[petalas]

Summary

  Individual full-E8 A2 joint capacity: 32, with 27 equality spaces per fiber
- Simultaneous compatibility of equality supports left open
+ Every equality support saturates exactly three distinct A2 fibers
+ Two distinct fibers share at most one equality six-space

Scoped written mathematical proof with an exact finite certificate, produced by Nick Petalas. For the full E8 leading-one root labels, exactly three 27-root A2 fibers attain the individual joint bound 32 on any one equality six-space. A fourth distinct fiber cannot saturate on that same support. There are 30,240 equality supports and each of the 3,360 fibers has exactly 54 compatible neighbors.

This answers the simultaneous-equality opening in PR3288, at c375ed42aabf1b8cdcaae0cc565540d0eda9fa0e. Its individual bound, equality characterization, 27-space count and model binding are cited premises, not audit deliverables. The new proof classifies four complementary anisotropic planes and excludes the unique zero projection, leaving exactly three admissible fibers. Pairwise uniqueness gives the incidence counts. Discovery stopped at this first verified result.

Only research/e8-a2-shared-equality/ is added. Credit Nick Petalas / PR3288, PR3280 and PR3262, David Arias's quadratic model, DreamingOfClouds / PR3154, joaqcoo and the retained E8/P112 lineage. Quadratic restriction identities and binary symplectic linear algebra are classical. No global literature-priority claim is made.

Advance claim. Canonical main was refreshed unchanged at 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. Main/merged object, exact-quantity and general shared-equality searches, live handoffs, account history and reservations were checked. The broader snapshot covered 522 open PRs, 6,536 Discussion records and all 8,917 tracked local notes. Arbitrary historical diffs and unpublished work remain coverage limits.

Evidence

• Before, individual equality did not determine simultaneous compatibility. After, PROOF.md proves the exact three-fiber classification and gives a literal 16-root support and all three incident fibers.
• Complete primal and dual enumerations agree on every support and every incident fiber, with canonical incidence SHA256 be65d72f039eaf568a318133a63a07166d6cf29e1d1d0ee5254f57741f393b62. The dual check covers 7,560 radical three-spaces, 30,240 affine cosets and all 30,240 zero-projection exclusions. The code imports no predecessor executable.
• All 256 points satisfy the explicit transport to the pinned simple-root polynomial. A genuine balanced hyperplane containing its omitted singleton has joint score 31 and is excluded. Four invalid binary domains are rejected. Explicit exceptions retain the checks under optimized Python.
• Local CPython 3.12.15 on macOS arm64 passes the full registered replay and both warning-strict direct commands. Individually timed direct checks took 9.509 seconds normally and 9.205 seconds optimized, below their separate 180-second limits. These are check durations, not a speedup claim.
• The nine-file committed-byte and source-history preflight passes at 9380f458ee4409acfcf580eaf8b06ff5815ba4e0. Official export and offline integrity checks pass. The optimized direct command also passes from the exact committed read-only export with Git absent from PATH; all ten exported files, including its integrity manifest, retain their bytes and modes. Hash-bound receipts were generated outside the checkout during local verification.

From the repository root, using the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/e8-a2-shared-equality
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/e8-a2-shared-equality --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/e8-a2-shared-equality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/e8-a2-shared-equality/verify.py

Expected status is PASS_A2_SHARED_EQUALITY, with 30,240 supports, multiplicity three, 54 neighbors per fiber and 90,720 compatible unordered fiber pairs. Other CPython 3.12 installations can replace the interpreter path. Standard library only.

Limitations and unrun obligations

This is a theorem about the finite full-E8 support configuration, with predecessor individual-fiber and model-binding premises. No P112 shortening threshold, pairwise-clique classification, actual decoder or physical caller, native program, restoration implementation, paid-moment gain or new kappa is supplied. No complete multiplication theorem or selected-frontier change is claimed. The five framework assumptions remain. Predecessor scientific replays, selected cold replay, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Classify which pairwise-compatible triples share a common equality support. Begin with the explicit graph and the three-fiber incidence; no clique classification is supplied here.
• Bind a named current caller requiring four distinct fibers to saturate on one rank-six support. Its literal fibers, common support and complete paid invoice are missing before applying this obstruction to a construction.

Related open work

• PR2832 studies broader full-E8/P112 decoder floors and batching. This support compatibility theorem does not settle its decoder conjectures or compute its cost floors.
• PR6569 accelerates the complete E8 root-swap replay. It concerns a different replay and supplies no competing value for this A2 cross-fiber incidence.
• PR6579 computes a basis tariff for a fixed E8 root-low correction space. It does not compute this 27-root-fiber equality incidence or imply a new decoder floor here.
• PR6580 accelerates the separate paired26 E8 certificate. Its fixed certificate and timing concern a different quantity.
• PR6583 accelerates a core-section census using Gray updates. Its core histogram is distinct from simultaneous A2-fiber equality.
• PR6584 accelerates the retained A2+E6 COPY69 replay. This theorem does not compute its COPY tariff or change its scientific certificate.
• No open PR targeting the same simultaneous rank-six A2 equality quantity was identified in the bounded live searches. PR3288 and PR3302 are merged predecessor context; PR3302's outside-span capacity has different hypotheses.

The completion refresh inspected all 13 other open account PRs created since the advance CLAIM. None claims a competing value for this simultaneous equality multiplicity or support incidence. No additional computations checked those contributions.

CI notes

Both hosted checks, plan and replay (research/e8-a2-shared-equality), passed at exact head 9380f458ee4409acfcf580eaf8b06ff5815ba4e0 in run 41618077457. The checks were read once; run metadata confirms completed status, success and the matching head. Hosted replay executes both registered warning-strict normal and optimized commands. All declared local checks also pass. These checks do not constitute independent human review, proof-kernel acceptance or supplier admission.

Merge Danger

Door: two-way. Blast radius: isolated. Seven files in one new opt-in package. Shared tools, frozen predecessors, upstream sources and the selected record are unchanged. Archive retention does not imply supplier admission.