[Matarener]

Requested by Mark

Change and mathematical scope

Before: PR #5442 splits native_all_columns for the selected #5413 caller into an algebraic half, which it closes, and a physical half: executing g^{⊗f} and the port adapters on a fixed number of tapes, for all f columns at once with one fixed primitive list, without disturbing spectators, at the retained tariff. SPECIALIZATION.md §2–§3 (the P1 specialization of #4610) states that realization in writing (≤171 records per nine-axis monomial, each a compact-control selected-bit addition or a phase record) but executes none of it on the actual inventory.

After: research/selected-monomial-tape-lift executes the record-level part of the physical half. Status: one written lemma (record realization, uniform in f) with exhaustive and sampled execution; no new κ, no selected-record change.

• Every one of the 896 compiled port wrappers of NORMALIZERS.json is a fixed list of at most 107 records (XROW/NOT movement records = compact-control selected-bit additions/translations; S/CZ/global-i phase records). The list does not depend on f and realizes the lift m^{⊗f} exactly, spectators untouched.
• The body's own 1,088 monomial words (283 distinct: 800 whole-job helper normalizers, 240 Z-map block maps, 32 C8 routes and aliases, and the provider's 8 mode output_N words that All-width certificate for the complete #5413 caller: group-algebra lifting lemma, 896 port identities, free body, abelian controller (no new kappa) #5442's mode-output certificate binds) are fixed lists of at most 76 records each, read off the exact engine as group elements and executed at f = 1 exhaustively and f = 2, 3, 4 sampled. (Added in 377c2c7; the first version covered only the port wrappers and wrongly said the helper aliases were among them.)
• The central translation X0 is one record; C0 and D are positive-rank Gaussians, i.e. paid children of the recursion (FINITE-PRECOMPILATION.md §1), not adapters.
• The literal pre_word/post_word of all 119 targets of WHOLECALLER-PLAN-v1 (16,748 SWAP_BITS events on 145 axes) are three XROW records each and realize the declared permutations; the 16 central X corrections are one record each.
• The tariff per movement record is the compact-control Proposition (notes/compact-control-movement.tex): O(V((f log p)^τ+1)) on fixed tapes for f ≤ p, A = O(p). That domain is the actual layer's own domain (f ≤ e ≤ b_in^ε, p = 6 b_in, SPECIALIZATION §3), so f ≤ p is not a gap.

Remaining obligations: the Proposition's own proof (four-update rotation scheme, overflow repair, chunk swaps) is not executed; the control-field reservations of compact-control-layout.tex for the 145-axis layout are taken as written; no numerical tariff.

Task and overlap

Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04; package commit 36167d06b. Inputs pinned in SOURCES.json: SPECIALIZATION.md, the two compact-control notes, NORMALIZERS.json and WHOLECALLER-PLAN-v1.json (the #5413 capsule), clifford_exact.py, prune_native.py. Task card formal-interface-lemma, scope refined (TASK.json). Nearest work: #5442 (algebraic half; its open question is exactly this interface), #4610/P1 SPECIALIZATION (written realization), #5443 (our record counts of the compiler and the completion/splitting pass; together they give the record-level bill of the compiled template, the price per record being the Proposition's tariff).

Validation

python3.12 -B scripts/research_package.py --package research/selected-monomial-tape-lift

PASS locally on CPython 3.12 (Windows), standard library only (about 100 s; controls under a second); check_research_commit.py passes. Evidence: all 896 wrappers at f = 1 on all 512 addresses with random spectators; 8 wrappers against the exact Clifford engine's basis action; 2 wrappers at f = 2 on all 262,144 addresses; all 896 at f = 2, 3, 4 on 48 random addresses each; every wrapper within the §2 bound (maximum 107 records: 69 movement, 38 phase); X0; all 238 literal route words at f = 2 on 145 axes. Controls (test_verify.py): an unlifted global phase fails at f = 2 (the §3 mutation); a dropped row addition, a removed swap, a single-column phase record and an asymmetric Q are rejected; spectators untouched; transposition chains. Finite executions corroborate the lemma; the all-f statement is the written proof in PROOF.md. Not run: the forward body's wrappers, the Proposition's internal scheme, Lean.

Credit

Cao Yifan / intqwq (#4610 and the P1 specialization), Cinna Davis (compact-control notes, controller/helper lineage), zxv (#4359 free-word substitution, #5428), Chris R / Clear1nsight (#5413), Dillon (#5402), Ali Romman (#5383/#5398/#5376), Dugongue and icekylinx (Clifford engine lineage), and the parallel #5442 census. Apache-2.0. Prepared by Mark (Matarener) with substantial Claude Code assistance; no independent human or formal-kernel review.

Open next questions

1. Execute the Proposition: a bit-level simulator of the four-update rotation scheme with dirty compact fields u, t, b, chunk swaps and overflow repair at small f, K, G.
2. Map the plan's active axes and stage planes onto the q_0, q_F, q_B chunk reservations of compact-control-layout.tex.
3. Replay the forward body's wrappers (review only).

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