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Mister-Meeseeks merged 3 commits into from
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Scoped negative result: orthogonal P120 shared-total pairs fail diagonals #1386

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41f06d1
Research seed 48 orthogonal P120 shared-total obstruction
1337isnot1337 Oct 13, 2026
94d9ec3
Record final seed 48 overlap refresh
1337isnot1337 Oct 13, 2026
cd29f53
Link seed 48 result to PR 1386
1337isnot1337 Oct 13, 2026
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44 changes: 44 additions & 0 deletions research/seed48-p120-orthogonal-shared-total/CLAIM.json
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{
"assignment_seed": 48,
"track": "structural decoder research",
"base_commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
"coordination": "No pinned sprint coordination issue was found; claim recorded locally. Refreshed overlap at 2026-10-13T23:43:55Z covered 847 open PRs and 94 discussions, complete pagination, no warnings. PR #1366 is the predecessor for one odd-pair pencil (e=L51,d0=L1). No claim for the seed-selected orthogonal pair e=L48,d0=L91 was found.",
"scope": "For the full P120 E8 root set and labels, set e=L_48 and choose d0=L_91 as the seed-selected 49th entry (zero-based index 48) of the sorted 63 orthogonal roots. Enumerate every target/nonzero F2^9 sink row in the calibrated one-row family, then every distinct-target pair with one shared d0 correction total.",
"changed_restriction": "Change the fixed relative geometry from PR #1366's odd pair <L_51,L_1>=2 to an orthogonal pair <L_48,L_91>=0, and use a separately implemented exact enumerator.",
"method": "Reconstruct the pinned root/label certificate independently; enumerate rows with exact Fraction arithmetic and bitset label supports; filter consistent rational calibrators and nonzero diagonals; test total availability, cross-target diagonal effects, and all row equations for every distinct-target pair.",
"optimistic_cost": "One rank-one shared total plus eight rank-eight coordinate totals costs 65 COPY-rank units in the complete local unit. This is seven below the P120 all-pairs floor 72, before claiming any full supplier or recursive improvement.",
"expected_deliverable": "A source-pinned research package with the independent exact P120 orthogonal-pencil census, proof of the pair obstruction or a fully checked exact witness, the P112 C.6 positive control, and mutation controls.",
"stop_rule": "Exhaust all 120 targets x 511 nonzero labels and all distinct-target pairs in the stated filtered family; stop after the exact census, control replays, and one final package check. Do not search other e,d0 geometries.",
"expected_finish_utc": "2026-10-14T00:15:00Z",
"preliminary_replay": "Using the predecessor's verifier functions as triage only for e=48,d0=91 found 65 nodes, 2,048 distinct-target pairs, 1,056 shared-total-available pairs, and zero exact pairs. Independent replay remains outstanding.",
"candidate_tasks": [
{
"task": "Orthogonal P120 pencil e=L48,d0=L91 with one shared total",
"decision": "chosen",
"reason": "This changes the root-pair geometry from the completed PR #1366 odd-pair case and has a fully priced local upside: 65 versus the 72-copy all-pairs P120 floor. The finite row/pair domain is small enough to exhaust exactly."
},
{
"task": "Global rank-eight masked rigidity on the P104 class of size 28",
"decision": "rejected",
"reason": "PR #1292 establishes only local rigidity and leaves this global question open, but classifying distant components is broader than the bounded session and the complete copy bill for a candidate component is not yet favorable."
},
{
"task": "Rank-nine single odd-entry completion on P104",
"decision": "rejected",
"reason": "The active claim #1360 studies arbitrary rank-nine factorizations of the corresponding P112 one-entry completion; PR #1320 and PR #1331 cover nearby P112 co-star and free-rectangle cases. This is too close to the active one-entry overcomplete line for this seed."
}
],
"predecessors": [
"PR #1366 / discussion result: calibrated two-target diagonal obstruction for P120 e=L51,d0=L1, plus P112 C.6 positive control.",
"PR #968: P120 all-pairs copy-rank floor 72 and P104 frame-pair geometry.",
"PR #1292: local masked P104 rank-eight rigidity; global off-orbit cases remain open."
],
"result": {
"status": "completed negative result",
"completed_utc": "2026-10-13T23:54:12Z",
"pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/1386",
"package_commit": "41f06d17bd5693bcd7529eed18f96cae06f3e647",
"finding": "For the seed-selected orthogonal P120 pencil e=L48,d0=L91, every one of 1,056 shared-total-available calibrated row pairs cancels at least one target diagonal; the exact pair count is zero.",
"verification": "Independent exact replay passed, including PR1366 odd-pair regression, full P112 C.6 positive unit, total-source deletion control, label mutation control, source pins, and package metadata."
}
}
201 changes: 201 additions & 0 deletions research/seed48-p120-orthogonal-shared-total/LICENSE
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9 changes: 9 additions & 0 deletions research/seed48-p120-orthogonal-shared-total/NOTICE.md
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`vendor/seed51/certificate.tsv` is copied from
`CrocSwap/integer-mult-bounds` commit
`21046cb8fa0d4f0e9b2f826ba71e01baaf9be02c`, the P120 root/label table used in
PR #1366. Its Git blob and SHA-256 are recorded in `SOURCES.json`. The source
contributor is Cinna Davis. The repository license and retained attribution
apply; this new package is under Apache-2.0.

