Realize the selected caller's monomials and J routes as fixed f-independent record lists (no new kappa)

research/selected-monomial-tape-lift/PROOF.md

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+# The f-column lift of the selected caller's monomials as a fixed record list
+
+Scope: the physical half of `native_all_columns` for the selected #5413 caller, at the level
+of records. PR #5442 closes the algebraic half (every identity of the caller lifts to every
+column width `f`) and leaves open the tape realization of `g^(tensor f)` with spectators and
+adapters at the retained tariff. `SPECIALIZATION.md` §2–§3 (the P1 specialization of #4610)
+states the realization in writing: any nine-axis monomial has a recipe of at most 171
+retained elementary movement/phase records, each a compact-control selected-bit addition or
+a per-record phase, acting on all `f` columns with one fixed list of primitive types. This
+package executes that recipe for the caller's actual monomials and routes.
+
+## Setting
+
+A field with `f` columns on `n` logical axes has addresses `X = (x_1, ..., x_f)`, `x_j` in
+`F_2^n`, next to arbitrary spectator bits. The retained monomial convention
+(`prune_native.monomial`) is `m = (G, b, Q, alpha, a)` acting on one column as
+
+```
+|x> -> i^(a + q(x)) |G x xor b>,   q(x) = sum_i alpha_i x_i + 2 sum_(i<j) Q_ij x_i x_j  (mod 4),
+```
+
+and its lift (SPECIALIZATION §3) as `|X> -> i^(a f + sum_j q(x_j)) |(G x_1 xor b, ..., G x_f xor b)>`.
+
+Five record types, each acting on all `f` columns at once and on no spectator bit:
+
+| record | action | retained primitive |
+|---|---|---|
+| `XROW(i,k)` | `x_{j,i} ^= x_{j,k}` for every column `j` | compact-control selected-bit addition, slots `k -> i` (movement-paid) |
+| `NOT(i)` | `x_{j,i} ^= 1` for every column `j` | selected-bit translation (movement-paid) |
+| `S(i)` | phase `i^(sum_j x_{j,i})` | unit-S phase record |
+| `CZ(i,k)` | phase `(-1)^(sum_j x_{j,i} x_{j,k})` | controlled-sign phase record |
+| `I` | phase `i^f` | global `i` record, lifted to `i^f` |
+
+A row swap is three `XROW` records. `SWAP_BITS(i,k)` of the plan is a row swap.
+
+## Lemma (record realization, uniform in f)
+
+For every monomial `m` the list `R(m)` = [`alpha_i` copies of `S(i)` for each `i`; `CZ(i,k)` for
+each `i<k` with `Q_ik = 1`; the reversed Gaussian-elimination list of `G` with swaps expanded
+to three `XROW`; `NOT(i)` for each set bit of `b`; `a` copies of `I`] does not depend on `f`,
+contains at most `96 + 9 + 27 + 36 + 3 = 171` records, and executes the lift of `m` exactly on
+every address and every `f >= 1`, leaving every spectator bit unchanged.
+
+*Proof.* Each record is the `f`-fold product of the same one-column operation, applied to
+the columns independently (`XROW`, `NOT` act column by column; `S`, `CZ`, `I` add a phase that
+is a sum over columns, with the global record contributing `f`). So the composition of the
+list on `X` is the `f`-fold product of its composition on one column, and the phase is the
+sum of the one-column phases plus `a f`. On one column: the phase records applied first on the
+input address give `i^(q(x))` (the diagonal `Q` is carried by `alpha`, exactly as in the
+convention); Gaussian elimination gives elementary matrices `E_1, ..., E_r` with
+`E_r ... E_1 G = I`, hence `G = E_1 ... E_r` (each elementary matrix is an involution), and
+applying `E_r` first and `E_1` last maps `x` to `G x`; the translations give `xor b`; the
+global records give `i^a`. The bounds are those of SPECIALIZATION §2: at most 72 additions
+and 8 swaps (24 records), 9 translations, 27 unit-S, 36 controlled-sign and 3 global
