Abstract
Compressing the nine rows modulo three converts the prescribed Legendre-pair chart into an exact length-\(37\) complementary-autocorrelation model over ten Eisenstein profile letters.
The shells \(n_9=6,5,4,3\) are empty. Shell two has exactly five orbits of sizes \(24,12,12,12,24\), while a separate dense-shell census has exactly eighteen compressed-profile orbits at \(n_9=0\) in its explicitly stated characteristic-two/modulo-nine scope. The intermediate \(n_9=1\) shell remains open.
For shell two, exact physical placements are primitive units in all six prime-\(167\) factors. Star pairing then yields three pair-resultant norm equalities in \(\mathbb F_{167^3}^{*}\), whose character factors have orders \(2\), \(83\), and \(28{,}057\). This is an exact reduction, but the estimated complete join remains too large for the present architecture.
Main results
Shell classification. The exact shell descent leaves five \(n_9=2\) orbits; the stated dense census leaves eighteen \(n_9=0\) orbits. No exhaustiveness claim is made for \(n_9=1\).
Placement geometry. The first labelled placement digit has rank \(18\), leaving an affine \(36\)-space over \(\mathbb F_3\); exact margins impose six further independent affine equations. The five shell-two profiles have maximum structured retraction dimensions \(4,3,3,3,4\).
Primitive-unit and norm gate. Every physical placement over every shell-two image is nonzero in all six primitive factors. Any solution must therefore match three exact pair-resultant norms between its two channels.
Complete \(3^9\) physical slices show strong contraction, but all audited one-coordinate character images are full and every triple image has affine rank three. The slices behave at approximately the random-model rate and exclude no profile.
Claim-to-evidence map
| Claim | Evidence |
|---|---|
| Five shell-two orbits | Frozen exact certificate, semantic hash, detached verifier, and direct orbit checks |
| Eighteen dense orbits | Released 729-shard production ledger, byte-reproducible aggregate, and replay of every retained profile and orbit |
| Physical placement dimensions | Exact affine-rank and quadratic-retraction censuses |
| Primitive units | Meet-in-the-middle audit of all 90 profile/factor instances |
| Three norm keys | Finite-field theorem verifier and an independent compiled/repository field bridge |
| Exact slice behavior | Frozen \(3^9\) slice certificate and character-count replay |
Novelty and scope
The order-three common multiplier subgroup is not claimed as new. A July 2026 multiplier theorem publicly leaves this subgroup open while excluding larger common fixed multiplier groups. This paper is narrower still: it additionally fixes the \(p=37,q=3\) quadratic-character compression.
The candidate contributions are the exact shell classifications in their stated scopes, labelled phase-lifting geometry, primitive-unit theorem, pair-resultant norm gate, and measured complete-search cost. The priority audit is incomplete and cannot establish worldwide novelty.
AI-assistance and verification disclosure
The mathematical exploration, program development, certificate analysis, manuscript drafting, and publication materials used heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a non-specialist and cannot independently certify every derivation or implementation. Exact artifacts improve auditability; they are not peer review or proof-assistant formalization.
Suggested citation
Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “Exact shell descent and finite-field norm gates in a prescribed-compression chart for Legendre pairs of length 333,” provisional research paper, 25 July 2026. Permanent page.