Niemeier research (no κ claim): operator factors and paid spinor circuits

[DominikScholz]

No κ claim. This PR records the Niemeier follow-up to #598: exact lattice factorizations, transformations of the scalar computation, constructed compensation subwords, and scoped explanations of why the tested routes do not yet improve the exponent. It adds only research/niemeier-lattices/; the production suppliers and selected exponent certificate are unchanged.

Results and scope

Result	Evidence and limitation
Census of all 23 rooted Niemeier systems; explicit D24+, E8³ and E8² scalar factors	Exact shell counts and identities. The symmetric Pauli factor for E8² has 816 centers and support bill 12,752. The remaining 21 glue-dependent systems are still open.
Rational 16-dimensional bit addresses for E8² and D16+	Each full shell has 30,960 ports and binary decoder rank 815. The complex E8² compensation I-K has full rank 30,960, so small kernel rank does not make compensation free.
A D16 spinor computation in a square-zero algebra	The transformed operator factors into 35 commuting shears, totaling 2,240 additions in that basis. Exact frame propagation and scoped all-stage bounds show why the direct full Möbius round trip consumes the saving. Partial and fused transforms are not excluded.
Fused clique/matching compensation with earlier cleanup	Exact dirty-state and frame replay lowers local rank mass from 1,024 to 832 per clique: 18.75% in this subword, not in κ. The final replay checks all 1,056 input columns of a representative 256-port block and rejects an omitted cleanup operation. Proper-endpoint wrappers, spectator banks and full supplier admission remain open.
Explicit native E8 tensor lift and paid invoice	Exact checks cover 224 embeddings, signed port transport, spectator complements and center descents. The native scalar word is inherited, not freshly replayed. Using pinned #600 profiles, the best tested odd stage count gives nominal κ ≈ 0.000872182222, about 46% below that historical comparison.
Scoped residual and alternative-kernel obstructions	The rank-1,268 residual needs at least 11,412 descent ranks under the specified rank-minimal/private-copy implementation, exceeding the 7,168 surplus. The fused alternative decoder has rank 1,024 with full-dimensional nonzero image columns and no center surplus. The full-shell hybrid has rank 1,851 and an exact all-odd-stage pressure bound below 0.000448646990529330. These are not universal exclusions of Niemeier suppliers.

The package also retains exact section and cross-factor experiments, a 135-clique E8 edge partition, conditional helper budgets, and bounded searches that failed. Sampled intermediate ranks are distinguished from final certificates. Zero-helper screens are explicitly conditional and are not presented as exponent improvements.

Package and reproduction

Based on main at 5996f9043c27bafa90b23e1c937de92a738f1a76. Requirements: Python ≥ 3.11, NumPy and SymPy. Z3 is only for optional clique discovery; it is not needed to replay the committed certificates.

python3 -B research/niemeier-lattices/verify.py --workbench
python3 -B -I -O research/niemeier-lattices/verify.py --workbench

Omit --workbench for the census, geometric factors and initial cost screens. The full driver checks the package manifest and inherited source pins, regenerates coordinate inputs, and compares saved receipts without modifying the checkout. NPZ equality is checked by arrays, not ZIP bytes; only the recorded elapsed-seconds field is excluded from JSON comparisons.

Validation: normal and optimized combined replays passed from a relocated package with only its six pinned external files; the final staged export was also replayed. The manifest inventory and corruption rejection passed, as did git diff --cached --check. make gen4-bank-verify passed all nine fresh stages and its omission controls (273 seconds). The broader make verify, run in an isolated checkout of the unchanged base, passed through the bank checks but did not complete: frontier596-verify / bit543 stopped because the C++ compiler could not find boost/multiprecision/cpp_int.hpp. Boost headers are absent from this machine's standard include locations. This is an environment prerequisite for the inherited verifier; no full-suite pass is claimed. No formal review or full multiplication-theorem verification is claimed.

What remains open

A gain still requires a joint routing/restoration word whose actual frame histories retain a positive center surplus, with all wrappers, spectator banks, dirty-helper cleanup and outer inequalities paid. The negative results leave redundant factorizations, shared arithmetic circuits, other endpoints and the glue-dependent Niemeier systems open. Nothing here proves worst-case O(1) indexing.

Comparison inputs are pinned historical claims, not a declaration of the current frontier: the initial costs use #592/#543; the native tensor invoice uses #600 at dfb7b827990d7b586b3cd778e3d750c050dd4778. The contemporary #604 proper-endpoint work is explicitly credited and is not claimed as a new idea here.

Credits and AI disclosure

The direction follows DominikScholz's guidance and the suggestion attributed to David Ryan / @davidryan59 in the supplied image. Classical lattice and code background credits J. H. Conway, N. J. A. Sloane, Gabriele Nebe, Shoyu Nagaoka, Sho Takemori, and the related authors listed in SOURCES.txt; their papers are not represented as proving these supplier deductions.

Inherited frame and Hamming-coordinate work credits icekylinx, with the OpenAI GPT-6 Astra assistance recorded in those sources. The accounting and helper profiles credit Jeremy Kothe / jeremykothe2, with disclosed Claude Opus 5.5 assistance in the omp harness, and DreamingOfClouds, with disclosed Claude Opus 5.5 assistance via Claude Code. The retained bit comparison credits eumemic and its disclosed Anthropic Claude assistance. NOTICE and the pinned source notices preserve the earlier lineage, including Zhihao Chen / jacklightChen, Jacob Sussman, the original OpenAI #109 framework, and community work maintained by Douglas Colkitt / CrocSwap.

AI disclosure and authorship: OpenAI GPT-6 Astra, running in T3 Code, developed the research, certificates, code and write-up under DominikScholz's direction. Apache-2.0, following the repository license. Inherited source files retain their attribution and are not presented as new authorship.