Prove odd signs on u keep the X_3 trace -48

research/c3-odd-sign-trace-slot/PROOF.md

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+# An odd minus on u keeps the trace -48 on (+1, +1)
+
+This note answers the second open question of the sector-3 payload-involution
+certificate: whether one other cross-coset pair, with a different odd sign,
+moves the trace `-48` of `X_3` onto another `(u, v)` eigenvalue. The selected
+conditional kappa stays `781451/250000000`. Nothing here is a characteristic
+polynomial, a basis, or a census of all 3072 cross-coset pairs.
+
+The claim is a **proof**. The joint trace formula is an identity for commuting
+involutions. The 512 addresses, the 16 odd signs, and the 16 characters below
+are finite sets, and the verifier evaluates the identities on every element of
+each set. That evaluation is exhaustive on those sets. It is not a heuristic
+search, and it is not a proof that an arbitrary change of mask keeps the trace.
+
+## The pair
+
+The input is the pinned 8-by-112 query, SHA256
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`.
+Bell block `c=3` and the payload involution `X_3`, translation of the address
+by 3, are the operators of the degree-881 certificate. The sign group `H` is
+the order-16 group of diagonal signs supported on the odd coordinates
+`{1, 3, 5, 7}` and positive on the even coordinates. Its characters are the
+16 homomorphisms `F_2^4 → {±1}`, indexed by coordinates `(1, 3, 5, 7)`.
+
+The explicit pair from the c=3 sign-split certificate is
+
+| name | permutation | minus signs |
+|---|---|---|
+| `u` | `(0 2)(4 6)` | coordinates 0 and 2 |
+| `v` | `(0 4)(2 6)` | coordinates 0 and 4 |
+| `uv` | `(0 6)(2 4)` | coordinates 0 and 6 |
+
+The mask of `u` is 5 and the mask of `v` is 17. These even signs are one legal
+opposite-cycle pattern: each 2-cycle is uniformly signed, and the two cycles
+carry opposite signs. The odd signs of this pair are all positive. On the
+trivial character, `X_3` has joint traces `-48, 16, 16, 16` in the eigenvalue
+order `((+1,+1), (+1,-1), (-1,+1), (-1,-1))`. The verifier recomputes that row,
+and the other published weight rows, before any sign is changed.
+
+## One odd minus
+
+Let `D` be the diagonal sign that is `-1` only on coordinate 1. Set
+`u' = u ∘ D`, with `D` applied first. The permutation of `u'` is still
+`(0 2)(4 6)`. Its minus signs are coordinates 0, 1, and 2, so the mask is 7.
+The product `u' v` is still `(0 6)(2 4)`, now with mask 67. Both operators are
+involutions, they commute, they commute with every element of `H`, they fix
+sector 3, and `X_3` commutes with every element of the group they generate.
+The group is the same set of 64 signed permutations as the group generated by
+the unvaried pair and `H`. In particular `u'` lies in the original coset `uH`.
+
+The joint trace on the character `χ` and the eigenvalues `(ε, δ)` is
+
+    tr(X_3 P)
+      = (1/64) sum_{h in H} χ(h)
+        [tr(X_3 h) + ε tr(X_3 u' h) + δ tr(X_3 v h) + εδ tr(X_3 u' v h)].
+
+As `h` runs through `H`, so does `Dh`. For the trivial character every
+coefficient `χ(h)` equals `+1`, and `{u' h : h in H}` equals `{u h : h in H}`
+as sets of 16 operators. The same holds for the product coset, because `D`
+commutes with `v`. The four unweighted sums are therefore the sums already
+computed for `(u, v)`. The trivial-character traces stay
+
+| `(u', v)` eigenvalue | dimension | trace of `X_3` |
+|---|---:|---:|
+| `(+1, +1)` | 216 | -48 |
+| `(+1, -1)` | 216 | 16 |
