Scoped negative result: classify E8's first decoder rank drop (no new kappa)

[RossRichardson]

Research note / scoped negative result; no new κ or physical supplier.

At 56 E8 root-line deletions the original binary reader code can first lose rank. This classifies every one of the 255 rank-loss supports and computes the complete minimum charge in the fixed star-reader model:

Deletion sets	Available reader dimension	Required live-target dimension	Minimum cost on 64 live ports
120	8	7	50
135	8	8	57

The distinction between available readers and required live-target rows matters at this boundary. In the seven-dimensional class, an eight-pattern basis using readers outside the required code is allowed and also costs 50. The proof prices both possible spans, so it does not obtain a false minimum by forbidding those extra patterns. Both ratios exceed 63/111. All other 56-deletion sets retain rank nine and cost at least 44 by a proved relaxation. Combined with PR939, this extends the fixed-star ratio exclusion through 56 deletions.

Only research/e8-first-rank-drop/ is added. It contains a proof, exact certificate with all punctures and basis witnesses, two independent implementations, adverse controls, source pins and the local sprint claim. The primary checks every rank table directly. The separate checker reconstructs signed roots, uses Fraction elimination on two representatives and explicit root isometries to check all 255 tables. This is a complete finite classification after a written reduction, not an enumeration of all 56-subsets or a formal multiplication theorem.

Reproduce and validation

Python 3.10+; standard library only; offline:

python3 -I -B research/e8-first-rank-drop/verify.py
python3 -I -B research/e8-first-rank-drop/independent_check.py
python3 -I -B research/e8-first-rank-drop/test_package.py

Expected primary marker: PASS_E8_FIRST_RANK_DROP cases=255 orbits=120+135 target_dimensions=7,8 costs=50,57 retained_lower=44 controls=7 no_new_kappa.

• CPython 3.14.6 and 3.12.12 pass, including isolated optimized execution.
• Seven mathematical rejection controls and all five package tests pass. Controls include incomplete target spans, underpricing, wrong principal dimension, omitted cases, source/evidence corruption, and output preservation.
• Literal scalar reconstruction checks all 64 live source columns for both basis types in both classes.
• Relocated package and staged Git-tree export pass both checkers; regenerated certificate bytes agree.
• make selected-record-check and staged whitespace checks pass. Full unchanged repository/supplier suites and hosted CI are not claimed.

Scope, provenance and credit

Base: 47481c8. OS-random seed 40, structural decoder track. Main refreshed to 007664306fff84fbb1c9a1b4053b69f439764034 with unchanged selected PR890 κ = 1661957774647029/10^18. No pinned coordination issue exists; the local claim records three candidates and the completed scope. The full overlap refresh covered 702 open PRs through #949, followed immediately before submission by recent arrivals through #953; no duplicate boundary result was found.

Consumed predecessors are pinned and retained verbatim: Chi Zhao / PR631 at fb893b5ee1317c2da3b297e6fe78eca0857d1a76, and Ross Richardson / PR939 at 3a8ec54d62fb8824c658dc7ed356f99423cac8cf. Their heads remained unchanged. Credit also Philipp / PR875, Joel Pulikkan / GamingPuzzled / PR620, DreamingOfClouds / PR595/602, eumemic / PR543 and the earlier E8 lineage. PR939 expressly stops before the rank-loss regime; this adds its exact first boundary, not another rank-nine replay.

Fixed root geometry, independent inputs, the fixed decoder matrix and additive star-rank pricing are essential. General fitting matrices, nonlinear or non-label readers, 57+ punctures and improvements to other supplier terms remain outside scope. All-size/compiler, common-label/mutable-source, hosting, precision/tape and analytic assumptions remain inherited. The selected frontier and upstream/ are unchanged.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance, including both implementations and proof development. Code independence is not independent human review. Apache-2.0.