Host one P104 frame-indicator row and charge its new monomials.

research/p104-one-frame-indicator-word/PROOF.md

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+# One frame-indicator row as a length-4 label word
+
+No new κ. The selected record stays `781451/250000000`. This note hosts one
+frame-indicator row and charges the monomials of that word. It does not host
+the other twelve rows and it does not compare a histogram with the selected
+profile.
+
+## 1. Hypotheses
+
+Ports and the odd graph are those of PR #6933. Two distinct ports are adjacent
+when their `F_2` dot product is 1. Frame 0 of the #7014 witness is the ordered
+8-tuple
+
+```text
+11, 21, 38, 56, 140, 146, 161, 191.
+```
+
+A **letter** is a root-line functional `λ_i(j) = (<r_i, r_j> + 1) mod 2` on the
+120 root lines. A **word** is a product of letters. The product is evaluated in
+`F_2`, so the value is 1 only when every factor is 1. The word **hosts** the
+frame-indicator row when, on all 120 lines, the product equals 1 on these eight
+lines and equals 0 on the other 112. Restriction to the 104 P104 lines is then
+the indicator of this frame.
+
+Labels use the inherited simple-root chart. If `B` is the doubled simple-root
+basis of `e8model.py`, the label of a root `r` is
+
+```text
+u(r) = (coordinates of r in B, reduced mod 2) with a ninth coordinate 1.
+```
+
+The symmetric form `S9` is the Cartan matrix of `B` on the first eight
+coordinates and 1 on the ninth diagonal entry. The chart is accepted only
+because `S9(u_i, u_j) = (<r_i, r_j> + 1) mod 2` holds for every pair of the 120
+lines. That identity is checked exhaustively; it is the pairing (1) of the
+PR968 model.
+
+Square-free monomials are written in `x0, ..., x7`, the first eight label
+coordinates. The relation `x_i^2 = x_i` is used throughout, so a function
+`F_2^8 → F_2` has one algebraic normal form.
+
+## 2. The word
+
+**Theorem.** Under the hypotheses above, no product of fewer than four letters
+hosts the row. Every product of four letters that does host it expands to the
+same 48 monomials
+
+```text
+(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7).
+```
+
+Eight of those monomials have degree at most 1: the constant `1` and
+`x0, x1, x2, x4, x5, x6, x7`. The other **40** have degree 2, 3, or 4
+(17, 17, and 6). Those 40 are the new monomials charged for this row.
+
+One such word, the lexicographically first length-4 subset in the frame order
+above, is the product of the letters of ports `11, 21, 38, 140`:
+
+```text
+λ_11 = 1 + x1 + x2 + x4 + x5,
+λ_21 = 1 + x0 + x1 + x2 + x4 + x5,
+λ_38 = 1 + x0 + x1 + x2 + x4,
+λ_140 = 1 + x0 + x5 + x6 + x7.
+```
+
+The support of this function on the chart `x8 = 1` is the 16-point flat
+
+```text
+x0 = 0,  x5 = 0,  x1 + x2 + x4 = 0,  x6 + x7 = 0
+```
+
+(`x3` is free). Exactly eight of those points are root-line labels, and those
+eight are the frame. On every root line the function is therefore the
+frame indicator. It is not the indicator of those eight points inside all of
+`F_2^8`.
+
+## 3. Proof
+
+**Factor restriction.** Suppose a product of letters equals 1 at a line. Each
+factor is 0 or 1, so each factor equals 1 at that line. The indicator equals 1
