Prove coupled E8 coordinate floors and the 28-neighbor partial-quotient case

[conorholds]

Change and mathematical scope

Conditional on the retained written premises from pending same-author
PR3755, proves
term-count-independent COPY ≥ 72 for new two-sided exact full-E8
factorizations PQᵀ = G/2 with left space inside C + Kδ_v, where C is
the eight-dimensional root-evaluation code. The dense binary COPY table is
fixed before scalar reduction, and transposes are ordinary.

The load-bearing premises are PR3755's singleton classification, rank-at-most-nine
theorem and coordinate-pair classification. Vendored hashes and finite
certificate replays preserve those dependencies; they do not independently
validate the written proofs. The low-budget partial-quotient argument also
uses PR3166 Theorem A, recorded as partially reviewed in the upstream ledger.

The reusable step is an actual-basis coupling lemma. The only unpaid
singleton/hole state marks a coordinate of the actual equality basis as −1;
the free-pivot cut forces that same coordinate to be zero. The argument does
not replace the given basis with a conveniently chosen minimum basis.

The resulting floors cover:

• Every full actual coordinate extension col(Q) = C + D_A, in
  characteristic zero, with each exceptional left ray occurring at most once.
• Raw coordinate-pair inclusion col(Q) ⊂ C + span(δ_a,δ_b) over every
  characteristic different from two.
• Raw arbitrary partial quotients supported on at most 28 neighbors of v,
  and supports of at most 13 coordinates whose binary label span excludes
  u_v, in characteristic zero.
• Inherited strict-floor corollary: any actual right space of dimension
  at least 24, in characteristic zero, under the same exceptional-left-ray
  condition. Its one-sided slack comes from PR2962's existing size relaxation.

Raw inclusion results handle proportional merging and the actual rank drops.
All actual spaces/ranks first discard zero-factor terms. Ambient inclusion
alone is not substituted for full actual right-space equality.

The partial-quotient extension uses two new sharp word bounds: nonroot
weight-eight words need 27 neighbor deletions, and root weight-seven
words need 29. An explicit 24-neighbor word already refutes the stronger
eligible-root-only classification. The invoice proof nevertheless reaches
28 by bounding the whole coefficient-space support at low basis budget;
it permits the nonroot weight-eight words appearing at 27. A further sharp
nonroot low-word threshold is 55, so the next 29–54 cases involve root
normals only.

For normalized actual ranks (9,n), n ≥ 10, the floor gives
Σ ρᵢ^q ≥ 1 + 8^q + 7·9^q for 0 < q ≤ 1. An eleven-term rank-(9,10)
fixture attains this. The expression and concavity method credit adamgall's
PR2062; the new result is its coupled-space coverage. A noncoordinate
partial quotient also attains COPY 72 and survives normalization on both
sides. A normalized rank-(9,17) eighteen-term fixture has an actual basis
bill of 73 but COPY 72, showing that the single rebate is sharp. Raw-rank
padding controls explain the normalization requirement.

Only research/e8-coupled-coordinate-floor/ is added. This does not close
general C′, G1-MFREE, arbitrary partial reader spaces of dimensions 10–23,
masked/punctured decoders, 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 scoped structural obstruction. This is an off-card partial result,
not the card's construction acceptance gate; maintainers may give it lower
priority than current priority-card work.

PR3154's
ARCHITECT / RL-s1 compensation questions and
PR3755's higher-rank
residual motivate this work. PR3755 already proves the one-coset budget and
both-ranks-at-most-nine case; it does not supply the coupled rank-(9,10+)
consumers proved here. Its exact geometry and written premises are retained
with hashes, so no pending commit is required for replay.

Arham-Tahir64's PR2798
already proves charged-pure compensation on its fixed P112 completion. Target rank
eight alone is not the distinction. The additional scope here is full P120,
every actual coordinate extension, and the normalization-stable partial spaces.
The L120 programme in PR3074/3191 is adjacent; no conditional completion of
that programme is used as a premise for these two-sided theorems.

