Record the characteristic polynomial of the 84-dimensional X_3=+1 subspace

research/c3-x3-plus-charpoly/PROOF.md

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+# Characteristic polynomial of the 84-dimensional X_3=+1 subspace
+
+This is a finite exact computation for one invariant subspace of Bell block
+`c=3` of the historical one-slot doubled centre operator. It is not a new
+minimal polynomial of the full operator, not a change of the selected
+conditional kappa, and not a census of payload involutions outside the named
+pair.
+
+The claim is a **proof**. The coordinate set has 4096 elements. The basis
+below classifies every coordinate. The characteristic polynomial is an exact
+determinant, the factorisation is checked by multiplication and an exhaustive
+integer search, and the minimal polynomial is identified by an exact
+annihilator. None of these steps is a stopped search.
+
+## Operator
+
+The input is the 8-by-112 query in `INTERFACE.json`, SHA256
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`.
+Its columns lie in `{-1,0,1}^8`, each has weight 4, the 112 port labels are
+distinct odd-weight vectors in `F_2^9`, and `R R^T = 56 I_8`.
+
+On `Q^8 ⊗ Q[F_2^9]`, of dimension 4096, the `c=3` block and its integer form are
+
+```text
+A_1 = (1/4) sum_q r_q r_q^T ⊗ P_q(3),
+B   = 4 A_1 = sum_q r_q r_q^T ⊗ P_q(3).
+```
+
+`P_q(3)` sends `e_a` to `e_{a+q}` when the `F_2` dot product `q·3` equals 1,
+and to `(-1)^{q·a} e_a` otherwise. Basis index `i*512+a` means query
+coordinate `i` and address `a`.
+
+Krzysztof Wojt's `research/exact-one-slot-gram/` gives the minimal polynomial
+of the full one-slot operator, of degree 881. This note does not replay that
+certificate. It uses the two published `c=3`, `s=+1` sector coefficient lists
+only for the divisibility comparison below. Those lists are the degree-126
+polynomials for `(t,s) = (0,+1)` and `(1,+1)`.
+
+## The subspace
+
+`H` is the order-16 group of diagonal signs that are positive on the even
+coordinates `{0,2,4,6}` and arbitrary on the odd coordinates `{1,3,5,7}`.
+The explicit pair from the c=3 sign-character certificate is
+
+| name | permutation | minus signs |
+|---|---|---|
+| `u` | `(0 2)(4 6)` | coordinates 0 and 2 |
+| `v` | `(0 4)(2 6)` | coordinates 0 and 4 |
+
+Each element is rebuilt from the query as a signed coordinate permutation.
+All 64 products are symmetries, they fix sector 3, and `u` and `v` are
+commuting involutions. The payload involution `X_3` translates every address
+by 3 and fixes the eight coordinates. It commutes with all 64 products and is
+not equal to any of them. `Z_508` is the diagonal character of `3 xor 511`.
+
+Let `V` be the set of vectors fixed by every element of `H`, by `u`, and by
+`v`, and fixed by `X_3`. This is the `X_3 = +1` subspace of the trivial
+`(+1,+1)` joint space. The same equations without `X_3` have an explicit
+basis of dimension 216. The trace of `X_3` on that 216-space, computed as the
+average of the address-and-coordinate traces of `X_3 g` over the 64 group
+elements, is `-48`. The projector onto `V`, averaged over the order-128 group
+generated by `H`, `u`, `v`, and `X_3`, has trace 84. The explicit basis of the
+same equations also has dimension 84, and the `X_3 = -1` equations have
+dimension 132. Imposing `v = -1` instead of `v = +1`, with `X_3 = +1`,
+produces dimension 116, so the basis is the `(+1,+1)` joint eigenvalue rather
+than one of the other three.
+
+The basis is canonical. The equations `v_{ρ(j)} = σ_j v_j`, as `ρ` runs
+through the 64 group elements and through `X_3`, partition the coordinates
+that are not forced to zero. In each component the representative is the
+least index, its coefficient is `+1`, and every other coefficient is the
+unique sign forced by the equations. Components are ordered by increasing
+representative. The basis is stored in `evidence/basis.json`. Support sizes
+are exactly 8, 32, and 64.
+
+`Z_508` preserves `V` and splits it into dimensions 42 and 42, for eigenvalues
+`+1` (`t=0`) and `-1` (`t=1`).
