Bound the E8 label pairing rank by 9 and screen it.

[unpaidmoon]

Change and mathematical scope

Negative screen. No new κ. Selected κ stays 0.001672660197076918.

On the 120 root lines of E8, the PR717 label pairing B_ij = (<r_i, r_j> + 1) mod 2 has F_2 rank 9. Every row reduces into a 9-row basis, so every support pair has pairing rank at most 9. The nine basis lines against all 120 columns attain rank 9. Witness lines are 0,1,2,4,6,8,10,12,56.

On the selected template (width 73, 896 families, nine stages, centre 64, local mass 5586, idle mass 1646, idle cap 51), per-unit cost at a = 1/99 is strictly decreasing in rank through 9. The cheapest local histogram under the cap is {9: 5586/9}. Its relaxation is strictly above 1, and its 1e-15 ceiling bracket is (1409316073/488281250000, 577255863501/200000000000000), below 1/99. The integer histogram {9: 620, 6: 1} also fails. A fractional rank-44 histogram passes the same one-point screen.

Frame, restoration and copy children that are not pairing-rank terms of B are outside this alphabet. Inside it, extra children of rank at most 9 cannot pull the moment back through 1. This is not a decoder, a block climb, or a multiplication theorem.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main, priority revision 6).
• Head: 5b74600f2296a0c6265e22a6dec46949ad9a6f89.
• Task: complex-rank-shape-decoder, scope refined. The catalogue method is a bound on effective child rank for one fixed support alphabet. The alphabet is the PR717 label pairing. Copies, frames and restoration are not given a rank above 9; inside the alphabet they cannot repair the failed screen.
• Handoff: Handoffs and requests for independent checks #1166 (comment)
• Package: research/e8-label-pairing-rank-cap.
• Fork draft: Bound the E8 label pairing rank by 9 on the selected template. unpaidmoon/integer-mult-bounds#53

Item taken: the child-rank bound, index (10-1) mod 8 = 1 of the eligible list below. The list has fewer than 10 entries, so this is a refinement of that handoff rather than a recomputation of the cap curve. Open PR #3701 already tabulates a_frac(s) for s = 1..51 and leaves the realizable rank blocked. Open PR #3895 screens one rank-16 E8 block itinerary. Open PR #4066 screens rank-16 delivery plus every legal rank-1 surcharge. None proves this pairing-rank cap.

Ordered eligible list. Chronological open next tasks on #1166, starting from the numbered list in PR #4142 and continuing through later released next tasks. Dropped when a later DONE, an open PR, or this round through #4157 answers or refines the named question:

1. The (7,18) / (7,20) functional census (comment 19021170, question 1).
2. This bound — largest child rank of one fixed support alphabet (comment 19021222).
3. Centre-copy continuation of the 96-output F17 demand (comment 19030434).
4. One downstream consumer that keeps the encoded P104 endpoint (comment 19032700, task 2).
5. Independent review of the research: consume P104 residuals in actual cleanup and price source return #3943 cleanup factorization (comment 19032700, task 3).
6. Changed all-role encoding for the PR research: derive P104 two-axis endpoint intertwiners and cost gate #3954 two-axis equations (comment 19034025).
7. One changed chronological caller with literal encoding and a favorable bill (comment 19034741).
8. Profile research/p112-ten-deletion-span's census.cpp (comment 19045344, lever 1).

Dropped before that cut, among others: the single-hub factorization (#4124, #4145), weight >= n-1 (#4148), the T488 invoice (#4150), the rank-8/u2 slack (#4151, #4156), the shear payload (#4126, #4135, #4139, #4146), the #3942 reduction and two-frame edits (#4134, #4122, #4128, #4144), the adjacent completed-call pair (#4133, #4136, #4140), the transported warp-0 frame (#4157; the later-warp remainder in comment 19057530 stays a refinement of that delivered item), fiber tags (#4154), the A1858 cleanup (#4152), source-normalization fusion (#4153), the tensor bridge and (N-P)⊗M consumer (#4125, #4131, #4143), the K/adjoint boundary (#4155), the many-axis guard (#4142), and event 58 (#4149).

Validation

Python 3.12.3, standard library only. Registered replay:

python3.12 -B scripts/research_package.py --package research/e8-label-pairing-rank-cap

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/e8-label-pairing-rank-cap outcome=negative-result. Both -I -B and -I -O -B print PASS_E8_LABEL_PAIRING_RANK_CAP.

verify.py rebuilds the 240 roots and 120 lines, reduces every pairing row into the 9-dimensional basis, attains rank 9, rejects a synthetic rank-10 row, and calls the pinned ceiling.py for the fractional and integer histograms. The rank-44 control must pass a = 1/99.

Not run: an eight-term decoder search, a width-10 block climb, frame or restoration pricing, the selected cold replay, Lean, or Docker. The rank cap is the row-space argument in research/e8-label-pairing-rank-cap/PROOF.md. The moment comparison uses the pinned enclosures.

Credit

themilkman216, research/complex-frame-pair-decoders, for the PR717 label alphabet and the recorded rank-9 pairing. Douglas Colkitt, with Claude assistance, for research/complex-local-word-ceiling. Nick Petalas, discussion comment 19021222 / PR #3701, for the open realizable-rank question. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review is claimed.

Open next questions

1. Does exactness of an eight-term all-pairs decoder force every term to pairing rank at most 8? First step: one dual-basis term of rank 9, or a proof that AB^T = 2I forbids it. Blocker: no decoder search is run here.
2. Can a block-climb child whose rank is a multiplication width exceed 9 under the selected coverage law? First step: one legal width-10 child with its frame and restoration bill. Blocker: no such itinerary is pinned here.

Workflow feedback (optional)

The #4142 numbered handoff list is still the stable order, but later DONE notes and parallel refinements move the cut. A single current-status line on each handoff, naming the one remaining question, would keep the tenth slot fixed.