+| `(-1, +1)` | 216 | 16 |
+| `(-1, -1)` | 216 | 16 |
+
+The value `-48` remains the trace on the eigenvalue `(+1, +1)`. The
+`+1` eigenspace of `X_3` on that 216-dimensional space still has dimension
+`(216 + (-48))/2 = 84`, and the `-1` eigenspace still has dimension 132.
+Those dimensions are the same arithmetic as in the unvaried pair. This package
+does not list a basis of either space.
+
+The sign change is visible on another character. On the weight-1 character
+that is `-1` on coordinate 1 and `+1` on coordinates 3, 5, and 7, the same
+reindexing multiplies the `u`-eigenvalue by `χ(D) = -1`. The traces move from
+`(36, -12, -12, -12)` to `(-12, -12, 36, -12)`, and the dimension 180 moves
+from `(+1, +1)` to `(-1, +1)`. The value `-48` does not occur among those four
+traces. The weight-1 character that is `-1` only on coordinate 3 is `+1` on
+`D`, and its traces stay `(36, -12, -12, -12)`.
+
+The coordinate trace of `u'` is 2, while the coordinate trace of `u` is 4, so
+`u'` is not the same operator as `u` on the whole block. The agreement on the
+trivial character is the character average, not an unchanged matrix.
+
+## Every odd sign on this u
+
+The same identity applies to every diagonal sign supported on
+`{1, 3, 5, 7}`. There are `2^4 = 16` such signs: one for each subset. The
+empty subset is the unvaried pair. The other 15 are the odd-sign patterns on
+this `u`, with `v` held at mask 17 and with the even minus signs held on
+coordinates 0 and 2. Each pattern preserves the permutation `(0 2)(4 6)`,
+keeps the product permutation `(0 6)(2 4)`, and uses the same 64 operators.
+On the trivial character, all 16 patterns have `X_3` traces
+`-48, 16, 16, 16` in the same eigenvalue order. For every character `χ`, the
+traces and dimensions equal the unvaried traces and dimensions at eigenvalue
+`(ε χ(D), δ)`.
+
+That is the complete list of odd signs that can be added to this `u` without
+changing its even signs. The verifier checks the signed-permutation identity,
+the fresh address trace of every element, and the joint formula, for every
+pattern and every character.
+
+## What is not claimed
+
+The argument uses this query, Bell block 3, the operator `X_3`, the group
+`H`, the permutations of the explicit pair, and odd signs added to `u` only.
+It does not change the mask of `v`. It does not evaluate the other legal
+even-cycle pattern on the same permutation, the pattern with minus signs on
+coordinates 4 and 6. Flipping the sign on the single even coordinate 0 is a
+different operation: that signed permutation still preserves the query, and
+it sends sector 3 to 470, so it is not an endomorphism of this block.
+
+The degree-16 polynomial of a weight-4 summand, and the question of whether a
+nontrivial odd-sign pattern splits that polynomial, are untouched. The
+characteristic polynomial of the 84-dimensional `X_3 = +1` subspace is
+untouched. No trace of `Z_508` was recomputed. A trace that stays equal to
+`-48` on one joint space does not say that every cross-coset pair has this
+trace, and it does not say that no other mask can move it. The selected
+conditional kappa is unchanged.
+
+The rejection controls are part of the verifier, not part of the theorem. The
+identity trace on the same 216-space is 216. Dropping one sign whose
+contribution to the `(+1, +1)` numerator is nonzero does not reproduce `-48`.
+Translation by 1 does not commute with the mask-7 representative. The
+coordinate transposition `(0 1)` is not a symmetry of the query.

