[unpaidmoon]

Change and mathematical scope

Real root-linear factorizations sum_k a_k b_k^T = 2 I_8 of the E8 root Gram have label-pairing copy cost exactly 9t after zero factors are deleted. Nine or more nonzero terms therefore cost at least 81 on all 120 lines, above the eight-term floor 72. After deleting any set of at most 13 lines, the live cost is at least 9t - p. The explicit nine-term split costs 81 on P120 and 79 after the coordinate-frame puncture, above the frame floor 64.

A one-coordinate excursion keeps the Gram identity, raises the left factor rank from 8 to 9, and drops one term from 9 to 8 (total 80). The exact count 9t needs both factors to stay in the root span.

This is a scoped obstruction: a proof for that rank-(8, 8) branch, not a construction, not a paid per-call profile, and not a kappa improvement. The selected record is unchanged. Still open: factor rank at least 9, E-graph rank at least 9, complex coefficients, and any frame, restoration, or centre invoice.

Task and overlap

Base ff3cc58, priority revision 7. Task complex-rank-shape-decoder. Scope changed: the card asks for a block-climb profile that beats the selected word by at least 5%. The decoder scope map leaves the copy bill of nine or more costly rank-8 terms undetermined, which is the overcomplete branch of that question. This package prices only the factor-rank-(8, 8) branch and shows it cannot undercut the eight-term floor. The old a/(1+a) ceiling is not the test. Details, the cited Weyl-chamber step, and the two open questions are in research/nine-costly-rank8-copy-floor/.

Nearest overlap:

• PR968 (themilkman216) proves the eight-term all-pairs floor and lists n >= 9 as not covered. Theorems 3 and 4 price the root-span case for every term count. The violator calculus is reused in a self-contained census; the eight-term proof is not replayed.
• PR1212 (Brett Halligan / burntout76) supplies the rank-8 E-graph gauge form. It is cited and not recomputed. Rank-(8, 8) factorizations of that gauge matrix inherit the same supports and floors on P120 and on a frame complement.
• research/ten-term-copy-cost attains copy cost 72 with ten terms at factor ranks (9, 9). That example is outside this hypothesis. The new excursion is the small control that makes the separation visible.
• research/e8-one-extra-direction and Prove weighted E8 query limits and low-redundancy obstructions #4967 price binary factorizations of the pairing matrix. Different field, not a premise.
• Unpaidmoon Bound the E8 label pairing rank by 9 and screen it. #4161, Record that the selected k8 word emits no paid child above rank 8. #4158, and Price the plane-port u=2 screen against the rank-8 cell's 16-child sl… #4156, and the fork branches cursor/e8-label-child-rank-*, bound E8 label or child rank. They do not price this Gram factorization.

Validation

From the repository root, CPython 3.12, standard library, about one second:

python3 -B research/nine-costly-rank8-copy-floor/verify.py
python3 scripts/research_package.py --package research/nine-costly-rank8-copy-floor

Both exit 0. The verifier ends with

PASS_NINE_COSTLY_RANK8_COPY_FLOOR rank88_cost=9t p120_nine=81 p112_nine=79 excursion_rank9_cost=80 no_new_kappa

The package runner ends with PASS_REGISTERED_RESEARCH_PACKAGE. check_research_commit.py matched commit 99dfc77.

Finite checks: 120 lines, pairing rank 9, violator sizes {56, 64, 120}, Cartan determinant 1, all 383 dominant lattice vectors of norm at most 128, the eight hyperplane sizes, and all 120 root profiles (1, 24, 25). Witnesses: the coordinate frame (cost 72), the nine-term split (81, and 79 on the frame puncture), a seven-term failure, a mutated split, an odd deleted pair of cost 7, and the rank-9 excursion (cost 80). Theorems 3 and 4 are written arguments from that census, not an enumeration of every real factorization. The Weyl-chamber transport (Bourbaki, Lie, Chapter VI, §1) and PR1212's Theorem 1 are cited and not recomputed. No Lean, Docker, or frontier replay was run.

Credit

DreamingOfClouds (PR717) for the E-graph model; themilkman216 (PR968) for the eight-term floor and violator calculus; Brett Halligan / burntout76 (PR1212) for the gauge form; Arham Tahir and starship-s for the decoder scope map; conorholds and the ten-term-copy-cost notes for the rank-(9, 9) contrast. Nick Petalas's binary one-extra-direction theorem is a different model and is not used as a lemma.

Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human review or proof-assistant verification is claimed. Apache-2.0.

Open next questions

1. Can a real factorization of 2G with a factor rank greater than 8 cost less than 72? Start from excursion_control in verify.py, or from research/ten-term-copy-cost, and either go under 72 or isolate a defect budget. Blocked by the lack of a normal form once a column leaves the root span.
2. Can a real E-graph solution of rank at least 9 on P120 or on a frame complement cost less than the eight-term floor? Start by writing one exact rank-9 solution and pricing its terms. Blocked by the lack of a gauge form, which PR1212 does not supply at rank 9.

Workflow feedback (optional)

The useful split was factor rank 8 versus factor rank 9. A ten-term cost of 72 looks like a counterexample until that rank is part of the statement. Enumerating dominant vectors of norm at most 128 was enough to certify the hyperplane list inside Hadamard's bound; enumerating the Weyl group was not.