+records. No record reads or writes a spectator bit. ∎
+
+Cost. Each `XROW` or `NOT` record is one compact-control selected-bit addition on the ordered
+pair of slots (`compact-control-movement.tex`, Proposition), costing `O(V((f log p)^tau + 1))`
+on a fixed number of tapes in the domain `f <= p`, `A = O(p)`, `K / log p -> infinity`, with
+the front/back control fields reserved as in `compact-control-layout.tex` ("the proposition
+applies to every ordered pair of slots"). A phase record computes a fixed degree-two residue
+from `O(f)` selected address bits per complete record and applies one of `{1, i, -1, -i}`
+(SPECIALIZATION §2); it is absorbed by the record fiber. The domain `f <= p` is the actual
+layer's own domain: `f <= e <= floor(b_in^epsilon)` and `p = 6 b_in` (SPECIALIZATION §3). The
+tariff per record is inherited from the Proposition, not re-derived here.
+
+## What is executed (`verify.py`)
+
+- The 896 compiled port wrappers of `NORMALIZERS.json` (112 ports x 8 orientations/paths):
+  the record list is executed on all 512 addresses at `f = 1` (and compared with the exact
+  Clifford engine's basis action on one wrapper per path), on all 262,144 addresses at `f = 2`
+  for two wrappers, and on 48 random addresses at `f = 2, 3, 4` for every wrapper, with random
+  spectator bits. Every wrapper is within the §2 bound; the largest list has 107 records
+  (69 movement, 38 phase).
+- The body's own monomials, which are not port wrappers: all 1,072 `MONOMIAL` operations of
+  `NEUTRAL-BACKWARD-v16c` (800 whole-job helper normalizers, 240 Z-map block maps, 16 paid C8 axis
+  routes and 16 mode aliases) and the 8 `output_N` words of the #5402 provider's mode boundaries,
+  which #5442's mode-output certificate binds (the 8 `input_K` words coincide with alias words).
+  These are 283 distinct words. Each is read off the exact engine as a monomial `(G, b, Q, alpha, a)`
+  on 46 basis states and certified to be the same group element (Pauli tableau and image of `|0>`);
+  its record list is then executed on all 512 addresses at `f = 1` and on 48 random addresses at
+  `f = 2, 3, 4`. The largest list has 76 records (60 movement), within the §2 bound.
+- The central translation `X0` is one `NOT(0)` record. `C0` (cgate on axis 0) and `D`
+  (Fourier on axes 1..8) are positive-rank Gaussians: in the retained framework they are paid
+  children of the recursion (`FINITE-PRECOMPILATION.md` §1, "one ordinary positive rank-r f
+  child on the actual owner's full stream"), not adapters, and no record list is claimed for them.
+- The literal `pre_word` and `post_word` of all 119 targets of `WHOLECALLER-PLAN-v1`
+  (32 main, 72 helper, 15 idle), 16,748 `SWAP_BITS` events in all, are executed as `XROW`
+  records on 145-axis addresses at `f = 2`; each word realizes the declared `permutation` or
+  `inverse_permutation` of its target with phase zero and spectators untouched. The 16
+  central `X` corrections of negative `Q` powers are `NOT(0)` records.
+
+## What is not claimed
+
+The Proposition's proof (the four-update rotation scheme with dirty compact fields, its
+overflow repair and the chunk-swap primitive) is not executed; the reservations of the control
+fields for the 145-axis layout are taken from `compact-control-layout.tex` as written. No
+numerical tariff, no new κ, no change to the selected record. The body monomials, including the
+400 helper alias pairs of #5442, are executed separately above (an earlier version said they
+were covered by the port-wrapper lists; none of them is a port wrapper). The forward body is not
+replayed.

