The end-cap wall: why copies are forced in the E8 skeleton's verifier model (no κ claim)

[DreamingOfClouds]

What is claimed, and what is not.
Scope. The k8 / #1018-type local word of the E8 complex unit: 112 targets with ports u_t ∈ F₂⁹ (distinct, odd weight, u·u = 1), frames that are subspaces of F₂⁹, the verifier rules R1 (an ADD climbs both operands to one frame), R3 (frames are monotone) and E (every target register ends in perp(u_t)); fixed ports, fixed decoder, fixed targets (gk_word.py:116, :102, :117; audit_local.py:137-138, :122, :115, :142). The deletion experiments run on the gauge-forward word of that skeleton (the stage before the port, cold44 and layer stages), regenerated by verify.sh from vendored inputs; the census, bound and window rows run on the #1018 word.
Result. (T, argued from the cited rule lines; instances checked) A direct read y_t += s at time τ forces the end frame of y_t to contain D_s(τ), the join of every frame register s had before τ; y_t must end in perp(u_t), so the read is legal only if D_s(τ) ⊆ perp(u_t), and otherwise one of four prices is paid: (a) a version of s (held or reconstructed), (b) a reorder that puts the offending operations after τ, (c) a FULL end or descent on the target (a rank unit per call), (d) a descent on s. (M) Deleting any of the 24 legal single S3 packet copies, the 16 hub copies or any of five larger families (up to 80 copies) leaves 4 to 64 targets ending FULL, one dimension outside perp(u_t), and no complement sink can host a FULL frame; all 30 distinct regenerated words pass the generator's own frame check and fail the verifier's end rule E (gk_word.profile run on two of them, the end-cap replay on all). (P) One unit costs −2.819% κ per call (linear to 32 units; no exact certificate at 35) against a helper worth ≈ +0.15% and a ceiling of +3.5152% for all 112 S4 copies, so a copy is about 19 times cheaper and the copies are economically forced where the obstruction fires. (E) The escapes we built are closed in their stated models: one-accumulator version reconstruction (0 orders, exhaustive over the 20-21 op cluster, four absorption variants), the snapshot hoist and slide (0 of 64, 0 of 86), rematerialising the 161 never-read helpers (0 of 65), coded target pairs and cold-slot holds (0 of 224), and the frame-conflict bound (111 helpers forced, 113 attained; #1140 proves a stronger 224 on the same word).
Not claimed. No κ gain, no word, and no proof that the 268 S3/S4 copies are necessary for every skeleton or op multiset. The theorem is a necessary-condition and pricing statement for the verifier model above. It does not exclude two accumulators (+1 helper per group; not searched), a new h = 8 family (needs a different decoder; C.6 gives none at window ≤ 9 for the current one), joint deletion and reordering at scale (only single hoists and slides were run), asymmetric pairings that unpin Lemma R, non-projection rational labels, or helpers (arbitrary ends via helper_final). No deletion word passes the chain, so no κ is computed for one.

Knowledge package research/end-cap-obstruction/, in the layout of #899: one verify.sh, a hard-failing step 0 (every file against MANIFEST.sha256, every vendored input against SOURCES.json: git blob ids, and for the #1018 word the hash that #1018 publishes), vendored pinned inputs, and every CHECKED row re-run from them.

python3 -m pip install -r research/end-cap-obstruction/requirements.txt
bash research/end-cap-obstruction/verify.sh /fresh/output/dir     # from a checkout or `git archive` of the whole repository

Results (26 rows: 19 CHECKED, 4 ARGUED, 3 RECORDED)

