Classify cube-zero local algebra isomorphisms (no new κ)

research/cube-zero-gorenstein-isomorphism/PROOF.md

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+# A complete restricted recovered-algebra isomorphism test
+
+Produced by Nick Petalas. Apache-2.0.
+
+## 1. Contract and multiplication form
+
+Let K be a finite field of odd order q. Let A be a finite-dimensional,
+commutative, associative, unital local K-algebra, with radical J, residue
+A/J=K, nonzero one-dimensional J^2 and J^3=0. Require its socle
+Ann_A(J) to equal J^2. Put r=dim_K J/J^2. Necessarily r>=1.
+
+Choose a nonzero s in J^2 and a K-linear complement V of J^2 in J.
+Then A=K*1 direct-sum V direct-sum K*s. Since J^3=0, there is a symmetric
+bilinear form B on V such that
+
+```text
+uv = B(u,v)s,   us = s^2 = 0,  for u,v in V.
+```
+
+The form is nondegenerate. A vector in its radical annihilates J, so would
+belong to the socle J^2 while lying in its chosen complement. Conversely,
+every nondegenerate symmetric r-dimensional B defines such an algebra A_B.
+Associativity follows because every product of three elements of J is zero.
+The socle contains no element with nonzero constant coefficient, and its
+intersection with V is zero by nondegeneracy. Thus it is exactly K*s.
+
+Adding multiples of s to the chosen lifts does not change B. A change of
+basis in V replaces B by a congruent form; rescaling s multiplies B by a
+nonzero scalar. These choices are the only ambiguity relevant below.
+
+## 2. Isomorphism is similarity of forms
+
+A unital K-algebra isomorphism preserves J and J^2. Between A_B and A_D
+it therefore has the shape
+
+```text
+1 -> 1,    u -> P u + ell(u)s,    s -> c s,
+P invertible, c nonzero.
+```
+
+Multiplication preservation is equivalent to
+
+```text
+P^T D P = c B.                                      (1)
+```
+
+Every such P,c and every linear ell give an isomorphism. In particular,
+set ell=0 to construct a transporter. Thus the complete invariant is the
+similarity class of the nondegenerate symmetric form, not its congruence
+class with a fixed socle generator.
+
+## 3. Elementary classification and constructive reduction
+
+We include the classical form argument to specify the maps exactly.
+Every nondegenerate symmetric form over odd K has an orthogonal basis.
+If every diagonal entry in a nonzero residual block is zero, a nonzero
+off-diagonal entry yields a vector e_i+e_j of norm 2B_ij != 0.
+Split off that vector and continue on its nondegenerate orthogonal
+complement. This terminates after r steps.
+
+For any nonzero a,b in K, the sets
+
+```text
+{a*u^2 : u in K},  {1-b*v^2 : v in K}
+```
+
+both have size (q+1)/2, so intersect. Choose u,v with au^2+bv^2=1.
+In diag(a,b), the vectors (u,v) and (-bv,au) have respective norms
+1 and ab and are orthogonal. Their basis determinant is one. Apply this
+two-coordinate reduction successively to obtain diag(1,...,1,d).
+The square class of d equals that of the original determinant, since
+congruence changes the determinant by a square. Scale the last vector
+to make d either 1 or one fixed nonsquare nu. This proves both existence
+and necessity of the two congruence classes, including r=1.
+
+Scaling the entire form by c changes its determinant square class by c^r.
+For even r it changes nothing. Therefore precisely two similarity classes
+remain, represented by diag(1,...,1) and diag(1,...,1,nu).
+
+For odd r, multiplying a nonsquare-determinant form by nu makes the
+determinant square; a square-determinant form needs no scaling. Apply
+the congruence reduction to get the identity. Hence all forms of that odd
+dimension are similar. The exact number of algebra isomorphism classes is
+
+```text
+1 if r is odd;
+2 if r is even, distinguished by the square class of det(B).
+```
+
+The distinction cannot be removed by changing lifts or the generator of
+J^2 when r is even, by equation (1). For odd r, retaining the determinant
+square class would falsely reject isomorphic algebras.
+
+The implementation returns Q and h with Q^T B Q=hD, where D is the
+canonical representative. The map from the canonical algebra to A_B sends
+its V basis to the columns of Q and its socle generator to h*s. The
+displayed equation verifies every product. Composing the two canonical
+charts gives a map between the original tables; every basis product is an
+independent checkable certificate. A negative answer uses the invariant,
+without a search over candidate maps.
+
+## 4. Recover the form from an arbitrary supplied table
+
+Validate the algebra laws and recover the unique unit by linear equations.
+For a finite-dimensional commutative local algebra, J is nilpotent and
+J^n=0 for n=dim A. One way to see the exponent bound is that the radical