The read-only October 19, 11:19 UTC refresh screened 177 updated open/closed
PRs since 01:44 UTC, followed plausible matches to their pinned primary
sources, and read all 53 changed comments/replies in six relevant Discussions.
Those threads were fully paginated: 931 comments and 158 replies were
captured. Main, PR3154/3755 heads and all four live input pins were unchanged.
No matching completion or dependency counteraudit appeared in that coverage.
Other threads, private work and arbitrary PR diffs remain outside this
bounded check; it is not a novelty guarantee. No public claim was posted.

The closest new work is Clear1nsight's
PR3973, bounding
residual-rank-seven normal directions under a common P112 cut, and
PR4012, strengthening
an eleven-piece P112 support-incidence floor to 178. These provide adjacent
geometry and historical caller exclusions, not the full-P120 actual-basis
compensation or COPY invoice proved here.
PR3951 audits inherited
capacity/A2 inputs; it does not validate the weighted root-swap or PR3755
premises on which this package depends.

The changed-32-port search in Discussion1587 was released without a sub-64
construction. Source-level inspection also distinguishes PR3876's rank-112
identity and full-port tariff from this rank-eight Gram target and line-label
tariff; the new fixed-P112 gauge screens retain their different target and bill.

Validation

python3.12 -B scripts/research_package.py --package research/e8-coupled-coordinate-floor

• Exact root/parabolic/Fourier inputs, 109 unit minors, 61,320 root/character
  checks and all 49 strict-dimension cases through two dynamic programs.
• Relative bound 27 through 15,575,280 parabolic comparisons and a separate
  complete scalar-kernel method. Root-low bound 29 through all 14,400 ordered
  root pairs and a separate 10,851-plane fixed-pivot exhaustion. Nonroot-low
  bound 55 through 30,480 parabolic/pivot cases and a separate full-support
  scalar-stratum algorithm.
• Exact products, actual ranks, normalization and binary tariffs over Q,
  F3, F5 and F7; 60 actual equality bases and 7,200 cuts; noncoordinate
  equality fixtures and genuine product/compensation failure controls.
• Registered replay, committed-byte/source-history checks, official export
  integrity, optimized Git-free replay, artifact/bytecode controls,
  attribution preflight and git diff --check pass on the frozen candidate.

The two relative-cut implementations share the Weyl/parabolic reduction
for the complement-character branch. The separate AI-assisted audit sampled
that branch; it did not independently exhaust it.

Five Init-only Lean 4.21.0 declarations check the additive contradiction and
critical-state/actual-basis arithmetic, with warnings as errors. Full E8
instantiation, geometric dependencies, field transfer and theorem assembly
remain written. Internal AI-assisted review is not independent human
certification. Hosted CI and repository-wide builds have not been run.
Lean RSS observations are sampled; sub-second runs do not establish a
measured peak-memory bound.
The candidate contains one focused commit.

Credit

The singleton-ray and all-odd pair inputs credit conorholds' PR3755;
the weighted root-swap, Fourier and parabolic inputs credit PR3166.
DreamingOfClouds' PR2962 supplies the inherited prefix-counting relaxation,
and PR3154 supplies the scoped C′ / RL-s1 obligations. adamgall's PR2062
supplies the concave profile and argument. Arham-Tahir64's PR2798 compensation result is
distinguished above. Earlier E8, projection, compatible-independence and
all-odd rank-witness credits are retained in NOTICE.md and the vendored
source. Assistance and licensing are disclosed there and in TASK.json.

Open next questions

Extend the actual-basis cut argument to unrestricted partial right spaces
of dimensions 10–23; account for mixed weight-seven words starting at 29
neighbor corrections; and formalize the E8/linear-functional and invoice
bridges. The 24- and 29-coordinate controls prevent silently reusing the
stronger classifications beyond their proved boundaries.