[JoyboyBrian]

The actual frame506 route in #4989 contains an ordered shift from high-coordinate address 0 to address 3 with coefficient 1/2. Modulo Q=q^(w-1), its multiplier is (Q+1)/2: repeating that many complete unit-shear calls would introduce a factor growing with the supplied width.

This package supplies one literal finite controller for y -> y+c*x+C mod Q, and its inverse, at every complete volume V=P Q^2 G S, with arbitrary positive binary extents and 0<=c,C<Q. Each generic direction has 877 local microinstructions, nineteen tapes and seven symbols. The all-size proof charges descriptor conversion, coefficient advances, cyclic resets, payload copying, every rewind and complete scratch erasure at 1024V transitions and 64V visited cells. Nonunit coefficients work because x traversal has a separate Q counter. Source/descriptors are preserved and output is separately owned.

The linked actual half leaf has 934 microinstructions on the same fixed bank. Its literal 47-instruction preparer derives the binary (Q+3)/2 = c+1 descriptor from Q; a final scan erases that owned descriptor and returns its head. The same physical bound therefore includes actual half-coefficient preparation and cleanup. No scalar inverse, random-access gather, ordinary-interchange oracle or runtime host arithmetic is free. General runtime coefficients remain supplied descriptors.

Three substantive commits separate the general controller/dual replay, actual half binding/resource proof, and stronger discriminating controls plus reproducible packaging. The lost-coefficient test initially used x=0 and was corrected to x=2; a payload mutant initially changed an unreached bucket branch and was corrected to cover both output branches. These verifier corrections remain visible in the history.

Validation on CPython 3.12.2 / macOS 27.0 arm64:

• All four registered normal/optimized sparse/dense commands pass: 610 directional executions, 38,850 payload bits, with identical exact transition/span/transcript observations. Complete small coefficient families include zero/nonunit coefficients, all Q=3 basis columns, translated origins and nontrivial spectators.
• 10 complete linked half calls and 107 descriptor cases, including 40-bit carry chains, pass; malformed positive/canonical extents, coefficient bounds and invalid half moduli reject. Seventeen semantic/physical control groups include missing coefficients/offsets, wrong direction, corrupted output, retained scratch, nonlocal movement, changed source and endpoint-preserving extra work.
• A separate read-only AI peer independently recounted the transfer, comparison, phase, cleanup and visited-space ledgers and checked literal half preparation/erasure and actual source argument order; no blocker was found. This is not independent human or proof-kernel acceptance.
• Metadata/pins, committed bytes/source history (21 files) and official export integrity pass. All four direct modes pass in a fresh Git-free export. Actual exported source, program-table and typed-receipt mutations reject, including integer 1 changed to Boolean true; original bytes were restored. The final export changes only README/TASK documentation relative to that cold replay; every executed program, receipt and source byte is identical.

python3.12 -B scripts/research_package.py --package research/fixed-tape-affine-shear

The actual generator, serialized program and interpreter are pinned and the source assignment is checked, so the mathematical half-leaf interface is bound to its real read/write direction. The finite executions compress spectator dimensions; they do not materialize the full native high matrix. This result does not install the separately owned output into the original in-place owner bank or complete unit_scale, reversed-field routing, native/raw-body/ordinary callees, ROW padding or P3 stock/precision/tape/recurrence admission. Output installation and original-source retirement require the caller's separately paid ownership protocol. No selected-frontier change or new kappa is claimed.

Scientific base: canonical bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7. Canonical claim and scope. Entry overlap covered 310 open PRs, 4,096 Discussion records and all 6,690 tracked local notes; metadata coverage is advisory and not a novelty certificate. Full proof, source pins, commands, limits and Open next questions are retained in research/fixed-tape-affine-shear.

Credit David Arias / #2776 for the finite local machine, counters and split/merge macros; Charles Bethin for the fixed unit-carry successor; Jose Manuel and the retained ordered-affine/selected-stream authors; ai-nanachi #4760/#4814/#4326, Chris R / Clear1nsight #4957, #4989 and the native/ROW/ordinary/stock/routing/precision/tape lineage. Predecessor files remain unchanged. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Apache-2.0. No Lean, Docker, unrelated historical replay or complete multiplication theorem is claimed.

Submission refresh (2026-10-23 06:22–06:24 UTC): canonical main f259dabc7d70e0db8d6d36bfda888f008bbfecc1 was fetched; governing guidance, selected record and declared scientific inputs are unchanged. Scientific source base bee95ce88f9f19802f0613f92838895cd0e14a1f is intentionally retained; changed paths are disjoint. Refreshed overlap covered 304 open PRs and 4,114 discussions; local inventory covers all 6,690 notes at the scientific base. Coverage is advisory, not a novelty certificate.

Supplemental AI peer review is local-only evidence; the committed written proof, source bindings and registered verifiers are the public reproduction artifacts. Peer-only observations do not constitute independent human or scientific acceptance.