+power chain strictly decreases until zero; equality persists and contradicts
+nilpotence. These are classical finite-dimensional local-algebra facts.
+
+Choose q^e>=n. The q^e-power map on A is K-linear and has kernel exactly J:
+it kills J, while the residue of a unit remains nonzero. Compute this kernel
+by linear algebra. Check codim J=1, dim J^2=1 and J*J^2=0. Choose s from
+J^2, extend it to a basis of J, and express all lift products in that basis.
+Reject if the resulting symmetric B is singular, which by section 1 is
+equivalent to the socle being larger than J^2. This recognizes exactly the
+declared class and constructs its chart. It does not recognize every local
+algebra.
+
+For recognition before locality is known, in any finite-dimensional
+commutative unital K-algebra the same power kernel is the nilradical: every
+nilpotent is killed by exponent n and no nonnilpotent has a zero power.
+Codimension one makes the reduced quotient K, so the algebra is local.
+This justifies using that check without a locality promise on the input.
+
+The mathematical algorithm works over any explicit odd finite field.
+Searching at most q^2 pairs for each binary reduction and at most q scalars
+for a square root gives polynomial arithmetic complexity in n for fixed K.
+No complexity polynomial in log q is claimed. The bounded implementation
+uses only odd prime fields 3<=p<=31 and table dimensions 3<=n<=10; the
+written proof has neither bound. The verifier exercises exact forms and
+arbitrary-basis algebra tables within those executable limits.
+
+## 5. Consequence for the recovered multiplication tensor
+
+Ali Romman's PR #5540 reconstructs a general commutative unital algebra and
+three explicit factor maps from a qualifying square prime-field tensor.
+That theorem and its full source lineage are cited premises here. Applying
+section 4 to its recovered table decides whether that table belongs to our
+class. Applying sections 2 and 3 to two such tables decides their algebra
+isomorphism and constructs the missing map when it exists.
+
+For completeness, independent input/output tensor equivalence introduces
+no further classes. Suppose invertible K-linear F,G,H satisfy
+
+```text
+F(x)G(y) = H(xy)
+```
+
+between two commutative unital algebras. Put a=F(1), b=G(1). Then
+F(x)b=H(x) and aG(y)=H(y). Since all maps are bijective, multiplication
+by a and by b is surjective, so a,b are units. Also H(1)=ab and
+H(x)H(y)=ab H(xy). Consequently phi(x)=(ab)^(-1)H(x) is a unital algebra
+isomorphism. Conversely any algebra isomorphism gives a tensor equivalence.
+This elementary unital-isotopy argument proves that the signatures above
+also classify the multiplication tensors. It does not perform or recheck
+the predecessor's nucleus reconstruction.
+
+## 6. Evidence and limits
+
+The proof is the all-dimension claim; finite checks corroborate its
+construction and class boundary. `verify.py` enumerates all symmetric forms
+for (p,r)=(3,1..4),(5,1..3),(7,1..2). For each nonsingular form it verifies
+the exact similarity certificate, with a permutation-sum determinant oracle.
+An independent complete orbit generator enumerates GL2 at p=3,5,7 and GL3
+at p=3, including every nonzero scalar. It compares the entire resulting
+partition with the new signature. New changed-basis algebra tables check
+the reconstruction and complete isomorphism maps on every basis product.
+
+Negative controls include even characteristic, composite primes, malformed
+coefficients/tables, noncommutativity, nonassociativity, absence of a unit,
+wrong residue, zero radical square, nonzero radical cube and excess socle.
+Transporter mutations change the image of the recovered unit, so are actual
+mathematical corruptions rather than arbitrary changes that might give
+another valid isomorphism. All acceptance checks survive optimized Python.
+
+This result supplies an algebraic interface only. It leaves physical
+coefficient preparation, native realization, storage, movement, dirty-state
+restoration, precision and a complete paid recursive caller unknown. It
+does not improve kappa, change the selected frontier or prove the complete
+multiplication theorem. Independent human and proof-kernel review are
+unclaimed. No historical scientific replay is used as the deliverable.
+
+## Credit
+
+The new specialization and implementation were produced by Nick Petalas.
+Credit Ali Romman #5540 for the explicit isomorphism handoff and recovered
+table interface, apoursh #5502, Ali Romman #5525, Nick Petalas #5252,
+David Ganz and their retained tensor/algebra predecessors. Finite-field
+symmetric-form reduction, radical theory and unital isotopy are classical.
+No general mathematical or literature priority is asserted.

