Research status
& verification
The manuscripts, datasets and atlas present one current research account. This page explains its classification scope, remaining questions and the limits of the available verification.
Audited continuation · 27 October 2026
R9_2_1607, R9_2_1975 and R9_2_1976 are excluded over the complex numbers. Seven rings remain open: two at rank six, four at rank eight and one at rank nine. All three exclusions passed a fresh mathematical audit.
The seven-case continuation adds restrictions, with no additional model or full ring exclusion. In particular, a separable algebra 1+X in a hypothetical pseudo-unitary R1660 category would produce R941; neither is constructed. Every GH(C6,2) Morita construction of R1768 is ruled out, for any associator and algebra. Independent human verification remains pending.
Current report · PDF · Individual proofs and exact replay ↗ · Frozen proofs and certificates · ZIP
26 October 2026 · rank twelve, multiplicity one
All 632 supplied rings are classified over ℂ: 480 general exclusions and 152 modeled rings, with none unresolved. Of these models, 151 are known unitary. R12_1_623 has a complex and pseudo-unitary model; its unitarity remains unproved. R12_1_561 and R12_1_628 share a new conductor-saturation exclusion. The construction of R12_1_623 extends uniformly to k congruent to 2 modulo 4.
Current manuscript · All 632 case records · Complete proofs, code and accepted replay
Verification scope
All source tensors, physical line IDs, structural filters, model scopes and exact coefficient maps are checked. Separate assistant proof reviews and the complete 32-check preceding audit passed; independent human verification and formalization remain pending. Complete source and producer logs are supplied, while full census enumeration was not independently repeated. That release had ten open cases; the 27 October continuation leaves seven.
Correctness revision · 25 October 2026
The current classification is unchanged. Exact replacement arguments identify R6_2_143, R8_2_820 and R8_2_869; a direct completion identifies R8_3_721 without census completeness. The revised manuscripts include these proofs. Code and replay defects are corrected and their scope is documented.
Findings, corrected proofs and exact commands ↗ · Original audit evidence · ZIP
Verification scope and remaining limits
All 1,433,410 supplied rank‑3–11 tensors passed exact fusion axioms and based-isomorphism checks. The original audit reviewed 11 main mathematical manuscripts, their operative criteria and model code, and all accepted routes. Three identification gaps were repaired rather than inferred from successful scalar calculations. The rank-nine command now covers the accepted baseline and phases two–five, with the omitted certificates supplied.
Historical failures and source identities are preserved. The original rank-six audit used a disclosed partly log-derived composite; the current correction route retains raw results and compatible immutable proof contexts. Complete upstream census enumeration and independent reproof of every external categorical theorem are not claimed. The audit checkpoint had ten open cases; the subsequent three exclusions leave seven. Independent human verification and formalization remain pending.
Audit and independent proof repair
The old TPE completeness premise for R8_3_436 remains withdrawn. A separate spherical-equivariantization and formal-codegree proof now establishes its general exclusion. Read the replacement proof and exact certificate →
Current classification · 7 open questions
Open the current case directory →
Classification counts and scope
Rank nine through multiplicity two has 2,225 fusion rings: 2,148 general exclusions, 76 unitary models and one unresolved case. Exact multiplicity two contributes 2,083 rings: 2,046 exclusions, 36 models and one unresolved.
Rank eight through multiplicity three has 4,403 general exclusions, 93 models and four unresolved cases. Its exact-multiplicity-three band has 3,507 exclusions, 16 models and four unresolved.
The other open cases are R6_4_155 and R6_8_1660. Rank ten is complete at multiplicity one, with 136 general exclusions and 58 unitary models. R10_1_189 is recognized inside a classical conformal-embedding module category. Rank seven is classified through multiplicity five, and rank eleven at multiplicity one has 155 exclusions and 27 models. Completeness and verification retain their explicitly stated scopes.
The remaining mathematical questions
Rank six · R6_4_155
Rank six · R6_4_155
The necessary center data determine a rank-52 candidate with a rank-26 modular factor. Every odd generalized pivotal rotation must factor through a cyclic spin character; all 1,144 tested blocks pass this condition. The all-order path-Clifford divisibility condition for the family R′(m) holds exactly when m is even, including m = 4. These constraints do not provide compatible associators or a categorification, and no general exclusion is established.
Rank six · R6_8_1660
Rank six · R6_8_1660
In the established positive spherical representative, saturated modular representations determine all indicator sequences and central phase masses. The eight spin objects form four dual pairs with a common restriction vector. Exact phase Gram constraints leave 640 labelled restriction-data profiles, in three branches of 68, 286 and 286. These are necessary data; they neither construct nor exclude a categorification.
