[petalas]

Summary

A backward safety reduction closes the arbitrary-plane question left open by Cinna Davis / 1337isnot1337's PR1996. Claim type: scoped proof with an exhaustive finite obstruction. On the exact historical PR1094 gauge44 word, no binary two-plane injected at one helper-helper gate improves the fixed-template ceiling under least common-frame closure and unchanged internal/terminal interfaces. No new kappa.

43,435 planes × 2,525 seeds = 109,673,375 pairs
  backward criterion: 919,876 legal pairs
  remove 718,317 no-ops: 201,559 changed closures
  exact pricing: 45 profiles, baseline uniquely best

The backward intersection lemma applies over every field and to arbitrary injected subspaces. The finite application covers binary planes only. The 44 nonbaseline profiles already have normalized moment strictly above one at 1690054109/10^12, below the baseline root. All induced climb ranks, the eight unchanged centre descents and rank-51 idle mass are included. The source scalar identity is inherited; the predecessor census and native auditor are not replayed.

Only research/complex-all-plane-rails/ is added. The contribution is produced by Nick Petalas. Credit Cinna Davis's temporal-rail question and contract, PR1094 and the retained gauge44/frame-coalescence lineage, and Douglas Colkitt's exact interval engine. Source notices remain unchanged.

Evidence

• Before: PR1996's coordinate-plane result explicitly left arbitrary planes open.
• After: a written necessary and sufficient backward criterion plus generated exact evidence for the complete arbitrary-plane domain. There are 108,753,499 inadmissible pairs, 919,876 legal pairs and 45 distinct changed-closure profiles. The rejection complement is not presented as a taxonomy of failure causes.
• Complete warning-strict normal and optimized registered replay passed on CPython 3.12.15/macOS arm64. Direct runs took 7.450 and 7.545 seconds respectively. Both printed PASS_ARBITRARY_PLANE_FRAME_RAIL_CLOSURE pairs=109673375 changed=201559 profiles=45 controls=28720.
• A separate full-frame forward propagation checks all 45 profile witnesses. Another 28,672 small schedule/plane cases check the backward criterion, with three further rejection controls. Assertions are not proof gates.
• Metadata, source pins, exact committed bytes/source history and official 12-file export/input-integrity checks passed at e1ec7b370176e14bb6309969968c0ae04c4aa556.

From the repository root, reproduce with the actual interpreter used locally; substitute an installed CPython 3.12 path on other machines:

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

Limitations and unrun obligations

This is a historical finite-word side result under the unchanged PR1996 single-injection/least-closure contract. It does not impose a ceiling on the selected P104 word, changed scalar programs, different boundaries, nonminimal closure, multiple injections or arbitrary finite networks. Historical interval-engine correctness and the source scalar identity are inherited. Full native operator, physical frame delivery, stock, rounding, hosting, compiler, outer and all-size theorem obligations are unrun and unclaimed. Independent human and formal proof-kernel review are unclaimed. No selected replay, Lean, Docker or historical suite was run.

Related open work

• #4993 explores joint retiming after carried swaps in a different BIT/native family. This result fixes the historical PR1094 complex word and one plane injection.
• #3729 optimizes integer idle packing for the same fixed baseline profile. This result compares arbitrary-plane frame variants under the historical fractional idle relaxation; the two results use different idle-packing models and answer different optimization questions.

The completion refresh found no open account PR created since the advance CLAIM. The account's two remaining open PRs, #6204 and #3729, both predate it; #6204 concerns the offline export checker. No new same-domain result or numerical disagreement was found in this refreshed scope. No additional predecessor computations were run.

Open next questions

1. Apply the backward criterion to one exact current-word event/barrier contract, with a complete signed invoice. The historical ceiling here cannot price that contract.
2. Test two coordinated injections with overlapping future cones, charging every induced climb. No interaction search or general frame-inflation obstruction is supplied here.

CI notes

Both hosted checks passed at exact head e1ec7b370176e14bb6309969968c0ae04c4aa556 in run 40964522070: plan and replay (research/complex-all-plane-rails). The completed checks were read once and confirmed successful. The maintainer has merged the PR.

Hosted replay certifies the registered normal and optimized checks at this head. Local metadata, exact source-history/committed bytes, complete direct replays and official export/input-integrity checks also passed. Each registered command has a 120-second timeout and finished locally in under eight seconds. No CI failure remains to report. Independent human and formal proof-kernel acceptance are unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen predecessors and the selected frontier are unchanged.