research/cube-zero-gorenstein-isomorphism/README.md

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+# Isomorphism of cube-zero local multiplication tensors
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+For an odd finite field K, consider commutative unital local K-algebras A with
+residue field K and radical J satisfying
+
+```text
+J^3 = 0,  dim_K J^2 = 1,  Ann_A(J) = J^2.
+```
+
+Put r=dim_K(J/J^2), which is positive. There is exactly **one isomorphism
+class for each odd r**, and **two for each even r**. Recover the symmetric
+multiplication matrix B on J/J^2 using any generator of J^2. When r is even,
+the two classes are distinguished by whether det(B) is a square. When r is
+odd, rescaling the socle removes that distinction. The same classification
+holds for isotopy of their multiplication tensors.
+
+This is a scoped written algebraic proof with exact finite corroboration.
+It answers one restricted part of the recovered-algebra isomorphism question
+in Ali Romman's [PR #5540](https://github.com/CrocSwap/integer-mult-bounds/pull/5540).
+That reconstruction theorem is a cited premise, not a result rechecked here.
+The new package supplies the specialization, explicit maps and bounded
+implementation. Symmetric-form classification and its elementary methods
+are classical; no general literature-priority claim is made. No new kappa.
+
+[PROOF.md](PROOF.md) gives the reduction, classification, constructive
+congruence algorithm and tensor implication. `algebra.py` accepts supplied
+tables in arbitrary bases over odd primes 3 through 31, in dimensions 3
+through 10. It checks the algebra laws and class hypotheses, computes a
+canonical signature and chart, and returns a literal isomorphism between
+accepted tables with the same signature. Different signatures return `None`;
+invalid or out-of-class inputs raise `ValueError`.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/cube-zero-gorenstein-isomorphism/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/cube-zero-gorenstein-isomorphism/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cube-zero-gorenstein-isomorphism
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cube-zero-gorenstein-isomorphism --check-source-history
+```
+
+Other machines may substitute their CPython 3.12 interpreter path. The direct
+checker prints `PASS_CUBE_ZERO_GORENSTEIN_ISOMORPHISM` in a JSON report with
+enumerated coverage and SHA256 hashes of every package source/document file.
+The default replay writes no files. No receipt is handwritten.
+
+The finite corroboration checks **51,200 nonsingular forms**. Independent
+full GL/similarity orbit generation covers 18, 100 and 294 forms in dimension
+two over F3, F5 and F7, and all 468 forms in dimension three over F3. It
+enumerates respectively 48, 480, 2,016 and 11,232 invertible matrices.
+The new 96 arbitrary-basis algebra presentations check 4,128 transported
+basis products. All 96 unit-image transporter mutations and 12 input/class
+rejection controls pass. These are new fixtures, not predecessor inventories
+or scientific replays. The proof supplies the universal quantifiers.
+
+## Scope and research decision
+
+The original `formal-interface-lemma` card is explicitly changed to this
+written algebraic side result. The current selected source-binding command
+passed; it is metadata orientation only. The actual caller is the recovered
+table interface of #5540. The contribution closes isomorphism for this
+specified class, without claiming the card's proof-assistant acceptance gate.
+Stop after the classification. Physical maps, costs and supplier admission
+remain unpriced and provide no basis for a construction expansion.
+
+The live overlap check included open PRs, this account's recent PRs, handoff
+comments, active Discussions, main-tree README searches and merged searches
+for the class and its invariants. The cached helper searched only a sparse
+subset of local notes; Git-object searches supplemented it. Absence of a
+match is not a global novelty certificate. The advance claim is
+[Discussion #1166](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19220065).
+
+The proof covers all odd finite fields and all positive r. The executable
+supports the bounded prime-field interface above. Characteristic two,
+nonsplit residue fields, larger socles, higher nilpotency, arbitrary algebra
+isomorphism, native/frame/ownership/restoration costs, precision, movement,
+complete recursive invoices and the multiplication theorem are outside scope.
+No selected cold replay, predecessor scientific suite, Lean or Docker build
+was run. Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+1. Extend this particular isomorphism interface to characteristic two. Start
+   with the alternating versus nonalternating multiplication form; the odd
+   pivot and square-class algorithm here does not apply. A separate proof
+   and field contract are needed.
+2. Price the returned maps for one concrete native caller. First identify
+   its legal coefficients, dirty state and required restoration, then charge
+   every conversion. The current algebraic maps have no physical tariff.
+
+## Credit
+
+Credit Ali Romman #5540, exact head
+`831b41da0fc09e3e4cb983c03662da5df7970f68`, for the recovered-table interface
+and explicit isomorphism question. Credit apoursh #5502 for the supplied-table
+recognition lineage, Ali Romman #5525 for the nucleus reconstruction lineage,
+and Nick Petalas #5252, David Ganz and the retained tensor/algebra sources.
+Their files and credit lines are unchanged. `SOURCES.json` records historical
+provenance at canonical main `19fdb086ea96af9a9f87936a2b888e10af4ed810`,
+priority revision 8. No predecessor code is copied or executed.
+
+## Implementation constraints
+
+All checks use explicit exceptions and remain enabled under optimized Python.
+There is no recursion, dynamic code execution or file-writing path. Loops
+are bounded by the validated dimensions and prime, or by the verifier's
+literal finite domains. The largest form enumeration has 3^10 candidates;
+the largest GL enumeration has 3^9 candidates. Functions stay below 60 lines.
+Python's managed allocation is a Power of Ten deviation: the table API caps
+dimension at 10 and prime at 31, and each exhaustive loop discards its current
+candidate. The finite orbit dictionary is bounded by its enumerated symmetric
+form space. No fixed-memory or general runtime-speed claim is made.

research/cube-zero-gorenstein-isomorphism/SOURCES.json

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