Prove new two-slot centre spectrum and degree at least 1093

research/equal-bell-two-slot-spectrum/PROOF.md

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+# Exact spectrum on the equal-Bell two-slot subspace
+
+Use the actual 112 port labels q in F2^9 and eight-entry root columns r_q
+from the pinned historical query. Put B_lambda=4 A_lambda, so
+
+```text
+B_lambda = sum_q r_q r_q^T tensor Gamma_q^tensor(lambda).
+```
+
+The root coefficient occurs once. The doubled representation, one-slot Bell
+formula and common positive-one vector of every Gamma_q are inherited from
+Ali Romman's PR4958 and the sources retained by PR5168. PR5168 proves that
+B_1 has minimal polynomial P of degree 881. Its complete proof is an inherited
+premise here. We authenticate its certificate and recompute its polynomial
+LCM and hash; we do not rerun its full upper/lower scientific replay.
+
+## A complete reduction for equal outer Bell labels
+
+In the one-slot Bell block c, the port acts on the remaining address a as
+X_q if q.c=1 and Z_q if q.c=0. Thus the two-slot joint block (c,d) is
+
+```text
+B_(c,d) = sum_q r_q r_q^T tensor P_q(c) tensor P_q(d).
+```
+
+For c=d, the payload factors commute. A second Bell basis is
+
+```text
+b_(u,v) = 2^(-9/2) sum_a (-1)^(a.v) e_a tensor e_(a+u).
+```
+
+The 512^2 vectors are orthonormal: distinct u have disjoint supports, and
+for fixed u the Walsh characters in v are orthogonal. X_q tensor X_q acts
+by (-1)^(q.v); Z_q tensor Z_q acts by (-1)^(q.u). Consequently every
+8-dimensional root space tensor b_(u,v) is invariant, and its exact matrix is
+
+```text
+M(c,u,v) = sum_(q.c=0) (-1)^(q.u) r_q r_q^T
+         + sum_(q.c=1) (-1)^(q.v) r_q r_q^T.
+```
+
+This gives all equal-label spectra, not just selected vectors. Let
+F(t)=sum_q (-1)^(q.t) r_q r_q^T. Then
+
+```text
+M(c,u,v) = (F(u)+F(u+c)+F(v)-F(v+c))/2.             (1)
+```
+
+The complete equal-label subspace has dimension 512^3*8=2^30. It is a
+proper invariant subspace of the full width-two space of dimension 2^39.
+Unequal outer labels are not classified by this formula.
+
+## Source symmetry and finite coverage
+
+PR5168 supplies 13 signed root permutations with orthogonal label maps T.
+The verifier reruns their exact all-port root/label correspondence and their
+512-label orbit computation. Orthogonality gives q.c=(Tq).(Tc), and the
+root sign disappears in r_q r_q^T. The same root permutation and two address
+permutations therefore conjugate B_(c,c) to B_(Tc,Tc).
+
+Every actual q has odd parity. Walsh Fourier on both remaining address
+spaces exchanges X_q and Z_q and conjugates (c,c) to (c+511,c+511).
+The source-verified orbit representatives and sizes are
+
+```text
+(0,2), (1,224), (3,224), (7,16), (30,14), (128,32).
+```
+
+This uses the predecessor's symmetries in a new two-slot implication; it
+does not claim a new symmetry discovery. Equation (1) evaluates all 512
+values of each plus and minus term. Duplicate matrices can be removed
+because every pair (u,v) is permitted independently. Thus the Cartesian
+product of the distinct plus and minus arrays has exactly the same spectrum
+as all 512^2 choices, with multiplicities omitted. It contains 190239 pairs
+across the six representatives. This is a complete finite cover of the
+specified equal-label subspace, not an exhaustive cover of width two.
