[aliromman]

This construction gives integer reciprocal matrices E=P R with zero diagonal in both E and E^-1, and sharp correction rank rank(E-I)=2, for every n>=3. For this family and n>=5, an invariant diagonal metric of dyadic units exists exactly when n=4+2^h. The actual 112-port construction has rank(E-P)=16 and rank(E-I)=32, and supports complete local native reciprocal words.

The proof, native contracts and costs, independent publication review, and source manifest give the exact claim and evidence. This does not improve the complete kappa calculation. Productive helper progress, paid source records, terminal access and restoration, recursive admission, and the complete precision and supplier calculation remain open. Correction rank is not a native cost.

The native words act on 640 arbitrary input fields and leave the 416 fields outside the two data banks unchanged. Removing unused native cuts reduces the child count from 1,632 to 1,292. It retains rank mass 1,792, direct scalar price 6,440, and 224 endpoint maps. The side-map upper bound falls from 3,264 to 2,584. The metric-wrapper price is 7,080. Physical P contains signed halves and is not a field permutation. The independent review checks every original and coalesced word, each scalar-frame cut, the per-field chain contraction, and all 206 new coalesced phases.

The supporting supplement preserves the exact rank-30 and rank-16 responses, reciprocal-large words, coded exits, and two scoped obstructions. The rank-30 cover optimum applies only to disjoint dyadic rank-one clique blocks of sizes 2, 3, and 5 on the fixed small support. Rank 16 permits a nonzero inverse diagonal and needs odd-grid primes 5 and 7 for inversion, with prime 31 also needed for ordinary normalization. Its raw native word does not supply those conversions. Coded exits retain literal W F9 Q_p^-1 order and leave W present; they do not give raw endpoints or a complete caller.

The ordered three-call application gives a residual rank of at most 48 arrays. A conditional clean-reflection variant has hidden rank at most 32 arrays. These are scoped algebraic statements. No native cleanR32 service or terminal decoder is emitted.

The publication base is b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Its selected record names PR #4470, with conditional kappa 0.00302015. This package proves no transfer to that construction and does not change the selected record.

The historical dependencies remain pinned: PR #3133 supplied the core; PR #3148 supplied the byte-equal base and extra provider words; and PR #2588 supplied the retained exact Clifford compiler. The source notice preserves the full predecessor credit and Apache-2.0 notices. Substantial OpenAI Codex assistance was used. No independent human review is claimed.

Use Python 3.10 or later and the standard library. From the repository root:

python3 -I -B research/reciprocal-native-responses/replay.py --out /tmp/reciprocal-native-responses-replay
python3 -I -B research/reciprocal-native-responses/replay_annex.py --out /tmp/reciprocal-response-annex-replay

Both output directories must be new. The expected final lines are:

PASS_RECIPROCAL_NATIVE_RESPONSES
PASS_LOW_DISPLACEMENT_RESPONSE_ANNEX

The core replay passed nine checks in normal Python and nine with -O; plain receipts matched exact bytes and regenerated gzip files matched decompressed bytes. The supplement repeats its independent matrix, scalar, and scope checks, and the publication checker verifies all 12 raw words, 16 coded words, and 352 changed-exit phases. See the core replay receipt, supplement replay receipt, and independent review commands.

Negative controls reject a deleted nonzero transfer in each principal direction, a changed inherited recipe, the wrong reciprocal direction, and omitted pair-code restoration.

Inherited raw phases and endpoints retain their pinned predecessor spatial reviews; that older complete spatial replay was not repeated. No selected cold replay, complete supplier replay, recursive admission, or full integer-multiplication theorem check ran. Open next questions identify the concrete remaining obligations.

The original-J rank and reader annex adds two distinct results. The rank proof and independent review classify reciprocal extensions E=J+KMR, G=J+KNR of the actual clean J=I-KR, where RK=14I_8. Dyadic corrections each have rank eight. Over the rationals the minimum rank sum is eight, and every equality case is M=(12/13)P, N=(12/13)(I-P) for a projection P. This measures the specified correction terms, not the native or query cost of every implementation. The later E=P R response is a different family.