+on the eight frame lines, so every factor of a hosting word equals 1 on all
+eight. Checking the 120 letters, the only ones with that property are the
+eight frame letters. Every hosting word is therefore a product of a subset of
+these eight letters, repetitions included. A repeated letter does not change
+an `F_2` value, so the 255 nonempty subsets exhaust the words.
+
+**Census.** Each of those 255 products was evaluated on all 120 lines. The
+products equal to the indicator are 56 of length 4, 56 of length 5, 28 of
+length 6, 8 of length 7, and 1 of length 8. None is shorter than 4. This is an
+exhaustive census of a finite domain of size 255, after the factor restriction.
+It is not a sample.
+
+**Reduction.** Write `p = 1 + x1 + x2 + x4`, `q = x0`, `r = x5`, and
+`s = x6 + x7`. The four letters are the functions
+
+```text
+λ_38 = p + q,  λ_11 = p + r,  λ_21 = p + q + r,  λ_140 = 1 + q + r + s.
+```
+
+On `F_2`, every function satisfies `f^2 = f`. Hence
+
+```text
+(p + q)(p + q + r) = (p + q)(1 + r),
+(1 + r)(p + r) = p(1 + r),
+(p + q)(p + r)(p + q + r) = p(1 + q)(1 + r).
+```
+
+The last step uses `p(p + q) = p(1 + q)`. Multiplying by `λ_140` gives
+`p(1 + q)(1 + r)(1 + q + r + s)`. The target form is
+`p(1 + q)(1 + r)(1 + s)`. Their difference is killed because
+
+```text
+(1 + q)(q + r) = r(1 + q),  so  p(1 + q)(1 + r)(q + r) = p(1 + q) · r(1 + r) = 0.
+```
+
+Thus the four-letter product equals `(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7)`
+as a function on `F_2^8`. The four factors use disjoint variables, so the
+square-free expansion has `2 · 4 · 2 · 3 = 48` monomials and no cancellation.
+The eight monomials in which at most one factor contributes a variable are
+`1, x0, x1, x2, x4, x5, x6, x7`. The remaining 40 are new relative to that
+degree-at-most-1 alphabet.
+
+**Same expansion for every shortest word.** The same evaluation shows that all
+56 length-4 hosting products determine one polynomial: each agrees with this
+expansion on all 256 points of the chart. Longer hosting products agree with
+the indicator on the 120 lines. Their chart polynomials are not charged here.
+
+**Pairing rank, not added.** On the eight frame lines the pairing matrix is
+the all-ones matrix, which has `F_2`-rank 1. That is the inherited copy-cost
+of the support of `χ χ^T`. It is a different unit from the 40 monomials and is
+not added to them.
+
+## 4. What the census does not decide
+
+The factor restriction uses products of root-line letters. A sum of such
+products, a coordinate change, or a circuit that is not a product can be a
+different hosting, and this note gives no lower bound for those larger
+alphabets. The 40 monomials are the expansion of this word in the fixed
+simple-root chart. They are not a count of Clifford monomial routes, Gaussian
+children, or selected-invoice primitives. No conversion into the selected paid
+histogram is claimed, so this bill does not beat the selected profile and does
+not change κ.
+
+The other twelve frames of the witness are not expanded. A monomial shared
+with one of them would make a joint bill smaller than the sum of the separate
+bills. That comparison is not made.