+
+## Exact minimal polynomial
+
+Every M is a real symmetric integer matrix. It is diagonalizable, so its
+minimal polynomial over Q is the square-free part of its characteristic
+polynomial. Taking the monic LCM over all matrices gives the exact minimal
+polynomial on the direct sum. The computed degrees are:
+
+| c | Distinct plus/minus arrays | Matrix pairs | Characteristic polynomials | Minimal degree |
+|---:|---:|---:|---:|---:|
+| 0 | 273 / 1 | 273 | 11 | 9 |
+| 1 | 194 / 202 | 39188 | 271 | 191 |
+| 3 | 200 / 192 | 38400 | 286 | 251 |
+| 7 | 185 / 199 | 36815 | 111 | 87 |
+| 30 | 201 / 187 | 37587 | 110 | 81 |
+| 128 | 202 / 188 | 37976 | 97 | 67 |
+
+The LCM E has degree 255. RESULT.json contains each representative polynomial
+and E in ascending integer coefficient order, together with hashes of the
+complete characteristic-polynomial histograms. The square-free argument
+proves minimality as well as annihilation; no numerical spectral tolerance
+or random rank witness is used.
+
+The verifier computes every characteristic polynomial by exact trace
+recurrences and checks the final full matrix Horner remainder is zero.
+For all 886 distinct characteristic polynomials across the representatives,
+a separate symbolic determinant on a literal port-sum matrix agrees exactly.
+That second path uses Python/SymPy integers and does not use the trace
+recurrence or equation (1) to construct its matrix. SymPy and NumPy are
+trusted arithmetic libraries, not proof kernels. FLINT was used only for
+candidate discovery and is absent from the clean verifier environment.
+
+## New spectrum and higher-width consequence
+
+At c=1, u=0, v=2, the second standard root vector is an eigenvector of M
+with eigenvalue 32. Exact evaluation gives P(32)!=0. Thus A_2 has eigenvalue
+8, absent from A_1. In particular, the complete one-slot annihilator does
+not extend to all widths.
+
+More fully, deg(E/gcd(E,P))=212 and deg(lcm(E,P))=1093. The predecessor's
+common fixed vector embeds B_1 in B_2 and embeds B_2 in every B_lambda for
+lambda>=2. Hence each such minimal polynomial must be divisible by E and
+P, and has degree at least 1093. This is not an upper bound for B_2 or for
+any larger width. The new 212 degrees come from a complete equal-label
+classification, not a small sampled increase of a Hankel minor.
+
+If alpha I+beta A_lambda is invertible in characteristic zero, beta!=0,
+and q(A_lambda) is its inverse, then
+
+```text
+((alpha+beta t)q(t)-1)(A_lambda)=0.
+```
+
+The displayed polynomial is nonzero and has degree at most deg(q)+1.
+For lambda>=2, it follows that deg(q)>=1092. This concerns scalar polynomial
+inverses only. It gives no lower bound for general ADD circuits, rational
+expressions or matrix-valued encodings, and no physical tariff.
+
+## Arithmetic bounds and limits
+
+The query satisfies sum_q r_q r_q^T=56 I. Every signed sum M therefore has
+operator norm at most 56. The implementation also checks symmetry and the
+conservative maximum absolute row-sum bound 112 before each trace recurrence.
+For a symmetric 8-by-8 matrix of norm at most 112, the kth recurrence matrix
+has spectral norm at most binomial(7,k)*112^k. This follows by diagonalizing:
+each eigenvalue is, up to sign, the kth elementary symmetric function of
+the other seven eigenvalues. Each product dot sum and the subsequent trace
+is bounded conservatively by 64*binomial(7,k-1)*112^k. For 1<=k<=8 these
+bounds are all below 2^63. Coefficient additions also fit. Every division
+by k is checked exact. Thus NumPy signed int64 arithmetic cannot overflow
+in the fixed accepted domain. LCM and determinant arithmetic use unbounded
+integers. The maximum batch contains 202 matrices of size 8 by 8.
+
+The literal Bell check covers 286720 actual port/address columns across five
+nonzero representative fixtures. The all-label conclusion follows from the
+written Bell identity and complete source symmetry cover. Nine controls
+exercise type/monicity, matrix dimension/symmetry/norm, swapped sign labels,
+wrong outer label, missing half projections and a changed annihilator.
+Source and typed-result corruption checks are separate integrity controls.
+
+No full predecessor scientific replay, selected native/supplier replay,
+Lean, Docker or independent human review is claimed. The actual historical
+query is preserved; it is not a new selected #4610 supplier. All native
+access, precision, workspace, dirty restoration and complete recursive
+cost obligations remain open. The selected frontier and kappa are unchanged.
