[aliromman]

This PR adds exact centre-entry, rational-code, and retained-reader services for two actual low-displacement responses. It supplies an arbitrary-state moving entry swap, a square code on the existing data bank, exact capture and decode words, and image tests for legal code transitions. It also supplies six joined D25 forward services: both providers for stage pairs (0,1), (0,2), and (1,2). A fixed-reader phase inverse remains available when the reader cannot change.

These are component constructions and enabling proofs. The complete kappa calculation does not improve. A complete routed caller, inherited helper host, terminal code removal, finite map price, and recursive precision bound remain separate obligations. The proof index, fresh replay, and independent publication review give the evidence.

For E=I+UV, B=VU invertible, and Q^4=I, the corrected entry uses J=Q^2 U and D=B^-1 VQ^2, so DJ=I. Its complete swap on arbitrary data a and a separate arbitrary centre bank z is ((I-JD)a+Jz,Da). It is an involution. One use stores old z inside the data; it does not clear or restore z. The rank-30 and rank-16 entries cost 5076 and 4448 real dyadic primitives, respectively, plus 224 translation calls. Rank 30 adds 22 real multiplications and 22 real divisions at each of 3 and 5; rank 16 adds 16 of each at 3.

The separate square code uses the existing 112 data fields. Its unnormalised dyadic variant requires dyadic U, V, and U_P^-1; a dyadic minor inverse alone is insufficient. For a scalar C with CEC^-1=K, the code moves through C Q^2, Q^-1 C Q^-1, Q^2 C, and Q C Q, then returns. Each actual advance must satisfy the stated commutation conditions. The final raw decode is Q C^-1 Q, and the physical cross-bank calls retain their odd phase. Neither the cycle nor the square code supplies a complete productive caller. The root-proof addendum states both required normalisation qualifications.

The rational services have complete local invoices:

Service	Old real scalar units	Additional real fixed-odd scans
Rank-16 whole-bank E or E inverse	3420	16 at 5 and 16 at 7
Rank-30 capture or inverse	2910	22 at 3 and 22 at 5
Rank-30 capture, G0 or G0 inverse, and decode	6090	22 multiplications and 22 divisions at each of 3 and 5

Forward code scans multiply; inverse scans divide. Valid-image decode is exact without odd root scaling. Generic inputs still need the exact grid. The rank-30 image has quotient (R0/3R0)^11 direct-sum (R0/5R0)^11 for R0=Z[i,1/2]. A complete intervening M must pass L_p M=S_p L_p on every data and helper column. An arbitrary ADD can fail this condition. The conditional whole-state theorem bounds odd height by (D+1) ell_p+c_p only after the caller proves matching code types, decoded child-entry bounds, and local-prefix bounds. It does not admit a current complete caller. These whole-bank rational codes are distinct from rational-unitary coordinate packing.

Each joined D25 service checks all 640 arbitrary input columns and preserves 416 outside contents. Its scalar cost is 17624, with 1520 children, native rank mass 1792, and a monomial call bound of 5776. The physical input level is t-1 even when the retained reader label is s=0. The complete regular cover must finish before the global query. The DATA partition does not provide the inherited typed helper host. The check binds the unchanged native prefix and all 35 phase-source files, then checks the new DATA phases and routed query. It does not repeat the earlier 4194304-coefficient spatial phase sweep. No separately packaged complete backward native service is supplied.

The fixed-reader appendix proves an inverse for every positive tensor width. Its physical kernel has 2712 terms, denominator 120, and infinity norm 88/5; odd factors 3 and 5 are necessary. Its complete section swap costs 208028 dyadic real primitives, 12072 translation calls, and 60 real multiplication-by-15 plus 60 real division-by-15 scans. Its reader is T^-1 VQ, whereas the earlier terminal needs VQ^-1. The moving construction changes the reader to remove that mismatch. All counts are local component invoices, not finite execution costs or multiplication exponents. The package description gives the exact contracts and next questions.

Reproduce with Python 3.11 or later, the standard library, at least 3 GiB of free disk, and a new output directory. No network is needed. From the repository root:

python3 -I -B research/centre-code-services/replay.py --output /tmp/centre-code-services-replay

Expected final output:

PASS_CENTRE_CODE_SERVICES

All seven checks passed in normal and optimized Python, and the independent review repeated them in a relocated copy. All 170 runtime and proof inputs kept their hashes. Compressed outputs match after decompression. JSON comparisons preserve every value except registered absolute source paths, which become package-relative paths. No checker source was patched. Controls cover arbitrary-state restoration, both inverse orders, exact division domains, syndrome failures, mismatched code repetition, and wrong DATA phase order. No archived producer, complete spatial phase suite, selected-frontier replay, full recursive supplier, or Lean formalization was rerun. The source record states these boundaries.