+
+## The matrix
+
+`B` preserves `V`. Two independent expansions, one by the 4096 columns and
+one by the 112 ports, agree on every basis vector of `V` and of both
+`Z_508` eigenspaces. In the ordered basis, the matrix `M` of `B` has integer
+entries in `-28..28`. It is stored in `evidence/matrix.json`. If `size_r` is
+the support of the `r`-th basis vector, then `size_r M_{r s} = size_s M_{s r}`.
+
+The trace of `M` is 0. The same trace is the average of `tr(B g)` over the
+order-128 group, so the matrix trace agrees with the trace formula. The two
+`Z_508` blocks have traces 168 and -168.
+
+The matrix of `A_1` on `V` is `M/4`.
+
+## Characteristic polynomial
+
+Let `chi_B(x) = det(x I - M)` and `chi_{A_1}(x) = det(x I - M/4)`. Then
+`chi_{A_1}(x) = 4^{-84} chi_B(4x)`, and both polynomials are monic of degree
+84 in `Z[x]`. The SHA256 hex digests of the JSON arrays of ascending
+coefficients are
+
+```text
+chi_A  c4bcbd4e52f92d13f3ad451d92c9d22256e7725e4be24ee203bea1c96c129648
+chi_B  185b51cfd084f95c1dbd814b0b211ce4ab35b2d59604d7795715d65a0a4e5e65
+```
+
+The irreducible factorisation over `Z` is
+
+```text
+chi_{A_1}(x) =
+    x^16
+  * (x - 14)
+  * (x - 12)
+  * (x - 4)^6
+  * (x - 2)^14
+  * (x + 2)^14
+  * (x + 4)^6
+  * (x + 12)
+  * (x + 14)
+  * (x^4 - 16x^2 - 8x + 16)^2
+  * (x^4 - 16x^2 + 8x + 16)^2
+  * (x^4 - 12x^3 + 20x^2 + 32x - 32)
+  * (x^4 + 12x^3 + 20x^2 - 32x - 32).
+```
+
+The two `Z_508` blocks have
+
+```text
+t = 0:
+    x^8 (x - 14) (x + 12) (x - 4)^6 (x + 2)^6 (x - 2)^8
+    * (x^4 - 12x^3 + 20x^2 + 32x - 32)
+    * (x^4 - 16x^2 + 8x + 16)^2
+
+t = 1:
+    x^8 (x - 12) (x + 14) (x - 2)^6 (x + 4)^6 (x + 2)^8
+    * (x^4 + 12x^3 + 20x^2 - 32x - 32)
+    * (x^4 - 16x^2 - 8x + 16)^2.
+```
+
+Their product is `chi_{A_1}`. Every irreducible factor of `chi_B` divides at
+least one of the two published degree-126 minimal polynomials of the `c=3`,
+`s=+1` sectors in `research/exact-one-slot-gram/CERTIFICATE.json`. The hits
+are recorded by `SECTOR_HITS` in `verify.py`. Those two polynomials are among
+the sector polynomials whose monic LCM is the degree-881 minimal polynomial
+of `B` in that certificate. This comparison reads coefficients. It does not
+recompute the LCM and it does not replay the degree-881 certificate.
+
+The degree-4 factors are irreducible over `Z`: the rational-root theorem
+gives no root, and an exhaustive search of integer pairs `(b,d)` with
+`bd` equal to the constant term finds no quadratic factorisation. The search
+does reject `(x^2+1)^2`, so the negative test is not vacuous. Linear factors
+are irreducible because they have degree 1.
+
+## Minimal polynomial and rational eigenvalues
+
+Every irreducible factor of a characteristic polynomial divides the minimal
+polynomial. The square-free product of the factors above has degree 25. The
+corresponding monic integer polynomial in `B`, obtained by sending each
+factor `f(x)` of degree `d` to `4^d f(y/4)`, evaluates to the zero matrix on
+`M`. Therefore that degree-25 polynomial annihilates `A_1` on `V`, and it is
+the minimal polynomial of `A_1` on `V`. The same argument on each 42-by-42
+block shows that the square-free part of that block's characteristic
+polynomial, of degree 14, is the minimal polynomial of the block.