Contributor: 1337isnot1337. Prepared with substantial OpenAI Codex assistance;
no independent human review is claimed.
101 changes: 101 additions & 0 deletions research/seed48-p120-orthogonal-shared-total/PROOF.md
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# Exact obstruction for the seed-selected orthogonal pencil

## Fixed certificate and pencil

The input is the source-pinned 120-row root/label table from PR #1366. The
checker verifies 120 distinct integer roots of squared norm four, integral
half-Gram entries in `{-1,0,1,2}`, diagonal Gram value two, and all 14,400
label-pair identities

\[
u_s\cdot u_t = 1 + G_{st}\pmod 2,\qquad G_{st}=\langle L_s,L_t\rangle/2.
\]

Set `e=L_48`. There are 63 indices `j` with `j != 48` and
`<L_48,L_j>=0`. Sorting those source indices and taking index `48 mod 63`
selects `d0=L_91`. The computed inner product is zero.

For each port `s`, define the exact integers

\[
p_e(s)=\langle L_s,e\rangle,\qquad
p_d(s)=\langle L_s,d0\rangle.
\]

## Row and total conditions

For target `t` and nonzero sink word `w` in `F_2^9`, let

\[
Z(t,w)=\{s:u_s\cdot w=1\pmod2\}.
\]

Keep rows with `t in Z(t,w)` and `w != u_t`. A calibrated row has one nonempty
set of rational ratios

\[
\left\{-p_d(s)/p_e(s): s\in Z(t,w),\ s\ne t,\ p_e(s)\ne0\right\}.
\]

Call its unique value `mu_t`, and require the uncorrected target diagonal
`p_d(t)+mu_t p_e(t)` to be nonzero. The off-target coefficients that the
`d0`-direction total must cancel are

\[
E(t,w)=\{s\in Z(t,w):s\ne t,\ p_e(s)=0,\ p_d(s)\ne0\}.
\]

For two rows `a=(t_a,w_a)` and `b=(t_b,w_b)`, the single total has source set
`Q=E_a union E_b`. It can be read at both sinks exactly when
`E_a subset Z_b` and `E_b subset Z_a`. The modified row coefficient is

\[
N_t(s)=p_d(s)+\mu_t p_e(s)-p_d(s)\,\mathbf 1_{s\in Q}.
\]

If `t_a in E_b`, then by definition `p_e(t_a)=0` and `p_d(t_a) != 0`; also
`t_a in Q`. Hence

\[
N_{t_a}(t_a)=p_d(t_a)+\mu_a\cdot0-p_d(t_a)=0,
\]

which violates the nonzero target-diagonal requirement. The same implication
holds with `a` and `b` exchanged. Thus every available shared-total pair must
avoid both cross-target memberships to be exact.

## Complete finite census

The checker tests all `120 * 511 = 61,320` target/word pairs and uses exact
`Fraction` ratios. It finds:

| Stage | Count |
|---|---:|
| Rows with target in sink support and nonbaseline sink word | 30,600 |
| Rows with one nonempty rational calibrator | 190 |
| Rows whose pre-total target diagonal is nonzero | 65 |
| Distinct-target pairs among those rows | 2,048 |
| Pairs for which the union total is available at both sinks | 1,056 |
| Available pairs with both targets contaminated | 1,024 |
| Available pairs with exactly one target contaminated | 32 |
| Available pairs with no target contaminated | 0 |
| Exact pairs | 0 |

The script checks the source/target equations for every available pair in
addition to the diagonal obstruction. Therefore the negative result is exact
for this finite calibrated one-total family; it is not inferred from sampling.

## Controls and boundaries

The same implementation replays PR #1366's `e=L_51`, `d0=L_1` control: 27
rows, 351 available pairs, all 351 with both target diagonals contaminated.
It also finds the published P112 C.6 positive control (`e=L_0`, `d0=L_76`):
targets 75 and 76, sink words 329 and 438, and a 24-source shared total. The
checker verifies the full 112-by-112 unit and confirms that deleting one
required total source breaks it. A mutated label is rejected by the pairing
identity check.

The result does not cover uncalibrated rows, other total vectors or scalars,
more than one total, another orthogonal pair, rank-nine factorizations, other
punctures, or the full complex supplier. The selected conditional κ remains
unchanged.
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