research/selected-monomial-tape-lift/README.md

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+# The selected caller's monomials and J routes as fixed, f-independent record lists
+
+**Scoped positive result; no new κ.** PR #5442 (`allwidth-caller-certificate`) splits
+`native_all_columns` for the selected #5413 caller into an algebraic half, which it closes,
+and a physical half: executing `g^(tensor 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. This package executes the record-level part of that
+physical half, following the written realization of `SPECIALIZATION.md` §2–§3:
+
+- every one of the **896 port wrappers** is a fixed list of at most 107 records (the §2 bound
+  is 171), each an `XROW` / `NOT` movement record (one compact-control selected-bit addition
+  or translation) or a phase record (`S`, `CZ`, global `i`); the list is independent of `f`
+  and realizes the lift `m^(tensor f)` exactly, spectators untouched;
+- the body's **1,088 own 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
+  #5442's mode-output certificate binds) are fixed lists of at most 76 records each, read off
+  the exact engine as group elements;
+- the **central translation** `X0` is one record; `C0` and `D` are positive-rank children of
+  the recursion, not adapters;
+- the **16,748 `SWAP_BITS` events** of the plan's 119 `J` routes (main, helper and idle
+  targets) are three `XROW` records each and realize the declared 145-axis 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)^tau + 1))` on fixed tapes for
+`f <= p`, `A = O(p)`. That domain is the actual layer's domain (`f <= e <= b_in^epsilon`,
+`p = 6 b_in`, SPECIALIZATION §3), so the restriction `f <= p` is not a gap. What remains
+inherited is the Proposition's own proof and the control-field reservations of the 145-axis
+layout (`compact-control-layout.tex`). [PROOF.md](PROOF.md) has the lemma and its scope.
+
+## Evidence (`verify.py`, standard library, CPython 3.12)
+
+| check | coverage |
+|---|---|
+| wrappers, `f = 1`, exhaustive | 896 x 512 addresses, random spectators |
+| wrappers, engine agreement | 8 wrappers x 512 basis states against `clifford_exact` |
+| wrappers, `f = 2`, exhaustive | 2 wrappers x 262,144 addresses |
+| wrappers, `f = 2, 3, 4`, sampled | 896 x 48 addresses each |
+| body and mode words | 283 distinct words (1,088 uses), engine read-off certified as group elements, `f = 1` exhaustive, `f = 2, 3, 4` sampled |
+| `X0` | one `NOT(0)` record, `f = 3` |
+| `J` routes | 119 targets x (pre, post) literal words, 145 axes, `f = 2`, 12 addresses each |
+| controls (`test_verify.py`) | unlifted global phase fails at `f = 2`; a dropped row addition, a removed swap, a single-column phase record and an asymmetric `Q` are rejected; spectators untouched |
+
+Record totals over the 896 wrappers and the maxima are in RESULT.json.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/selected-monomial-tape-lift
+```
+
+## Relation to `lockstep-completion-pass`
+
+That package counts the records the compiler and the completion/splitting pass add per unit
+instance (normalizer pairs, relocations, child calls, bank scans) without pricing them. This
+package gives the record list and count of each normalizer and route of the selected caller.
+Together they give the record-level bill of the compiled template; the price per record is the
+Proposition's tariff in its domain.
+
+## Open next questions
+
+1. Execute the Proposition itself: a bit-level simulator of the four-update rotation scheme
+   with dirty compact fields `u, t, b`, the chunk swaps and the overflow repair, at small
+   `f, K, G`, checking exact XOR at the `f` selected positions and restoration of the fields.
+2. Verify the control-field reservations of `compact-control-layout.tex` for the 145-axis
+   plan layout (front fields before, back field after every active slot pair).
+3. Replay the forward body's wrappers (review only; the selected controller runs the backward body).
+
+## Credit
+
+Cao Yifan / intqwq (#4610 and the P1 specialization), Cinna Davis (compact-control notes
+and the 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. Prepared by Mark (Matarener) with
+substantial Claude Code assistance. Apache-2.0. No independent human or formal-kernel review.