+
+On this matrix, each rational eigenvalue of `A_1` has geometric multiplicity
+equal to its algebraic multiplicity. For eigenvalue `λ`, the verifier
+exhibits `m` integer vectors in the kernel of `M - 4λ I`, checks that each
+image is zero, and checks that their free coordinates form a diagonal
+submatrix, so they are independent. The algebraic multiplicity is an upper
+bound, so the geometric multiplicity is exactly `m`:
+
+| eigenvalue of `A_1` | multiplicity |
+|---:|---:|
+| 0 | 16 |
+| 2 and -2 | 14 each |
+| 4 and -4 | 6 each |
+| 12 and -12 | 1 each |
+| 14 and -14 | 1 each |
+
+These nine eigenvalues account for 60 of the 84 dimensions. The remaining 24
+dimensions belong to the four quartic factors. Because each repeated quartic
+occurs to the first power in the minimal polynomial and to the second power
+in the characteristic polynomial, its primary part is two companion blocks of
+degree 4. Each simple quartic is one companion block of degree 4. Each
+rational eigenvalue is a diagonal of 1-by-1 blocks. This is the rational
+canonical form of `A_1` on `V`. The package does not list cyclic generators.
+
+## Verification
+
+`verify.py` uses CPython 3.12 and the standard library. It rebuilds the
+matrix from the query, computes each characteristic polynomial by the
+Hessenberg recurrence modulo primes whose product exceeds twice the
+Gershgorin binomial coefficient bound, and lifts by the Chinese remainder
+theorem. Determinants of `tI-M` at `t=0,1,2,3,5` agree with that polynomial
+modulo the first prime.
+
+Rejection controls refuse the coordinate transposition `(0 1)`, the `c=1`
+block operator on this subspace, translation by 1 in place of `X_3`, a
+flipped group sign that still claims dimension 84, a one-entry change of `M`
+that preserves the polynomial modulo `1000003`, a missing factor, and a
+one-coordinate sign error in a linked basis vector.
+
+Not run: Lean, Docker, the degree-881 coefficient replay, and the selected
+cold replay. Selected-baseline orientation returned
+`PASS_RESEARCH_BASELINE_BINDINGS_ONLY` for kappa `781451/250000000`.
+
+## What this does not say
+
+An 84-dimensional invariant subspace does not replace the degree-881 minimal
+polynomial. The selected conditional kappa stays `781451/250000000`. The
+polynomial is for this query, this sign group, this explicit pair, and the
+eigenvalue `X_3 = +1`. It does not compute the 132-dimensional complement, and
+it does not move the odd signs on the pair. A factor that divides a published
+sector polynomial is not a new factor of the degree-881 polynomial.

research/c3-x3-plus-charpoly/README.md

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+# Characteristic polynomial of the 84-dimensional X_3=+1 subspace
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+
+For the historical 8-by-112 centre query, the trivial `(+1,+1)` joint space of
+the explicit c=3 pair has dimension 216. The payload involution `X_3` cuts it
+into dimensions 84 and 132. On the 84-dimensional eigenvalue-`+1` subspace,
+the monic characteristic polynomial of `A_1` in characteristic zero is the
+explicit degree-84 polynomial in [PROOF.md](PROOF.md). Its minimal polynomial
+has degree 25 and is the product of the distinct irreducible factors.
+
+No new kappa. The selected conditional kappa stays `781451/250000000`.
+
+## Result
+
+Write `B = 4 A_1` on the `c=3` block. On this subspace,
+`chi_{A_1}(x) = 4^{-84} chi_B(4x)`, and both polynomials lie in `Z[x]`.
+The irreducible factorisation is
+
+```text
+x^16 (x-14) (x-12) (x-4)^6 (x-2)^14 (x+2)^14 (x+4)^6 (x+12) (x+14)
+  * (x^4 - 16x^2 - 8x + 16)^2
+  * (x^4 - 16x^2 + 8x + 16)^2
+  * (x^4 - 12x^3 + 20x^2 + 32x - 32)
+  * (x^4 + 12x^3 + 20x^2 - 32x - 32)
+```
+
+Ascending coefficient SHA256 values are
+
+```text
+chi_A  c4bcbd4e52f92d13f3ad451d92c9d22256e7725e4be24ee203bea1c96c129648
+chi_B  185b51cfd084f95c1dbd814b0b211ce4ab35b2d59604d7795715d65a0a4e5e65
+```
+
+The basis is the least-index signed orbit basis of the joint equations for
+`H`, `u`, `v`, and `X_3 = +1`, pinned in
+[evidence/basis.json](evidence/basis.json). The integer matrix of `B` in that
+basis is pinned in [evidence/matrix.json](evidence/matrix.json). Its entries
+lie in `-28..28`, and its trace is 0.
