Prove the uniform E8 one-coset budget and two-sided rank-nine floor

[conorholds]

Change and mathematical scope

Proves the arbitrary-family one-coset deficiency budget on the full 120 E8
root lines. If C is the scalar evaluation code, λ ∉ C, and nonzero
p_i ∈ C + Kλ are pairwise nonproportional, then

Σ_i (9 − w(p_i)) ≤ 9,

where w is binary support rank. The family may have any finite size. The
theorem holds over every field of characteristic different from two, with
the dense binary COPY table fixed before scalar reduction.

Compatible-character independence and a grouped-relation argument make the
support annihilators a direct sum in F₂⁹. This supplies the term-count-independent
packing step beyond the prescribed-frame results. After proportional merges,
the two budgets and Sylvester give COPY ≥ 9(m−s−t) ≥ 72 whenever both actual
factor ranks are at most nine. Nonzero cost-zero atoms are retained and final
ranks are recomputed after merging.

The genuinely two-sided consequence is the (s,t)=(1,1) exact-P120 cell for
every term count. A normalized rank-(9,9), ten-term decoder attains 72.
Necessary equality conditions are proved; they are not a coefficient classification.

The one-sided corollaries are scoped cases of L120, the minimum-basis-cost
bound for spaces containing C: all dimension-nine extensions and the specified
dimension-ten spaces C + span(δ_a,δ_b). The latter have a sharp COPY floor of
72 when the other factor lies in C. A sheared rank-(10,8) equality example
is pairwise nonproportional on both sides and remains ten terms after
normalization. These corollaries are proved directly in every characteristic
different from two, without assuming the broader L120 completion chain.

Only research/e8-one-coset-deficiency/ is added. This does not prove unrestricted
L120, arbitrary higher-rank factors, masked or punctured floors, or a new κ.

Task and overlap

Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. TASK.json
records the handoff-driven change from complex-rank-shape-decoder to this
structural obstruction.

PR3154's ARCHITECT
blueprint separates B′ / L120 from the two-sided C′ node and explicitly asks
for an “m-free deficiency budget”. This closes the exact (1,1) slice of C′;
the one-sided results are dimension-nine and sparse dimension-ten slices of B′.
For Discussion 1587,
the coverage is its nine-term P120 item and the ranks-at-most-nine portion of
m ≥ 10, not its other realization or higher-rank residuals.

The characteristic-zero L120 programme includes
PR3074's sufficiency
reduction and draft PR3191's
conditional ledger completion. At the 2026-10-18 19:29 UTC refresh, PR3191
remains an open draft at 9c9e094 with its second-implementation slot unfilled.
conorholds' PR3166
supplies C7 in that programme. A completed L120 proof would include these
one-sided characteristic-zero slices; it does not supply the arbitrary-family
theorem or the two-sided (1,1) floor proved here.

The read-only refresh checked 27 PR updates since 18:40 UTC and the relevant
Discussion records, including full pagination of the handoff thread. No matching
completion appeared within that coverage. petalas' PR3701 uses the same original
card on a different local-word rank question. This is a bounded overlap check,
not a guarantee about every PR diff or unpublished work. No public claim was posted.

Validation

python3.12 -B scripts/research_package.py --package research/e8-one-coset-deficiency

• Exact certificate: 109 power-of-two minors, complete orbit coverage and integral
  reflection transport. A separate model checks all 511 characters and 130,305 pairs.
• Eight rational exact factorizations and 24 reductions over F3/F5/F7 check
  products, ranks, COPY, normalization and the strict rank-(9,9) equality example.
• Both 29,524-ray F3 sparse-pair spaces are exhausted. F5/F7 use planted controls,
  not exhaustive censuses. Twelve sparse-pair equality fixtures include six
  normalized rank-(10,8) examples, six collapsing controls and exact-product
  rejection when the reader compensation is omitted.
• Registered replay, committed-source/history checks, official export integrity,
  Git-free optimized replay, disposable-export adversarial controls, attribution
  preflight and git diff --check pass for the revised artifacts.

The independent audit reconstructed the mathematics and finite inputs on the
previous frozen candidate using CPython 3.12. The CPython 3.14 replay is separate
author evidence. The new sheared construction has a written all-odd-field proof
and targeted checks. Review is AI-assisted; no independent human certification
is claimed.

The two unchanged Lean 4.21.0 files retain their passing checks: six partial
Init-only declarations cover the clique/transversal core, concrete binary laws
and final arithmetic implication. The E8 instantiation, field transfer and full
consumer assembly remain unformalized. Hosted CI and repository-wide builds
have not been run. The submission contains one focused commit.

Credit

Credit joaqcoo's PR1224
for the projection/annihilator framework and pair census; mightyOil's
PR1248 for the A8
budget and compatible-edge independence method; adamgall's
PR2062 for the A7+A1
family; and bbMisk's PR1633 for excess ranks and normal forms.

CubeJerry's PR3095
already proves the full-E8 single-character rank fact over every characteristic
different from two using power-of-two minors. That fact is reconstructed here,
not claimed as new. philippeVV's
restricted lift-family credit-at-most-nine argument
is an adjacent method antecedent.

DreamingOfClouds' PR2816/2862 supply singleton/root-swap antecedents; their
historical lift floors should be read alongside the later L120 programme above.
The E8 COPY lineage in PR446/717, PR968, PR1077, the punctured result PR2854,
and the rank-changing exact reduction PR2832 are retained. The sheared sharp
fixture was suggested in the local AI-assisted audit. No pending catalogue,
retraction or L120 completion assertion is a proof premise.

Open next questions

Extend the floor beyond the stated rank and coordinate-pair restrictions;
the six-ray counterexample rules out simply doubling the uniform budget.
Complete the E8, dimension and consumer bridges in Lean. General masked and
punctured decoders remain separate obligations.