The literal coordinate rank-four words cover both responses, orientations, and historical providers. Each word uses 3,511 native children and rank mass 5,105. Forward words cost 165,594 dyadic real primitives and backward words 165,618, with 16 real division-by-13 scans priced separately. All 408 logical helpers are restored. C0 returns zero at I; C1 through C7 retain their arbitrary inputs. Clearing those seven owned outputs costs 14 more primitives. These rational endpoints are not completed dyadic recursive providers. Ordered physical q factors, controller composition, hosting, a complete precision guard, and the full invoice remain obligations.

The reader-cut proof and independent review instead use the later signed-metric E=P R model. They identify the ordered readers ell0 x, ell1 P0 t, and ell2 P1 P0 w at actual F/B/F cuts. Their returned frames are F9, F9 q_anchor, and F25; the middle record is a hole record. Each formation costs 136 complex ADDs, or 1,088 inherited real scalar units, if sixteen separate typed accumulators are supplied. It retains an arbitrary accumulator baseline. The count excludes record supply, accumulator views, preparation, retention, transport, leases, and late erase. No reader native-event replay or complete native invoice is claimed.

The fresh receipt records twelve normal/optimized checks: the scalar rank cases, conditional grid checks, eight original-J local words, reader producer, independent cut review, and literal binding. The fresh separate agent review records its independent replay and proof limits. The original-J word review binds changed centre calls to the retained native phase evidence; it does not repeat the 3,036-class phase sweep. The annex checks reject wrong reciprocal coefficients or noncommuting pairs, a missing entry translation, and a false FULL middle-record return. Reader results with absolute source paths use complete canonical JSON comparison after raw payload-hash binding. All other fields are retained.

With Python 3.11+, the standard library, at least 3 GiB free space, and a new output directory:

python3 -I -B research/reciprocal-native-responses/annexes/original-j-rank-readers/replay.py --out /tmp/original-j-rank-readers

Expected final line:

PASS_ORIGINAL_J_RANK_READERS_ANNEX

All 73 frozen sources and all 152 prior package files stay unchanged. The notice preserves the campaign credit, PR #3133, PR #3148, and PR #2588 source lineage, and the stopped-product grid credit to @icekylinx. Historical commits and Apache-2.0 notices stay fixed. Substantial OpenAI Codex assistance was used. The refreshed main context remains b58ad9ef; this annex proves no transfer to its selected PR #4470 construction and gives no complete kappa improvement.

The signed rank-eight coefficient annex adds a complete classification for the fixed signed 48-port family E=G-K^T Lambda diag(beta_v)K. The proof and independent review show that the full response is a dyadic unit exactly when beta_v=(1-(-2)^n_v)/12 for integers n_v. Requiring the full response and both specified principal blocks to be dyadic units restricts each beta to 1/4, 1/8, -1/4, or 1/16. This gives exactly 65,536 distinct responses, closed under inversion. Principal units alone permit more choices: beta=1/2 has both principal units but full determinant 5^8. The later E=P R family is outside this fixed-family classification.

The six-step ordinary gauge acts on existing arbitrary data without an initial zero or additional centre field. Its reverse gives the ordinary inverse under the principal-unit condition. This proves a representation, not a scalar invoice for every coefficient choice. The ordered interface has determinant one on the 48-port part; including the fixed 64-port complement gives full 112-port determinant 2^-64. No new native schedule, phase replay, helper-restoration word, routed composition, terminal, precision bound, or complete supplier invoice is supplied. No complete kappa improvement follows.

The fresh replay runs both unchanged checkers in normal and optimized Python. All four receipts match their frozen bytes. The producer checks six full inverse cases and all 65,536 coefficient tuples through the proved formulas. The independent checker uses Bareiss determinants, nine matrix cases, all-column gauge and interface checks, and a separate count recurrence. The infinite classification rests on its proof, not the bounded exponent controls. A fresh separate agent review records the final scope and independent replay.

With Python 3.11+, the standard library, at least 3 GiB free space, and a new output directory:

python3 -I -B research/reciprocal-native-responses/annexes/signed-rank-eight-coefficients/replay.py --out /tmp/signed-rank-eight-coefficients

Expected final line:

PASS_SIGNED_RANK_EIGHT_COEFFICIENT_ANNEX

The notice credits the coordinating, controller, bridge-review, oblique-native, and budget-geometry contributions and retains the parent native source lineage. All thirteen frozen inputs, accepted parent files, historical source pins, and Apache-2.0 notices remain unchanged. Substantial OpenAI Codex assistance was used.