+
+`Z_508` splits the same 84-space into dimensions 42 and 42. The
+characteristic polynomial of the `t=0` block is
+
+```text
+x^8 (x-14) (x+12) (x-4)^6 (x+2)^6 (x-2)^8
+  * (x^4 - 12x^3 + 20x^2 + 32x - 32)
+  * (x^4 - 16x^2 + 8x + 16)^2
+```
+
+and the `t=1` block is
+
+```text
+x^8 (x-12) (x+14) (x-2)^6 (x+4)^6 (x+2)^8
+  * (x^4 + 12x^3 + 20x^2 - 32x - 32)
+  * (x^4 - 16x^2 - 8x + 16)^2.
+```
+
+Every irreducible factor of `chi_B` divides a published `c=3`, `s=+1` sector
+minimal polynomial from `research/exact-one-slot-gram/CERTIFICATE.json`.
+Those two degree-126 polynomials are part of the certificate whose monic LCM
+is the degree-881 minimal polynomial of `B`. The comparison reads
+coefficients. It does not replay the degree-881 certificate.
+
+Each rational eigenvalue of `A_1` on this matrix has geometric multiplicity
+equal to its algebraic multiplicity: 16 for 0, 14 for each of `±2`, 6 for
+each of `±4`, and 1 for each of `±12` and `±14`.
+
+## Scope
+
+Catalogue card `formal-interface-lemma` asks for a formal interface binding.
+The live question is open question 1 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).
+That pull request proves the trace split and states that the trace formula
+does not list a vector. This package builds the basis and computes the
+polynomial.
+
+It does not redo the payload-involution split, the signed-permutation
+classification, the 16-dimensional or 12-dimensional c=3 polynomials, or the
+216-dimensional c=1 polynomial. It does not change the odd signs on the pair.
+
+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-x3-plus-charpoly/verify.py
+python3 -B -O research/c3-x3-plus-charpoly/verify.py
+python3 -B scripts/research_package.py --package research/c3-x3-plus-charpoly --check-only
+```
+
+The marker is `PASS_C3_X3_PLUS_CHARPOLY`. Acceptance uses explicit checks, so
+`-O` does not skip a theorem or a rejection control. On the measured host one
+run takes about five seconds. No Lean, Docker, degree-881 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
+2026-11-10T10:24:45Z; 7013 discussion records, pagination complete, fetched
+2026-11-10T10:25:50Z. Closed PRs were searched through local READMEs, task
+metadata and the ledger only.
+
+The phrase `84-dimensional` matches open PR #7020, which records the
+dimension and says the characteristic polynomial is uncomputed, and open PR
+#7009, whose `84` is a c=1 character dimension. No open PR body gives this
+polynomial. This search is not a novelty certificate.
+
+## Open next questions
+
+The structured list is in [TASK.json](TASK.json).
+
+- The characteristic polynomial of the 132-dimensional `X_3 = -1` complement,
+  inside the same trivial `(+1,+1)` joint space, is still uncomputed. The
+  same basis routine applies with the opposite eigenvalue. The published
+  comparison for that half is the pair of degree-88 sector polynomials.
+- One 66-dimensional intersection of that complement with a single eigenvalue
+  of `Z_508` is the smaller block. Its polynomial is not determined by the
+  degree-84 factorisation above.
+
+## Credit
+
+JoyboyBrian's handoff recorded the sign-character dimensions. The explicit
+pair and the signed-permutation boundary are upstream PR #6965. The trace
+`-48`, the dimensions 84 and 132, and the open polynomial question are
+upstream PR #7020. Ali Romman's PR4958 and PR4720 retain the query and the
+Bell action. Nick Petalas's PR5120 is the degree-410 lower bound. Krzysztof
+Wojt's exact one-slot package defines `X_3` and `Z_508` and is the degree-881
+annihilator and the sector coefficient list used for the divisibility
+comparison. The doubled representation comes from PR1719. The 216-dimensional
+c=1 polynomial in `research/a1-joint-216-charpoly/` is a different block; its
+Hessenberg recurrence is the model for the determinant. New work is
+Apache-2.0.

research/c3-x3-plus-charpoly/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"
+    },
+    {
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+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "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/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "96c3dc7a8bca73fb35b8a9ec49a17856308a029d",
+      "sha256": "29e0b0d86c7fd8929ad36d9de48c2b8b55500491576dc5ac1d0b1ff076d27483"
+    }
+  ]
+}