or coframe change improves the signed moment. This
+package does not change a word, a coframe, or a moment. It answers the odd-character
+half of the window question left open by #6933: an odd-character decoder mask
+of this type cannot inherit the characteristic-2 window 9, and the best
+explicit pattern here has ratio 36, or 38 in characteristic 3.
+
+The lower ends 13, 14 and 15 are #6933's. They are not reproved. The upper
+ends are proved here. Base commit `7c49fe32c55de02c2c873655c804b16e4c3fef0a`,
+priority revision 8.
+
+## Reproduce
+
+From the repository root, CPython 3.12, standard library only:
+
+```sh
+python3 -B research/p104-odd-char-window/verify.py
+python3 -I -O -B research/p104-odd-char-window/verify.py
+python3 -B scripts/research_package.py --package research/p104-odd-char-window
+```
+
+Expected final line: `PASS_ODD_CHAR_WINDOW rank36=36 char3_rank=38`, then
+`PASS_REGISTERED_RESEARCH_PACKAGE` with outcome `partial`.
+
+The verifier enumerates all `C(120,2)` pairs, reduces `3I-A` modulo four odd
+primes, solves the 120 characteristic-3 linear systems, and rejects a planted
+orthogonal entry, the characteristic-3 reduction of `3I-A`, the all-ones
+matrix, and the linear-only feature map on eight sampled columns. No Lean,
+Docker, selected-baseline replay, or solver certificate was run.
+
+## Open next questions
+
+1. Is `w(P104; F)` equal to 13 for some odd characteristic, or is 36 (38 in
+   characteristic 3) the wrong shape of construction? The E8 argument of #6933
+   is the matching lower bound. A first step is one explicit factorisation of
+   width 13 over `F_5`. Nothing in this package exhibits one, and the
+   Bose-Mesner pattern `3I-A` cannot, because its rank is 36.
+2. Does characteristic 3 admit a pattern of rank 36? The matrix that works for
+   every other odd characteristic is illegal there. The degree-2 factorisation
+   has rank 38. Closing the gap of 2 needs a different column space.
+
+## Credit
+
+DreamingOfClouds, PR #6933, for the window definition, the port list, the
+characteristic-2 Gram rank, and Theorem 4's lower bounds `n/8`. The strongly
+regular census and both upper bounds are new. Prepared by unpaidmoon with
+substantial Cursor Grok assistance. New work is Apache-2.0.

research/p104-odd-char-window/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ]
+}

research/p104-odd-char-window/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Upper bound the odd-character window of the P120 odd graph and of every nonempty subset, including the P104 port set of PR 6933. No word, moment, or kappa.",
+  "scope_change_reason": "The card asks whether a decoder or coframe change improves the signed moment. This package does not change a word or a moment. It answers the odd-character half of the window question left open by PR 6933: the block-mask rank H_f is at most 36 in every characteristic other than 3, and at most 38 in characteristic 3. An odd-character mask cannot inherit the characteristic-2 window 9.",
+  "bottleneck_link": "A k-block decoder mask on these ports is the fractional Haemers window of the odd graph. The characteristic-2 window is at most 9. The odd-character ceiling shows that moving the same mask to an odd field does not undercut that window; the proved patterns sit at 36 or 38. This is a scoped rank obstruction, not a supplier admission.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after one explicit pattern whose rank is checked over Q and over F_3. Do not search for a rank-13 factorisation or a physical decoder in this package.",
+  "outcome": "partial",
+  "evidence": "PROOF.md Theorems 1 and 2. verify.py prints PASS_ODD_CHAR_WINDOW rank36=36 char3_rank=38. Lower bounds 13, 14 and 15 are cited from PR 6933 Theorem 4 and are not replayed.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance.",
+  "open_questions": [
+    {
+      "question": "Is w(P104; F) equal to 13 for some odd characteristic?",
+      "why_it_matters": "PR 6933 proves the lower bound 13. Equality would make the odd-character window sharp; the pattern proved here has rank 36, or 38 in characteristic 3.",
+      "first_step": "Search for one 104 by 13 factorisation over F_5 with zeros on orthogonal pairs and nonzero diagonal.",
+      "blocked_by": "The Bose-Mesner matrix 3I-A has rank exactly 36, so a sharper pattern needs edge weights outside that algebra. No such factorisation is known."
+    },
+    {
+      "question": "Does characteristic 3 admit a 1-block pattern of rank 36 on P120?",
+      "why_it_matters": "Every other characteristic not equal to 3 has the rank-36 pattern. Characteristic 3 is the only odd characteristic where that matrix is illegal.",
+      "first_step": "Decide whether the column space of the degree-2 factorisation can be replaced by a 36-dimensional space that still solves the orthogonal equations.",
+      "blocked_by": "The degree-2 matrix constructed here has rank 38. Eight linear-only samples are inconsistent, which does not rule out a different 36-dimensional model."
+    }
+  ]
+}

research/p104-odd-char-window/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exhaustive strongly regular census of the 120-vertex odd graph, rational eigenvalue arithmetic, rank 36 of 3I-A modulo 5, 7, 11 and 10007, induced P104 and P112 patterns, characteristic-3 degree-2 factorisation of rank 38, and controls that reject a planted orthogonal entry, the characteristic-3 reduction of 3I-A, the all-ones matrix, and linear-only features."
+    }
+  ]
+}