Add eisenstein-phase-cost-screen: certified necessary cost screen for the F4/Eisenstein decoder route

[Maximooch]

This PR adds a research package that price-checks the Eisenstein decoder route before anyone builds one.

It proves that any Eisenstein-supplied decoder that beats the current selected construction would need total copy cost at most 31 (and at most 29 for the best-known window), while every "clean" support costs exactly 40 — so the whole remaining upside lives in a narrow non-clean window.

Kappa does not change. No supplier is built.

What was checked

The task was: PR #831 reopened the F4/Eisenstein phase path (alternating F4 trace forms no longer block phase compilation), so if we build an Eisenstein supplier, does it beat the current selected complex supplier? This package answers the cost side under the repo's inherited accounting — the format-limits profile accounting from PR #801 and the abstract pairing-rank charge from PR #798 — before any construction is attempted.

All checks are exact (integers and rationals, or explicitly outward-rounded rational interval enclosures from research/format-limits/scripts/cert.py), standard library only, no randomness, no seed. python3 -B research/eisenstein-phase-cost-screen/verify.py passes in ~21s and is registered in VERIFY.json; the registered replay (python3 scripts/research_package.py --package research/eisenstein-phase-cost-screen) prints PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial.

1. Geometry replay (research/eisenstein-clique-copy-bound/geometry.py): 240 roots of E8, order-three complex structure J, 40 nonorthogonal lines (degree 27), full flat census {(0,0):1, (1,1):40, (2,2):240, (2,4):90, (3,5):360, (3,12):40, (4,40):1}, every line-spanned flat nonsingular, and a 7-clique exists — so any pattern-legal decoder B satisfies B[Q,Q] = I_7 on a 7-clique, hence rank_F2(B) >= 7 unconditionally (the screen does not rely on the archived minrank-10 SAT; that census only sharpens the numbers).

2. Clean-class floor = 40. An ordered flat pair (R, A) is clean when every cross pair of distinct lines is orthogonal. For clean pairs the per-rectangle pairing-rank charge is exactly |R ∩ A| (the functionals x ↦ ⟨x, t⟩ − ⟨Jt, Jt·x⟩ for t ∈ R ∩ A are independent on X_R by positive definiteness of the real form and pairwise orthogonality of R ∩ A). The verifier checks this on all 7033 clean ordered flat pairs (tight everywhere), and gives two independent pattern-legal witnesses at total cost exactly 40: ten pairwise-orthogonal full-span 4-line frames (block-diagonal decoder, F2 rank 10), and the star decomposition of the canonical pattern B0 = (all-ones + orthogonality) mod 2 into 40 clean stars of cost 1 each. Consequence: the PR Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620-style 40-unit construction is exactly the clean-class price, and no clean alphabet can beat the selected supplier.

3. Certified threshold screen. For a supplier with window h and odd stage count s, the format-limits skeleton (skel.profile, k=1, x = C/40, v = 40 Eisenstein ports) gives a pressure whose root X must exceed the selected complex-side saving b = 67018507072324140917/40000000000000000000000. The copy slot enters the profile as mass s·x at cap h, so the beat condition is exactly linear in C and the threshold has the closed form C*(s,h) = 40·((s−1) − F_0(b)) / (s·(m/h)^b) with m = 1 + s(h−1). The verifier certifies rational brackets on C*(s,h) for every h ∈ [7,40] and every odd s, confirms non-beat directly at each window's closure point, and closes all larger s with an exact rational increment bound (binomial-series chord bound; exact partial sums to P = 200, exact quadratic minimum beyond; the closed-form slot masses are cross-checked against the engine at 25 points). Results:

   window h	max integer threshold	argmax s	viable s range
   7	31	7	3..61
   8	30	7–15	3..49
   9	30	7–11	3..41
   10	29	7–11	3..35

   Larger windows only hurt (the s=9 screen dies at h ≥ 24). So any beating supplier needs C ≤ 31 at some s ≤ 61; with the minrank-10 window it needs C ≤ 29 at s ≤ 35.

4. Prize cap. The selected assembly's bit side is X_bit = 0.001733427039589844 (PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 audit, bit_lower), above the complex side b. Every screen argmax point clears X_bit, so any complex-only improvement is capped by the bit side: kappa ≤ X_bit/(1+X_bit) = 1733427039589844234749/1001733427039589844234749 = 0.001730427469823602, i.e. +3.45% over the selected record 836330098538459/500000000000000000 = 0.001672660197076918 — the uniform prize no matter how cheap the decoder is.

5. Negative controls (4): closed-form/direct cross-validation at eight boundary points; doubling b flips the argmax beat verdict; a tampered nonorthogonality changes the clean census; the naive consecutive-block partition is illegal (the frame witness needs orthogonality) and a star without its self-loop costs 0 (the star witness is not vacuously constant).

Relation to prior work

• PR Research note: Eisenstein E8 copy cost ≥22 and sharp clique-method limit #798 (copy floor 22, all supports): the floor is used as the screen's lower edge; this package is the complementary upper edge.
• PR Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620 (40-unit construction): shown to be exactly the clean-class price under this accounting; it cannot improve kappa.
• PR Research: certified format ceilings for Eisenstein/Gaussian/Hurwitz E8 copy floors (no κ claim) #801 (format ceilings): supplies the profile accounting and certified interval machinery; no Eisenstein threshold map existed there.
• PR Research audit: F4 alternating trace is not by itself a phase-compilation obstruction (no new kappa) #831 (alternating-phase interface): removes the interface objection; this package shows the remaining obstruction is economic, and localizes it.
• PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 (frontier audit): supplies the selected record and the bit-side cap.

Not done / limitations

• No supplier was built; no kappa changed; the selected record is untouched.
• The screen is a necessary condition only. It does not prove that a C ≤ 31 non-clean decoder exists, nor that none does.
• "Clean" is a sufficient hypothesis for the 40 floor, not a proven dichotomy: a non-clean decoder between 22 and 31 is exactly the open window.
• The screen prices the copy slots under the inherited k=1 skeleton; a real supplier's frames, adapters, restoration and residuals are additional charges not modeled here (they only make the route more expensive).
• The h range [7,40] is justified structurally (7-clique floor; 40×40 pattern), not by exhaustive search over exotic support alphabets.
• The tail closure uses a certified increment bound with exact partial sums to P = 200 and an exact quadratic minimum beyond; it is exact arithmetic, but the bound itself is not tight.

AI assistance: Penguin (OpenAI) coding/math agent assisted with the mathematics, implementation, checking and writing, as in PR #831; every claim is backed by the exact verifier output.

[Mister-Meeseeks]

Batch review at 148762722bb39c7ce2fd3fd0d42ef25d945aae4d.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact E8 clean-pair census and sharp40-cost witnesses; threshold formula and rational increment/tail closure bound all odd stages in the inherited format. Non-clean copy cost<=31 is necessary there, not a supplier. Fixed complex-only prize excludes changes to the bit side.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

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