Rank seven · census and verification
Rank seven · multiplicity five
Every supplied rank-seven ring through multiplicity five has a recorded general exclusion or unitary model. Read the complete rank-seven manuscript and its exact verification data.
The 5,805 new tensors and agreement with the earlier 4,201 tensors passed independent computational checks. Exhaustive enumeration completeness relies on the producer completion record; the optimized enumeration was not independently rerun.
Rank eight · current results and research history
Rank eight · multiplicity three
The 3,527 exact-multiplicity-three rings have 3,507 general exclusions, 16 unitary models and four unresolved cases: R8_3_928, R8_3_941, R8_3_1154 and R8_3_1768.
These four admit sphericality and Galois pseudo-unitarity reductions and have no braided Morita representative. Their necessary center data do not establish categorifiability. R928 has categorical Morita rigidity. For R1768 the entire GH(C6,2) Morita family is excluded, for every associator and algebra; the twisted precursor is not a source-ring model. R8_3_436 is separately excluded by its new spherical-equivariantization proof; the old TPE parameterization remains withdrawn.
Current manuscript · Rank-six/eight open tensors · Proof repair and reconciliation · Current attempt records
Rank eight · earlier inner-square obstruction
Rank eight · inner-square rigidity
A categorification of R8_2_24 would supply a tensor autoequivalence whose square is conjugation by a specified invertible object. Forced Frobenius–Schur signs constrain its action on two-dimensional fusion spaces. The short weights of all associator coefficients then force a continuous family of inequivalent tensor autoequivalences, contradicting tensor-functor rigidity. The argument applies to arbitrary complex categorifications after the justified Galois reduction.
The rank-eight manuscript gives the full proof. Separate AI reviews checked the categorical signs, common choices of bases, conductor reduction and rigidity argument. Exact checks bind the source tensor and verify the finite weight calculation. Independent human verification and Lean formalization remain pending.
Rank nine · multiplicity two
Rank nine · multiplicity two
All 2,083 supplied tensors have source-bound decisions: 2,046 general exclusions, 36 unitary models and one unresolved ring. The new exclusions are R9_2_1607, R9_2_1975 and R9_2_1976, each retained after a fresh audit.
Only R9_2_137 remains unresolved. Its forced nonpointed algebra returns to the known central-involution Morita module. Both TY signs retain exact feasible necessary-system witnesses after imposing the positive second indicators. These data do not construct compatible associators. General existence reduces to pseudo-unitary existence by the stated spherical/Galois step; braided existence is excluded.
Current manuscript · One open case · Current proofs, audits and exact replay
Rank ten · multiplicity one complete
All 194 rings are decided: 136 general exclusions and 58 unitary models. R10_1_189 is the charge-zero module subcategory from SU(4)_6 inside SU(10)_1; all 1,000 tensor coefficients match. This recognizes a classical construction.
Rank eleven · multiplicity one
Rank eleven · multiplicity one
All 182 supplied rings are classified: 155 general exclusions and 27 unitary models. This is a multiplicity-one classification.
What the checks establish
The verification separates mathematical hypotheses and cited theorems from exact tensor calculations, finite certificates, source hashes and publication checks. A computation can verify an identity; the categorical argument explaining why the identity applies requires a mathematical proof.
Exact checks cover fusion-ring axioms, classification-ledger consistency, certificate replay and manuscript/download checksums. The verification guide gives commands, prerequisites and the scope of each check. The verification report records the detailed coverage.
Rank-six computational coverage
The current correction route retains all original raw exclusion verdicts, complete reference-backend retries and the exact post-loop checks. The eleven immutable multiplicity-nine proofs run in their source-compatible frozen context, with current tensor and certificate identities checked. This is an explicitly source-bound combined campaign, not a single uniform-backend invocation.
The original pre-revision audit had partly log-derived coverage after raw-output loss. That limitation and the initial failures remain preserved. Current commands, receipts and scope ↗
Manuscripts and reproducible materials
The manuscript shelf links the current PDFs. Current publication checksums bind the published manuscripts and their download copies. All 7 open fusion tensors use the original source bases. The rank-nine M2 proof and replay packet accompanies the new manuscript.
The current audit and research-status report explains the correction. The historical lower-rank proof archive retains its original replay inputs and scope. The current lower-rank document manifest binds the corrected PDFs. Audit corrections and current verification →
Independent review is still needed
The most delicate arguments concern categorical coherence, simultaneous diagram conventions, sphericalization and conductor-equality refinements. Passing arithmetic checks is not a substitute for establishing those steps. The verification report identifies specific expert-review priorities and the remaining limits.
If you find an error, a repository issue naming the ring, theorem and precise failing step will support a precise mathematical review.