[GoGoKo699]

The field route had a paid finite decoder but no uniform native word for its division-dependent recursive maps. This revision changes the source representation explicitly: a moving Pauli helper query appears on both sides, and two multiplication-XOR batches pay each recoupling level. The complete source/probe/actual-adjoint echo returns every moving helper and private mask on arbitrary inputs.

The uniform decoder has at most 16(p−1) T bands, T count O(mp³), dirty width O(mp²) and elementary-gate count O(m²p²). The direct indicator pays four decoder/adjoint appearances and two field queries, with at most 64(p−1)+16 bands and an explicit shared workspace pool. These counts fit the precise relaxed fusion budgets; the linear depth does not meet the sublogarithmic target. R96 and selected Result D remain unchanged.

The revision also determines why the proposed fixed-frame batching approach cannot remove that depth dependence:

• For a full field intertwiner with exact Clifford sectors and a linear Pauli helper query, the helper shift rank and helper-dependent symplectic degree are at least p−1, including moving residual core helpers. The argument survives full-intertwining error below 1/√2. The augmented construction attains degree exactly p−1.
• A product of helper-affine symplectic stages in a fixed helper frame therefore needs at least p−1 stages. An explicit nonlinear helper-motion example guards the scope: this is not an unrestricted native T-depth lower bound.
• A separate weak complete-probe theorem restricts the Pauli domain without requiring a full Clifford endpoint. Independent four-wire cells have error floor 1/12 for p≥3, strengthened to 1/4 for p≥5, even after arbitrary joint outer Clifford walls. Projectively additive Pauli helper queries are included with literal sign handling.
• The finite coupled allocation uses 148 T gates, at most 39 bands, and six returned dirty helpers, with 1,110 forward and 1,166 actual-inverse gates. The 172-T, 83-band allocation without extra helpers remains available.

Validation completed on head 378e23c02a6c03d5f9fe8d694fb5a733c4ffce86, tree fc1fb46d64268a0cb8cf9694f36a9ac45dfb9e37:

• All 2,087 tests passed locally, including the 33 new exact tests. The final ten-command gate passed on Python 3.12.14 / NumPy 2.3.5; the test suite took 1903.124 seconds and the whole gate took 1937.662 seconds. The tree remained unchanged throughout.
• All four numerical CI jobs passed, each with 2,087 tests, all 33 new tests, all four fault-tolerant suites, and all 16 steps. Both pull-request jobs checked the synthetic merge c8efd1b47378fb4bc23fd54eb12151edf700ebdd, whose tree exactly equals the reviewed tree and whose parents are the recorded base and head. Both push jobs checked the reviewed head directly.
• The new fixtures cover native private-mask return, shared-target batches, every computational-helper column at two and three address bits, actual inverses, weak-probe phases, signed tableaux, group-algebra coefficients and scope counterexamples. The optional tableau audit checks all 32,768 helper sectors through five address bits; scalar phases remain certified by the analytic operator proof and bounded full-column fixtures.
• Focused presentation checks passed for 13 pages, five SVGs, 597 unique math expressions and 761 protected occurrences, with zero failures. Eight desktop/narrow views were manually inspected; navigation and all 21 presentation guards passed.

CI event	Python	NumPy	Tests	Test time (s)	Evidence
pull request	3.11.17	2.4.6	2,087	2414.229	complete job
pull request	3.13.16	2.5.4	2,087	1910.702	complete job
push	3.11.17	2.4.6	2,087	1756.473	complete job
push	3.13.17	2.5.4	2,087	2593.792	complete job

Full presentation CI passed all 9 steps. Its head-bound presentation artifact is presentation-previews, 350,124,016 bytes, digest sha256:497a6f0aa4b529dc2d244419126e979c8c05f4413a753910727023b6cd9cb5aa.

Exact-tree receipt and complete-log SHA256 hashes

local gate receipt: 6373f954d4cf20e2c7867da7007fe5c678905e9fa652dfc6da9efb84b6dbf410
CI qualification receipt: a385cbd3bf7314ee3337a09df1812444a8b15b99dd602b0261df087a225b715b
manual visual review receipt: 910816abaaafce2319297a23273b16b179e61bef68cc1b810a104892e52732ed
focused presentation report: 5a142c0551b8df82fa3072eec2f4e37093818742ceabc92b2c5470fccb19c1e1
pr-python311 complete log: 27a7278a68d0cc05168695bc2896fa971fe3b15cd5c4f51c612c2c2a480e68fb
pr-python313 complete log: 1417da63dcf53050a5987ad77a28efe2cf539a86af744a6c13efea17c7a328c6
push-python311 complete log: 0c0d6906f803fbb31c5d72d1159df6bff86f1311bab9ffd2f5066bacd6b05d40
push-python313 complete log: 4dcd1c14bf3044eb3fc34fb8344468b88fd5684a6f62901733828421031ce305

The working checkpoint, research routes and verification catalogue now point to single proof homes and the next construction obligation: leave the fixed-frame helper-affine stage grammar through a paid complete native word. Existing anchors and prior allocations are preserved.