research/p104-one-frame-indicator-word/README.md

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+# One P104 frame-indicator word
+
+**No new κ.** The selected frontier stays `781451/250000000`. This package hosts
+one frame-indicator row as a legal product of root-line labels and charges the
+new monomials of that word. It does not host the other twelve rows and it does
+not price a signed profile.
+
+## Result
+
+Frame 0 of the vendored 104-by-13 witness is the odd-graph clique
+
+```text
+11, 21, 38, 56, 140, 146, 161, 191.
+```
+
+A letter is the pairing of one root line with all 120 root lines. In `F_2`, a
+product equals 1 only when every factor equals 1, so every factor of a hosting
+word is identically 1 on these eight lines. The only such letters are the eight
+frame letters. The 255 nonempty products of those letters were evaluated on all
+120 lines. The products equal to the indicator begin at length 4: 56, 56, 28,
+8, and 1 products of lengths 4 through 8. None is shorter.
+
+Every length-4 hosting product expands, in the inherited simple-root chart, to
+the same 48 square-free monomials
+
+```text
+(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7).
+```
+
+The lexicographically first such word uses ports `11, 21, 38, 140`. Eight
+monomials have degree at most 1 (`1` and `x0, x1, x2, x4, x5, x6, x7`). The
+other **40** have degree 2, 3, or 4, in counts 17, 17, and 6. Those 40 are the
+new charges for this row. The proof of the factor restriction and of the
+disjoint-variable identity is in [PROOF.md](PROOF.md). The support of the
+function on the chart is a 16-point flat; exactly eight of those points are
+root-line labels, and those eight are the frame.
+
+The pairing matrix on the eight frame lines is the all-ones matrix and has
+rank 1. That copy-cost is a different unit and is not added to the 40.
+
+## Evidence class
+
+The factor restriction and the reduction to the four-factor form are written
+proofs. The census of 255 products, the check that only the eight frame letters
+are admissible, the 120-by-120 chart identity, and the agreement of all 56
+length-4 expansions are exhaustive finite checks. This is not a heuristic and
+not an incomplete search inside that alphabet.
+
+It is not a lower bound for sums of letters, for a different chart, or for a
+circuit that is not a product. It is not a selected-invoice histogram. It does
+not beat the selected profile.
+
+## Scope
+
+Catalogue card `complex-rank-shape-decoder`, priority revision 8, pinned at
+`17469fff78c22941280f04d39dc5a1faee09c840`. The card asks whether a decoder or
+coframe change improves the full signed moment. The live handoff in open PR
+[#7014](https://github.com/CrocSwap/integer-mult-bounds/pull/7014) asks a prior
+question: host one frame-indicator row and charge every new monomial before
+touching the other twelve. Scope is **changed** to that first step, for frame 0
+only. The connection to the bottleneck is recorded in [TASK.json](TASK.json):
+a later joint bill cannot be priced until each row's new monomials are known,
+and this package stops at the first row.
+
+`data/frames.json` is the frame partition from that open pull request (head
+`949c0ada285aae619468ae49317c8d9f4df130f7`), vendored and hashed inside
+`verify.py`. This package does not reprove the rank-13 factorisation. Ports and
+the E8 line map are the #6933 inputs pinned in [SOURCES.json](SOURCES.json).
+The label chart is the simple-root chart of the inherited `e8model.py` pairing.
+
+## Reproduce
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python3 -B research/p104-one-frame-indicator-word/verify.py
+python3 -I -O -B research/p104-one-frame-indicator-word/verify.py
+python3 -B scripts/research_package.py --package research/p104-one-frame-indicator-word
+python3 -B scripts/check_research_commit.py --package research/p104-one-frame-indicator-word --check-source-history
+```
+
+The verifier prints `PASS_P104_ONE_FRAME_INDICATOR` with `word=11,21,38,140`,
+`monomials=48`, and `new=40`. Failure controls delete one letter of that word,
+count the 14 length-4 products that miss the row, delete the constant monomial,
+and reject a letter from outside the frame.
+
+## Open next questions
+
+1. Host frame 1 by the same census and compare its degree-at-least-2 monomials
+   with these 40. A shared monomial would make a joint bill smaller than the
+   sum, which is the first place the thirteen rows can interact.
+2. Decide whether a sum of letters, rather than a product, can host this same
+   row with fewer than 40 new monomials. The product census does not speak
+   about sums.
+3. Name these 40 monomials as primitives on the selected #6030 invoice before
+   asking whether thirteen rows beat the profile. No conversion is claimed here,
+   and no κ change is available until that naming exists.
+
+## Credit
+
+Frame partition: unpaidmoon, PR #7014. Ports and odd-graph window: PR #6933.
+Label pairing: the simple-root chart used by the complex-frame decoder notes.
+Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is
+Apache-2.0.