research/c3-odd-sign-trace-slot/README.md

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+# An odd minus on u keeps the X_3 trace -48
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+
+On the explicit c=3 pair from the sign-split certificate, `X_3` has trace
+`-48` on the trivial `(+1, +1)` joint space. Adding the minus sign on
+coordinate 1 to `u`, and leaving `v` and the even signs unchanged, keeps that
+trace on the eigenvalue `(+1, +1)`. The same holds for every odd sign added
+to this `u`. The selected conditional kappa stays `781451/250000000`.
+[PROOF.md](PROOF.md) separates the trace-formula proof from the facts that
+are not claimed.
+
+## Result
+
+The live input is the query pinned in [SOURCES.json](SOURCES.json), SHA256
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`.
+The unvaried pair is
+
+    u = (0 2)(4 6) with minus signs on coordinates 0 and 2,
+    v = (0 4)(2 6) with minus signs on coordinates 0 and 4.
+
+Its product is `(0 6)(2 4)` with mask 65. The assigned variation is the
+composition of `u` with the diagonal sign on coordinate 1. The new mask is 7,
+the product permutation is unchanged, and the product mask is 67.
+
+On the trivial character, the four joint traces of `X_3` remain
+`-48, 16, 16, 16` for eigenvalues
+`(+1,+1), (+1,-1), (-1,+1), (-1,-1)`. The value `-48` remains on
+`(+1, +1)`. That space still has dimension 216, so the `X_3` eigenvalues
+there are still dimensions 84 and 132. The same four traces occur for all 16
+diagonal signs supported on `{1, 3, 5, 7}`, with this `v` and these even
+signs held fixed. The verifier enumerates those 16 signs, the 16 characters
+of `H`, and all 512 addresses.
+
+The sign change moves a different trace. On the weight-1 character that is
+`-1` on coordinate 1, the trace 36 and the dimension 180 move from
+`(+1, +1)` to `(-1, +1)`. The value `-48` does not occur on that character.
+The weight-1 character supported on coordinate 3 keeps the unvaried row
+`(36, -12, -12, -12)`.
+
+This is a proof. The address counts are exhaustive on 512 elements, once by
+enumeration and once by an affine solve, and the two counts agree. The odd
+signs are all `2^4` subsets of the four odd coordinates. It is not a stopped
+search, and it is not a claim about every cross-coset pair.
+
+## Scope
+
+Catalogue card `formal-interface-lemma` asks for a formal interface binding.
+The live question is open question 2 of
+[PR #7020](https://github.com/CrocSwap/integer-mult-bounds/pull/7020),
+recorded in discussion comment
+[19514155](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19514155).
+The first step named there is to repeat the trace with one odd minus sign
+added to `u`. The masks and the unvaried pair come from
+[PR #6965](https://github.com/CrocSwap/integer-mult-bounds/pull/6965).
+
+This package does not redo the payload-involution split. It does not compute
+the characteristic polynomial of the 84-dimensional subspace, and it does not
+factor the degree-16 weight-4 polynomial of
+[PR #7021](https://github.com/CrocSwap/integer-mult-bounds/pull/7021).
+The mask of `v` stays 17. The other legal even-cycle pattern, minus signs on
+coordinates 4 and 6, is not evaluated.
+
+Selected-baseline orientation returned `PASS_RESEARCH_BASELINE_BINDINGS_ONLY`
+for kappa `781451/250000000`.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+python3 -B -W error research/c3-odd-sign-trace-slot/verify.py
+python3 -B -O research/c3-odd-sign-trace-slot/verify.py
+python3 -B scripts/research_package.py --package research/c3-odd-sign-trace-slot --check-only
+```
+
+The direct marker is `PASS_C3_ODD_SIGN_TRACE_SLOT`. Acceptance uses explicit
+checks, so `-O` does not skip them. On the measured host the direct run takes
+about two seconds. No Lean, Docker, degree-881 coefficient replay, or
+selected cold replay was run.
+
+## Overlap
+
+`scripts/find_overlap.py` was refreshed on 2026-11-10 against upstream
+`17469fff78c22941280f04d39dc5a1faee09c840`. Coverage: local notes 10178 of
+10178 HEAD paths, complete; 339 open PRs, pagination complete, fetched
+2026-11-10T10:28:46Z; 7013 discussion records, pagination complete, fetched
+2026-11-10T10:29:46Z. Closed PRs were searched through local READMEs, task
+metadata and the ledger only. The search terms were `trace -48`,
+`odd minus`, and `cross-coset`.
+
+The matches are the predecessor statements and one unrelated use of the words
+"cross-coset". PR #7020 records the trace `-48` and leaves other cross-coset
+sign patterns unevaluated. PR #7021 asks whether a nontrivial odd-sign pattern
+splits a degree-16 weight-4 polynomial; that polynomial is not this trace.
+No open PR body reports the joint trace of `X_3` after an odd sign is added
+to this pair. This search is not a novelty certificate.
+
+## Open next questions
+
+The structured list is in [TASK.json](TASK.json).
+
+- One odd minus added to `v`, with the mask of `u` held at 5, was not
+  evaluated. The reindexing above uses a sign added to `u`. The smallest
+  check is the same joint formula with the minus on coordinate 1 added to
+  `v` instead.
+- The other legal even-cycle pattern on `(0 2)(4 6)`, with minus signs on
+  coordinates 4 and 6 and no odd minuses, was not evaluated. It is one of
+  the 32 stabilizer patterns of PR #6965, and it is not an element of `H`.
+
+## Credit
+
+The explicit pair, the free odd signs, and the joint-dimension table are the
+certificate in upstream PR #6965. The trace `-48`, the 84+132 split, and the
+question answered here are upstream PR #7020, head
+`6025743bc95e096845af0962ba0c4c37cce89c4e`. Krzysztof Wojt's
+`research/exact-one-slot-gram/` defines `X_3`. Ali Romman's #4958 and #4720
+retain the query and the Bell action. Nick Petalas's #5120 is the degree-410
+lower bound. JoyboyBrian's handoff recorded the sign-character dimensions.
+The doubled representation comes from #1719. New work is Apache-2.0.
+
+## Workflow feedback
+
+The named open question fixed the masks and named the one sign to add, so
+the trace stayed a sum over the existing group of 64 operators.
+https://github.com/CrocSwap/integer-mult-bounds/discussions/1200

research/c3-odd-sign-trace-slot/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "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/c3-centre-sign-split/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "8da44a79e0e3f7d3cfcd7ebb8c46cc46146a9869",
+      "sha256": "6d9446760eb90201b96a73c78a04fdc0320c0c3973dc9d96f05a7fd135a45558"
+    },
+    {
+      "path": "research/centre-polynomial-cache-limits/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json",
+      "usage": "input",
+      "git_blob": "4201e9a04c12273b919e2cae60a6b6482c8150f3",
+      "sha256": "5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3"
+    },
+    {
+      "path": "research/exact-one-slot-gram/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "91854803c13d19d7962aafedf37aca0004827514",
+      "sha256": "30ccfa6d6623313a3c73c7107b7d9c3872eeffc07d2062e7e9b797162585e9b4"
+    }
+  ]
+}