id	result	status
T1-T3	Read-drag lemma, the end-cap theorem with its four prices (a)-(d) and the last-hop argument, the version corollary (a snapshot is not enough when s is written between the drag and τ). The rule lines they use are read from the vendored files at the cited lines (19 anchors).	ARGUED; anchors CHECKED
T4	No complement sink helps: every violating y ends at dimension 9 = FULL with u_t in the frame (30 words), one dimension out; independent of Lemma R.	CHECKED
M1	The generator regenerates the pre-chain base word (hash recorded): no violator, native profile {1: 1928, 2: 736, 3: 336, 4: 168, 5: 168, 6: 32, 8: 8}; the #1018 tail reproduces its profile and κ = 1.67166528321877e-3.	CHECKED
M2	24 legal single-copy deletions: each has NVIOL 0 and exactly 4 targets end outside perp(u_t), all FULL.	CHECKED
M3	16 hub copies: 16 violators (y160..y175), frame violation at y160; for the CF = 0 copy the drag of packet 283 (dimension 5 → 7, 8 → 9) puts 170 -= 591/2 at dimension 9 > 8.	CHECKED
M4	Five larger families: 48, 48, 64, 48, 64 violators, all FULL (the 84-word table of the cards is RECORDED).	CHECKED; table RECORDED
M5	The generator's own check does not see the final cap; without the lazy widening the deletion is refused at register 299.	CHECKED
P1	Unit price −2.819% (1), −11.275% (4), −45.061% (16), −90.022% (32); no certificate at 35.	CHECKED
P2	Ceiling of all 112 S4 copies +3.5152%; a deletion pays only if it breaks ≤ 0.0156 caps per copy; the 80-copy word breaks 64.	CHECKED
P3	Two helpers removed, freed mass at rank 48: +0.298% (≈ 0.149% each); other conventions on the cards give 0.04-0.14%.	0.149% CHECKED; range RECORDED
Q1	Port geometry: block ranks 4; cross-block pairs/triples/quads have common-perp dimension 7 / 6 / 5 (4,080,384) or 6 (19,712 zero-sum quads); 24,276 zero-sum quads; orthogonality degree 52.	CHECKED
Q2	Today's sharing is already the minimum span (54+25+24 by pairs, 16+16 by triples, 56 by quads); regrouping by port removes no copy (net helpers 0).	counts CHECKED; bound ARGUED
Q3	Copy census: 248 helper-from-register copies followed by reads of both registers; after the copy s fails the ancestor floor for the readers in 213/248 (versions in all, drag in 208); 35/248 pass the necessary test.	CHECKED
Q4	Second encoding: 136 registers read by targets of ≥ 2 blocks, op-excess min 1 / median 3 / max 13, 0 fit; the deleted packet 283 has excess 22, 103 mass for one copy (1.8-4% against a credit of 0.08-0.13%).	excess CHECKED; conversion ARGUED
E1	Version reconstruction with one accumulator: reconstruction is value-exact with no per-operand floor (4 terms, each inside perp(u_58), perp(u_63)); absorbing into 591 or hoisting the non-capped readers leaves 10 resp. 8 violators; exhaustive search over the 20-21-op cluster: 0 orders for all four absorption variants; the floor is the cross-pair join.	CHECKED (cluster model); join reading ARGUED
E2	Snapshot hoist and slide on #1018: 172 → 64 windows → 0 frame-feasible; slide 86 candidates, 0 positions.	CHECKED
E3	The 161 never-read helpers: 96 rotation-locked, 65 algebraically removable, 0 of 65 frame-feasible (single, joint-65, joint-53).	CHECKED
E4	Frame-conflict bound: ≥ 111 helpers forced (CP-SAT OPTIMAL; greedy 113; 124 with non-helper holders); one per Y target, not the 480; weaker and fixed-order compared with #1140 (about 256 helpers remain unexplained beyond its 224).	CHECKED; reading ARGUED
E5	Cold-slot capacity: a coded pair on a cold target 112/112, warm 0/112, a second pair on a target 0/224; value gate 112/112.	CHECKED
E6	All 78 mergeable early-cleanup splits are blocked by a capped partner; lifting a single gate is legal for 2 of 116 (both Δκ = 0).	CHECKED; scalar rewrites RECORDED
E7-E9	Chain port stage cannot carry cold-hold words (B245); cross-block sharing census 0 (B255); asymmetry verdict: valid, no uniformity rule, windows ≤ +0.787% (B240).	RECORDED
E10	The cap is three stacked rules (swap contract SPEC, w_t = u_t convention, no descent convention); the two conventions are inert on every window.	ARGUED

Open problems and escapes

1. Two accumulators: terms join only inside their pair, y170 reads both; +1 helper per group (owner re-plan), estimated to eat most of the credit; rescue check is a second accumulator in cards/B256/dfs.py (not run).
2. A new h = 8 family (puncture: merge two source lines): one merge fixes one pair of a reader triple; needs a different decoder.
3. Joint deletion and reordering at scale (cap-ascending drag ordering); only single hoists/slides and one cluster are closed.
4. Join-frame relay (copy at the Y-partner frame), reader-set-aware decoder (change the u_t), decoder/motif redesign; the helpers the frame bound does not explain (about 256 beyond the 224 of Research: sharp k8 factory register bound and target frame barriers (no new kappa) #1140; value/algebra side).

Tested

Fresh git archive of the commit, every assertion PASS, exit 0: 91 PASS lines (2 in step 0), 0 FAIL, about 8 minutes wall on 12 cores of a shared machine (the four exhaustive orderings searches of group G5 set the wall time; everything else is under 90 s).

Credits

Builds on #595, #717, #446 and #899 (format limits, invoice prices, window/deficit theorems, helper minimality), #593/#599 (hyperplane-ended helpers; jeremykothe2, DreamingOfClouds), #555/#589 (bit supplier; eumemic, DreamingOfClouds), the k8 line #1005, #1018, #1031, #1065, #1087 (jeremykothe2; words and owner packings, cited and analysed) and #987 (rohanarun, the slot search), #645 (Gauge44; icekylinx) and #635 (emitter; jeremykothe2); #717, #446 and #899 are open. Prior observation: #651 (higherbeing; the next read lifts the frame out of the cap on the K-triangle word). Same word: #1140 (aliromman; at least 224 helpers, op multiset not fixed). Adjacent: #1143 (woklerx; commuting ADD moves and frame runs), #1186 (mdumitrean; exact frame solver with target caps), #620/#1035 (P112 asymmetric-sink screen), #1212 (burntout76; rank-8 decoders gauge-rigid). Overlap check: #698 (CK B frames at fixed boundaries; a different object), #1037 (asymmetric block rank 9d per profile; h stays 9; the h = 8 escape here is a different object), #1023 (width-7 bit headroom, bit side; window 0 for κ), #1032 and #1033 (factory coefficient work, not in the paid profile): no row duplicated.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 50a1c992ce28c95d83d11bc691353aaa289d9d65.

This remains open after scoped triage. Next review step: Large end-cap package has seven independent exhaustive/solver groups and universal wording to audit; verify the full generated-word inventory and exact model boundaries, not only stored negative-search claims.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.