research/selected-monomial-tape-lift/RESULT.json

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+{
+ "status": "PASS",
+ "bindings": 12,
+ "wrappers": {
+  "count": 896,
+  "max_records": {
+   "XROW": 60,
+   "NOT": 9,
+   "S": 22,
+   "CZ": 15,
+   "I": 3,
+   "movement_records": 69,
+   "phase_records": 40,
+   "total": 107
+  },
+  "sum_records": {
+   "XROW": 19548,
+   "NOT": 2380,
+   "S": 8960,
+   "CZ": 1204,
+   "I": 1120,
+   "movement_records": 21928,
+   "phase_records": 11284,
+   "total": 33212
+  },
+  "bound_s2": 171,
+  "histogram_total": {
+   "16": 19,
+   "19": 133,
+   "20": 25,
+   "21": 19,
+   "22": 156,
+   "23": 58,
+   "24": 52,
+   "27": 6,
+   "28": 44,
+   "31": 104,
+   "36": 7,
+   "39": 14,
+   "40": 7,
+   "43": 14,
+   "47": 11,
+   "50": 4,
+   "51": 14,
+   "54": 6,
+   "56": 2,
+   "58": 15,
+   "59": 3,
+   "60": 3,
+   "61": 2,
+   "62": 18,
+   "65": 2,
+   "67": 8,
+   "69": 18,
+   "71": 5,
+   "72": 5,
+   "73": 23,
+   "76": 24,
+   "78": 9,
+   "80": 6,
+   "82": 7,
+   "85": 15,
+   "86": 4,
+   "88": 8,
+   "91": 5,
+   "99": 6,
+   "104": 7,
+   "107": 8
+  }
+ },
+ "central": {
+  "X0_records": {
+   "XROW": 0,
+   "NOT": 1,
+   "S": 0,
+   "CZ": 0,
+   "I": 0,
+   "movement_records": 1,
+   "phase_records": 0,
+   "total": 1
+  },
+  "C0_and_D": "positive-rank children (cgate, fourier), not monomials"
+ },
+ "routes": {
+  "targets": 119,
+  "swap_bits_events": {
+   "main_targets": 3776,
+   "helper_targets": 10116,
+   "idle_jobs": 2856
+  },
+  "xrow_records": {
+   "main_targets": 11328,
+   "helper_targets": 30348,
+   "idle_jobs": 8568
+  },
+  "central_X_records": 16,
+  "conventions": {
+   "pre:permutation": 81,
+   "post:inverse_permutation": 81,
+   "pre:inverse_permutation|permutation": 23,
+   "post:inverse_permutation|permutation": 23,
+   "pre:inverse_permutation": 15,
+   "post:permutation": 15
+  }
+ },
+ "body_words": {
+  "occurrences": 1088,
+  "distinct": 283,
+  "occurrences_by_first_kind": {
+   "block": 248,
+   "mode_output_N": 8,
+   "originalmodecanonicalC8inputalias": 32,
+   "whole_job": 800
+  },
+  "max_records": {
+   "XROW": 60,
+   "NOT": 6,
+   "S": 21,
+   "CZ": 19,
+   "I": 3,
+   "movement_records": 60,
+   "phase_records": 29,
+   "total": 76
+  },
+  "sum_records_over_occurrences": {
+   "XROW": 9575,
+   "NOT": 96,
+   "S": 16317,
+   "CZ": 536,
+   "I": 399,
+   "movement_records": 9671,
+   "phase_records": 17252,
+   "total": 26923
+  }
+ },
+ "checks": {
+  "f1_exhaustive_body_words": 283,
+  "f1_exhaustive_wrappers": 896,
+  "f2_exhaustive_wrappers": 2,
+  "sampled_f": [
+   2,
+   3,
+   4
+  ],
+  "engine_agreement_wrappers": 8,
+  "route_f": 2,
+  "route_addresses": 12
+ }
+}

research/selected-monomial-tape-lift/SOURCES.json

@@ -0,0 +1,72 @@
+{
+  "schema_version": 1,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
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+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
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