[petalas]

Summary

Scoped proof with an exact finite certificate, produced by Nick Petalas. For the fixed adjacent A8 roots v=(0,1), a=(0,2), the natural correction space D=span(delta_a,E_a restricted to H_v) has exact minimum reader-basis tariff 78 over every characteristic-zero field.

29-neighbor root-low correction
  -> complete 511-character kernel table
  -> deficient words lie in three planes
  -> filtration dimensions 1,1,1,1,1,1,3,3,10
  -> exact basis profile (1,7,7,9,9,9,9,9,9,9)

Every basis has sorted weights at least that profile. The same basis therefore minimizes every nondecreasing tariff, with value F(1)+2F(7)+7F(9). Under PR4052's singleton-encoder and exceptional-ray hypotheses, its strict-floor corollary gives COPY >= 77 for this actual reader space. This is a lower bound, not an exact COPY optimum.

This answers one concrete candidate at conorholds' PR4052 29-neighbor opening. The source's threshold and COPY proofs are cited premises, not audited or replayed. Credit conorholds' PR3755/PR3166, DreamingOfClouds' decoder programme, adamgall's concave accounting and the retained E8 lineage. The new contribution is the explicit quotient classification and exact basis tariff. No new kappa or selected-frontier change.

Evidence

Before, PR4052 leaves mixed weight-seven reader directions beginning at 29 neighbor corrections outside its general 28-neighbor argument. After, this natural two-dimensional correction space has the exact profile above, and cannot reach the 72-unit baseline under the stated coupled model.

• All 511 nonzero character kernels agree with rational nullspace construction and a separate exact annihilation/modulo-101 rank certificate. The prime supplies a deterministic rational-rank lower bound, not a probabilistic inference.
• All seven nonempty unions of the three disjoint supports, the attaining ten-vector basis and seven failure/boundary controls pass. The universal coefficient and field quantifiers follow from the written kernel, disjoint-support and scalar-extension arguments.
• CPython 3.12.15, macOS arm64, standard library only. Warning-strict normal and optimized direct runs pass, taking 1.35s and 1.18s in the recorded local check. Both registered commands passed locally and remain far below their 60-second limits. No performance improvement is claimed.
• PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES with declared source ancestry, and official six-file export integrity pass. Normal and optimized direct replays also pass in a read-only Git-free export with empty PATH; all exported files remain unchanged. An occupied receipt path is rejected.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/e8-root-low-plane/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/e8-root-low-plane/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/e8-root-low-plane
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/e8-root-low-plane --check-source-history

Other installations can substitute their CPython 3.12 path. Direct replay emits PASS_E8_ROOT_LOW_PLANE in a generated JSON receipt bound to all six package hashes; optional --receipt PATH creates a new receipt. No receipt is hand-written or committed.

Limitations

Exactly one fixed correction space and root pair are covered. Arbitrary 29-site partial quotients, positive characteristics, masked targets and shared/dynamic delivery remain outside scope. The conditional COPY consequence retains the predecessor singleton classification and normalization premises. Physical access, restoration, native realization, compiler, stock, precision/tape and outer invoices remain open. All five selected framework assumptions remain. Selected cold replay, Lean, Docker, stronger unavailable static analyzers, independent human and formal proof-kernel review are unclaimed.

The current card is explicitly changed to this algebraic side result. Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8 and selected PR6030 are recorded. Live overlap included 522 open PR records, 6,535 Discussion records, recent account contributions, targeted merged/main formula searches, and a fresh precommit delta. Sparse notes, arbitrary historical diffs and unpublished work remain limits.

Open next questions

• Can a different correction space containing a 29-neighbor root-low word have basis tariff at most 72? Start by adjoining one independent correction and deriving its character-kernel filtration. No larger candidate or complete physical budget is supplied here.
• What is the exact coupled COPY optimum for this actual reader space? Start from the lower bound 77 and the source's exceptional-ray equations. An attaining factorization or stronger cut argument is missing.

Related open work

• PR2832 studies the broad full-E8 decoder-floor problem and reductions. This result resolves one explicit reader space under the narrower singleton-encoder contract.

• PR3045 concerns arbitrary fitting tables and square root-code factorizations. This fixes the dense table and permits a ten-dimensional reader space outside the root code.

• PR3040 concerns rank-eight pattern rigidity after deletions. It does not price this corrected reader space.

• PR5064 searches masked nine-factor completions. This proves an all-basis tariff for one explicit full-E8 ten-dimensional space, with no comparison of its search minima.

• PR6569 accelerates PR3166's complete root-swap replay. This adds a new quotient result and does not change or rerun that scientific command.

• PR1931 constructs a source-reflection supplier boundary. Its encoded-source contract and physical invoice differ from this local basis-price problem.

• PR647 supplies a paid-copy exact-RAM E8 realization. This is an algebraic obstruction for one different reader space, not a competing supplier or runtime bound.

• PR6582 classifies simultaneous rank-six A2 support equality in the full E8 label configuration. This result prices a different ten-dimensional scalar reader space; no common numerical quantity is computed.

• PR6583 accelerates the orthogonal-core functional-section consumer. Its fixed section histogram differs from this corrected-space character-kernel filtration.

• PR6584 accelerates the complete A2+E6 decoder replay. This result adds a quotient basis tariff and does not change or evaluate that decoder's COPY quantity.

CI notes

Both hosted checks, plan and replay (research/e8-root-low-plane), passed for exact head 277ff2c230937460aae5d2b19d83be37d0d60a5b in run 41617872127. The completed check list was read once; run metadata confirms the same head and successful conclusion. The registered replay covers the complete normal and optimized scientific checks. All registered commands also passed locally. These results do not establish independent human or proof-kernel acceptance.

Merge Danger

Door: two-way. Six new files under research/e8-root-low-plane/; no shared code, predecessor bytes or selected records change.

Blast Radius: package.