research/p104-one-frame-indicator-word/SOURCES.json

@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "17469fff78c22941280f04d39dc5a1faee09c840",
+  "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"
+    },
+    {
+      "path": "research/p104-port-window-haemers/data/e8_line_map.json",
+      "usage": "input",
+      "git_blob": "19ea2d553aff60e53af50dd9582ab70730e93450",
+      "sha256": "b37530e8e9b2b3bd5be418d5ed3060a42ceb66e60ed3789f8ee79f29dabde437"
+    },
+    {
+      "path": "research/p104-port-window-haemers/data/ports_p104.json",
+      "usage": "input",
+      "git_blob": "8037fd10877f49063a1c8a530b40c27f2453dac0",
+      "sha256": "691c7b7d212112115b1bfc76db89f90d304ec448dc8c4a0c43d9a5954317146c"
+    }
+  ]
+}

research/p104-one-frame-indicator-word/TASK.json

@@ -0,0 +1,87 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "17469fff78c22941280f04d39dc5a1faee09c840",
+  "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": "Host frame 0 of the #7014 P104 factorisation as a product of root-line labels and charge every new monomial of that word, before touching the other twelve frames.",
+  "scope_change_reason": "The catalogue card asks whether a decoder or coframe change improves the full signed moment. Open PR #7014 asks a prior physical question: host one frame-indicator row as a legal label word and charge every new monomial before touching the other twelve. This package answers that first step for frame 0 only. It does not price a changed itinerary, so it cannot improve the signed moment.",
+  "bottleneck_link": "A complete signed target bill needs the new charges of any legal producer that is added. This package records the 40 new monomials of one frame-indicator word and stops. Those charges are a prerequisite for pricing the thirteen rows together; they are not themselves a signed histogram or a supplier admission.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the length-4 word for frame 0, its 40 new monomials, and the statement that the other twelve frames are unexpanded. Do not convert the monomials into a selected-invoice histogram and do not claim a kappa change.",
+  "outcome": "completed",
+  "evidence": "verify.py prints PASS_P104_ONE_FRAME_INDICATOR word=11,21,38,140 monomials=48 new=40. Evidence class: a written F2 factor restriction and a written reduction to (1+x0)(1+x1+x2+x4)(1+x5)(1+x6+x7), plus an exhaustive census of the 255 products of the only eight admissible letters. Not a heuristic. Not a selected-profile comparison. Selected kappa remains 781451/250000000.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance. The census, chart identity, and polynomial reduction are rechecked by verify.py.",
+  "open_questions": [
+    {
+      "question": "Which degree-at-least-2 monomials of frame 1 coincide with the 40 monomials charged for frame 0?",
+      "why_it_matters": "A shared monomial makes the joint bill of two rows smaller than the sum of the separate bills, which is the first interaction among the thirteen rows.",
+      "first_step": "Rerun the same census with frames[1] in place of frames[0] and intersect the two sets of degree-at-least-2 exponents.",
+      "blocked_by": "none in this alphabet; the other frame has not been expanded"
+    },
+    {
+      "question": "Can a sum of root-line letters host the frame-0 indicator with fewer than 40 new monomials in this chart?",
+      "why_it_matters": "The 40-count is exact for products. A sum is a larger alphabet, so the product census is not a lower bound for it.",
+      "first_step": "Search linear combinations of the 120 letters, reduced in the simple-root chart, for a function that matches the indicator on all 120 lines and has fewer than 40 monomials of degree at least 2.",
+      "blocked_by": "the product census does not cover sums"
+    },
+    {
+      "question": "What is the selected #6030 invoice price of these 40 monomials, with no other rows attached?",
+      "why_it_matters": "Until each monomial is named as a paid primitive, the word cannot be compared with the selected profile and cannot change kappa.",
+      "first_step": "Match each of the 40 square-free monomials to a primitive the #6030 invoice already prices, or record that the match is undefined.",
+      "blocked_by": "this package does not identify those primitives"
+    }
+  ]
+}