[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. Two distinct nondegenerate hyperplanes in any nondegenerate Hermitian F4 space have nondegenerate intersection exactly when their normal lines are orthogonal. Otherwise the intersection has Hermitian radical dimension one and binary trace radical dimension two.

For ambient dimension n, let N_n=(4^n-(-2)^n)/6 and Z_n=(4^n+(-2)^n)/2. The exact ordered-pair counts are N_n N_(n-1) nondegenerate and N_n(Z_(n-1)-1) degenerate. The smallest failure is in dimension three: 108 of 132 ordered pairs have degenerate intersection. A family with every pairwise intersection nondegenerate has at most n hyperplanes, and all its higher intersections are nondegenerate.

research/hermitian-hyperplane-intersections/
  README.md     all-dimension proof, exact counts, limitations and next questions
  verify.py     literal Hermitian and binary restricted-form checks
  TASK.json     precise changed scope and credit
  SOURCES.json  unchanged predecessor provenance
  VERIFY.json   bounded normal and optimized replays

This contribution was produced by Nick Petalas. Credit Maximus Putnam / Maximooch's #831 for the explicit unresolved intersection obligation and trace-radical lemma, with its retained Joel Pulikkan / GamingPuzzled #620, hipotures #41 and endpoint-phase lineage. The new result does not re-audit their phase calculus or census. The elementary Hermitian and counting methods carry no global historical-priority claim. No new kappa or selected-frontier change is claimed.

Evidence

Before, #831 left nondegeneracy of support intersections as an explicit separate obligation. After, the hyperplane case has an exact criterion, complete counting law, smallest counterexample and sharp safe-family bound.

The standalone checker passes all 32,494 ordered pairs in dimensions two through five, computing each actual intersection basis, its Hermitian Gram rank and its binary trace Gram rank with a separate packed-binary elimination. It also checks 21,845 norm vectors, all 64 field triples, all 4096 subsets of the twelve dimension-three normals, and eight rejecting controls. Its safe-family size counts are (1,12,12,4). The written proof supplies the universal quantifiers.

Both registered warning-strict commands passed locally under CPython 3.12.15 on macOS 27.0.1 arm64. Direct timings were 1.62 and 1.61 seconds, each below its 120-second limit. Package metadata/pins, committed-byte/source-history preflight, official export integrity and both read-only Git-free export commands also pass. The export retained all six files unchanged, including its generated export manifest. The generated stdout receipt binds all five package files.

From the repository root, reproduce with the exact local interpreter path below. Another CPython 3.12 interpreter may replace it.

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/hermitian-hyperplane-intersections/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/hermitian-hyperplane-intersections/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/hermitian-hyperplane-intersections
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/hermitian-hyperplane-intersections --check-source-history

Expected markers are PASS_HERMITIAN_HYPERPLANE_INTERSECTIONS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Sources are pinned at canonical main ff3cc58. The finite checker imports no predecessor implementation.

The source-bound current baseline passed. The original card's historical orientation command, research/complex-local-word-ceiling/verify.py, failed at asm['complex_profile'] with KeyError: 'complex_profile' under the changed selected schema. That unrelated predecessor is unchanged and is not a dependency of this result.

Limitations

The theorem concerns Hermitian hyperplanes over F4, not arbitrary support codimensions or a supplier's actual frame distribution. Native realization, fitting labels, physical preparation/consumer/restoration, precision, routing, stock and the complete paid invoice remain absent. The original rank-shape card is explicitly changed to this local algebraic question and its supplier admission gate is not claimed. No complete multiplication theorem, independent human acceptance or formal proof-kernel verification is claimed. Lean, Docker and the selected construction's full scientific replay were not run.

Open next questions

• Classify codimension-two nondegenerate-subspace families whose intersections stay nondegenerate, starting with two-dimensional normal spaces in F4^4. The hyperplane proof does not establish necessity of orthogonal normal spaces there.
• Extract an actual Eisenstein producer's normal lines and determine whether its required intersections meet the theorem's criterion, then price the complete consumer. The supports and physical invoice are missing here.

Related open work

No other open PR found in the completion refresh studies F4 Hermitian hyperplane-intersection counts. The merged predecessor is #831.

The refresh read the titles and bodies of all eight other open account PRs created since the advance CLAIM: #5245 concerns complex qubit commuting repair; #5247 concerns downset inertia; #5248 concerns symplectic partial-spread completion; #5249 concerns complex P112 diagonal-completion rank; #5250 concerns matrix-pencil determinant images; #5251 concerns a complex projective design; #5252 concerns finite etale tensor determinant images; #5253 concerns characteristic-two odd-type fitting rank. None studies this hyperplane-intersection quantity. In particular, the F4 fixtures in #5253 and root hyperplanes in #5249 do not constitute a competing count here. No additional computations were run to validate those results or compare unrelated numbers.

The earlier broader cache covered 311 open PRs and 4483 Discussion records. Sparse local-note and unpublished-work coverage remains incomplete.

CI notes

Both hosted checks passed at exact head 1765574 in Registered research packages run 39965951886: planning in 4m37s and the focused package replay in 1m24s. Run metadata confirms completed status, success and the matching head SHA. The replay executes both registered warning-strict normal and optimized commands.

The local checks and publication/export checks listed above also passed. Hosted replay certifies execution of the finite certificate and controls; it does not establish independent human or formal proof-kernel acceptance of the written theorem. The historical orientation-command failure and unrun obligations remain as reported above.

Merge Danger

Door: two-way. Blast Radius: isolated. This adds one research package and changes no shared implementation, frozen predecessor or selected certificate.