The upstream base is b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. At that refresh, the selected record names PR #4470, with conditional kappa 60403/20000000 = 0.00302015. These historical constructions do not change or claim admission into that selected result.

The consumed predecessor pins remain fixed. PR #3133 supplies the actual coupled core at 51eb9c4ee9e4722010d308aa3309e039d3940cc2; its source credits @dcharb78 for the folds. PR #3148 supplies the provider words at f2927a4b8afcf0701a6b945e14f30de1d94f5176, which credits Jeremy Diviney (@jeremydiviney) for its packing exchange and integration.

The recipe lineage from PR #2588 retains its notice at 8cd9dc88e24f062e2ca2fa7f4844a3d3319381fb, including Cinna Davis (@1337isnot1337)'s dirty-borrowing method, Rohan Arun (@rohanarun)'s package work, and Jeremy Kothe (@jeremykothe2)'s retained methods. Both referenced PR #2226 notices are preserved in the package notice. No compiler author is inferred from a submitter account. The retained common-grid source at bce54fb3f895cbe5cd4c0eb477c3afeed100190f and upstream notice credit @icekylinx for the stopped product-ring and exact grid method. Apache-2.0 and contributor notices are retained. Substantial OpenAI Codex assistance was used for the proofs, constructions, checks, and package.

The independent publication review was performed by a separate Codex agent; no human peer review is claimed.

The append-only pivot-code proof publishes the original literal common-entry invoice that the earlier services package used only at the matrix level. The code uses sixteen pivots and 96 complement fields within the existing 112-field bank. It needs no zero input, new field, or native child. It conjugates the actual response to -5 on eight pivots, -7 on eight pivots, and identity on the complement. The source pins, fresh replay, and independent review bind this local result. The accepted 277-file core remains unchanged.

Each direction costs 470+664+944=2078 old real scalar units, plus sixteen real multiplication-by-three scans forward or sixteen division-by-three scans inverse. Stored-prefix norms are at most twelve forward and eight inverse; the inverse adds at most three binary fractional bits and one odd-three level. Its eight disjoint block divisions do not add eight levels. The literal entry C L applies 112 translations before C; the inverse has another 112 maps. All scan, descriptor, map, root conversion, and complete precision costs remain separate.

On a valid C image, every inverse division has the required factor three and needs no additional odd root scale. A generic inverse has the actual coefficient 1/12, so it is not a dyadic unit on arbitrary code coordinates. A middle operation needs its own image-preservation proof. Generic inverse access instead requires the retained common-grid premises on every numeric stream, completed dyadic children, and exact outer unscaling. The formal output type q C q diagonalizes the physical response by adjacent cancellation, but supplies no productive centre advance, native output cut, raw decoder, or complete caller. There is no complete kappa improvement or automatic applicability to the selected construction at base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51.

Six normal/optimized runs passed. The unchanged producer checks all 640 inputs and 528 unchanged roles, both scalar and spatial inverse orders, the exact invoice/grid, and the wrong-side-square control. The unchanged prior review checks the actual pivot algebra with a separate engine. A small no-import check confirms the invoice and all eight valid-image divisions on 112 columns; the generic 1/12 witness rejects a broader dyadic-unit claim. The prior review receipt is compared in full after only registered source-path conversion; compressed programs match after decompression. Other receipts match byte for byte. Native phase compilation, scan bit loops, and a complete recursive supplier were not rerun. Use Python 3.11 or later and its standard library, one worker, at least 3 GiB free disk, and a new output directory:

python3 -I -B research/centre-code-services/annexes/pivot-code-invoice/replay.py --out /tmp/pivot-code-invoice-check

Expected final output:

PASS_PIVOT_CODE_INVOICE_ANNEX

The notice preserves the original construction and review roles, licenses, native lineage, and the retained common-grid credit to @icekylinx. Grid source pins retain commit bce54fb3f895cbe5cd4c0eb477c3afeed100190f. These are existing AI-assisted campaign results, with substantial OpenAI Codex assistance in packaging and checks. No independent human review is claimed.