[Lean] Hyperplane-end helpers: multi-gate cleanup tails and the cold specialisation (stacked on #593, no κ claim)

[DreamingOfClouds]

No κ claim. This PR extends #593's hyperplane-end Lean (jeremykothe2) in two directions:

• multi-gate cleanup tails, with several helpers at once;
• the cold specialisation, where the helper enters at E = ⊥.

It is stacked on #593 and adds one commit, touching only research/hyperplane-end-lean/.

What is proved (HyperplaneEnd/Multi.lean)

• hpe_labels_multi: the word a ++ mtail ++ b ++ posts is well labelled, and every hyperplane-ended helper ends at (done, j). This holds for a general cleanup_tail: any number of gates t += q • s or t *= q, each at its own frame, where each operand climbs to that frame unless it is already there. A helper may be target or partner of several tail gates, and several helpers may interleave. Hypotheses:
  • the last tail gate touching j runs at term j;
  • the rest of the cleanup never moves j;
  • the entrance frame is arbitrary.
• hpe_word_multi: Lemma A on streams for every such helper, given the per-helper (N), as in [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593.
• hpe_labels_tail: Tail.lean's one-gate word is the one-gate instance when both operands start away from U.
• exact_cold, cold_hyperplane_width: given T_E = id, which holds for the canonical frame of E = ⊥, (N) reduces to T_U = L (C ⊗ id) R, and ⊥ ≤ U with dim U = 8 gives width 8. [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593's hpe_word already accepts an arbitrary entrance frame, so the Lean content specific to E = ⊥ is just this specialisation.

Scope

• Not checked against emitted words: the shipped hyperplane-ended helpers of Conditional κ = 0.001609915765452068: hyperplane-ended helpers on the PR588 witness #591/Conditional κ = 0.001612719427075359: hyperplane-ended helpers with ck/atm3 dispatch #592 have one tail gate each. Multi-gate and scale tails are not checked here against emitted words. This PR does not change the instance checker check_hpe_pairs.py.
• Hypotheses outside Lean: as in [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593 §4, (N) for each pair (E, U) used, the unit row hS, hrec, and (O′)/(W) are hypotheses checked outside Lean.
• E = ⊥ check: (N) was checked locally on all 511 hyperplanes of F₂⁹ with clifford_exact from main's direct-delivery504-global/vendor (tableau and global phase). That script is not shipped here.
• Possible clash: Multi.lean derives DecidableEq for XLab. If [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593 later derives it in Labels.lean, delete this line.

Checks

• Verify: python3 research/hyperplane-end-lean/verify.py passes. All 27 named results ([Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593's 13 plus 14 new) depend only on propext, Classical.choice and Quot.sound. There is no sorry, admit or axiom.
• Fresh build: an independent audit rebuilt LemmaG, LemmaA and HyperplaneEnd from a fresh git archive, reusing prebuilt Mathlib, and verify.py passed there.

Credit: #593 (jeremykothe2) is the base package and the single-gate theorem; #509 / #516 (DreamingOfClouds) are the Lemma G / Lemma A packages it builds on.

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

🤖 Generated with Claude Code