+
+## Attribution
+
+Krzysztof Wojt, with OpenAI Codex assistance, supplied PR5168's one-slot
+polynomial, source symmetries and executable model. Nick Petalas supplied
+PR5120's preceding exact lower certificate. Ali Romman's PR4958/PR4720
+supplied the actual query, Bell identity and fixed-vector lift; PR1719
+supplies the doubled representation. JoyboyBrian's released gate supplies
+prior symmetry findings. Preserve icekylinx's Gauge44 credit and the full
+native/helper/compiler and upstream lineage in the retained NOTICE.
+
+New equal-label reduction, spectrum classification and higher-width bound:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. No literature
+priority is claimed for Bell diagonalization or the trace recurrence.

research/equal-bell-two-slot-spectrum/README.md

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+# New centre Gram spectrum at two doubled slots
+
+**Exact equal-label classification; no new kappa.** The full equal-Bell
+subspace at width two has minimal-polynomial degree **255**. Its spectrum
+adds **212 degrees** to the inherited one-slot degree-881 polynomial.
+Every width at least two therefore has minimal-polynomial degree **at least
+1093**; every defined nonconstant affine scalar-polynomial inverse needs
+degree **at least 1092**. This is not a full width-two upper bound.
+
+The [proof](PROOF.md) reduces the 2^30-dimensional equal-label subspace to
+8-by-8 signed root sums and proves the six-representative cover. The
+[receipt](RESULT.json) contains exact polynomials, coverage counts and hashes.
+The [implementation](spectrum.py) checks all 190239 compressed matrix pairs,
+with an independent literal port-sum/symbolic determinant check for all
+886 distinct characteristic polynomials. A concrete new eigenvalue is **8**
+for A_2, at c=1, u=0, v=2 and the second root coordinate. It is absent from A_1.
+
+This answers the equal-label part of Krzysztof Wojt's
+[PR5168](https://github.com/CrocSwap/integer-mult-bounds/pull/5168) higher-width
+handoff in [Discussion5142](https://github.com/CrocSwap/integer-mult-bounds/discussions/5142#discussioncomment-19154307).
+The predecessor's full one-slot theorem is inherited, not independently
+reproved. Its exact certificate and source model are authenticated. Nick
+Petalas's PR5120 and Ali Romman's PR4958/PR4720 retain their roles; see the
+proof and unchanged predecessor notice for the complete source lineage.
+
+## Reproduce
+
+Use Python 3.12, `numpy==2.4.1` and `sympy==1.14.0`, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/equal-bell-two-slot-spectrum
+```
+
+The runner performs normal and optimized checks. Direct, Git-free commands:
+
+```sh
+python3.12 -I -W error -B research/equal-bell-two-slot-spectrum/verify.py
+python3.12 -I -O -W error -B research/equal-bell-two-slot-spectrum/verify.py
+```
+
+Expected marker: `PASS_EQUAL_BELL_TWO_SLOT_SPECTRUM`. The receipt is compared
+with exact JSON types. `--write` creates a new receipt exclusively and cannot
+overwrite one. Normal replay makes no files. The source digest is checked
+before execution. The verifier uses no floating-point spectral decisions
+and no FLINT dependency. Candidate discovery used python-flint 0.8.0 in a
+separate temporary environment; the complete clean verifier recomputes the
+polynomials without it.
+
+The scope is one exact finite spectral classification with a written
+all-positive-higher-width implication. No independent human or kernel review
+is claimed. No native physical decoder, storage, movement, precision, dirty
+restoration or complete supplier is priced. The current source-bound baseline
+passes on selected #4610; that metadata check is separate from this historical
+query. No selected frontier or multiplication exponent changes.
+
+## Open next questions
+
+1. Classify the unequal outer-label blocks (c,d), c!=d. The equal-label Bell
+   transform no longer diagonalizes all payload factors. Start with their
+   exact commutator structure and retain the root coefficient once. A full
+   width-two upper polynomial remains unknown.
+2. Determine whether a scalar-polynomial decoder is useful after all physical
+   costs, or provide a different decoder with explicit source/target and dirty
+   restoration contracts. This algebraic degree result supplies neither one.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance, Apache-2.0.