Research: certified format ceilings for Eisenstein/Gaussian/Hurwitz E8 copy floors (no κ claim)

[Rohansguliani]

Research note and tooling. No κ claim and no word. Contributed by Rohan Guliani (@Rohansguliani), with substantial AI assistance.

Builds on the extension-ring E8 floors already proved: #620 (screen), #703 (Hurwitz cost exactly 10), #738/#770/#767 (Gaussian floor 40), #798 (Eisenstein cost ≥22, open in [22,40]) and #765. It does not re-prove them. It turns them into certified B_K format ceilings using the unchanged research/format-limits-corrected engine.

Findings

• The outer assembly is saturated. The selected κ equals a/(1+a) for the binding complex supplier to a relative 1.4e-16. So ring, transform length, resampling and Fürer/SS/HvdH-type outer changes can gain at most a factor 1+a ≈ 1.0016.
• Eisenstein bit side. A decoder of cost ≤24 would give a ceiling above 1/99 (cost 24: X ≤ 1.150e-2; cost 25: X ≤ 1.0053e-2). Cost ≤28 still beats the realised E8 bit unit (6.000e-3). Coordinate-total decoders cost at least 28 (new exact clique bound), so only non-coordinate decoders inside Research note: Eisenstein E8 copy cost ≥22 and sharp clique-method limit #798's open interval could matter.
• Gaussian floor 40 gives X ≤ 7.688e-3. Hurwitz gives no positive saving for s ≤ 13.
• Complex side (argued). An Eisenstein complex unit has an F₄ window of at least ω = 7, which gives X ≤ 8.07e-3 even at cost 22. The current complex unit caps at 5.19e-3. So the label-field route cannot give κ > 0.01 in the B_K format, though it could still give a several-fold ceiling rise if a cost ≤24 bit decoder exists.

Reproduce

python3 -B research/label-field-census/census.py --json out.json runs in about 4 minutes with numpy only and reproduces RESULT.json. Not covered: s ≥ 15, non-coordinate decoders in the clique table, realisability of Q(ω) frames, and helper/hosting/analytic obligations.

[Mister-Meeseeks]

100-PR batch review at f0758653a5a1f72c3b59e58d6fe8d658db20b50b.

This remains open after scoped triage. Next review step: Label-field census and conditional format ceilings need a dedicated proof/pricing audit; no target-strength construction inferred.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.