Research: fuse native BIT dyadic shear pairs with three children (no new kappa)

[VictorLiwentao]

Change and mathematical scope

The completed DATA-target shears of PR677 normally pay four native queries to produce x ^= (h^a+h^-a)y. For PR521's exact unipotent ADDRESS map and a=+/-2^j, this supplies the literal fused word h^-a; XOR_y; h^(2a); XOR_y; h^-a. A six-affine/one-child lowering of every signed dyadic query works over every odd residue ring. The completed pair pays 3 ordinary children instead of 4, 18 affine events instead of 24, and 2 XOR scans instead of 4, with arbitrary dirty DATA/spectators and no full private row.

Status: recursive-interface research construction; no new kappa. The written proof also establishes a sharp three-query minimum in the stated power-query/XOR model, even with arbitrary dirty row spectators and integer query powers. This is not a minimum over arbitrary tape programs or a physical runtime comparison. The first-difference two-query theorem of PR677 remains intact.

The complete typed recurrence prices actual full-row children at volume V/3 and width N/3, coefficient preparation, all affine/XOR scans, primitive temporary streams, source access, routing and head returns. The target requires exclusive access until completion; the source never changes. The new middle coefficient differs from the baseline and streaming constants depend on fixed j. The seed-selected a=2^54 fixture is exact coefficient evidence, not a practical speedup. A centered DATA shear cannot replace ordinary ADDRESS interchange without a separate compiler; supplier/frame/owner invoices and all-size/tape/analytic transfer remain open.

Only research/dyadic-query-pair/ is added. Exact task base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261. Consumed source heads are PR521 a190fdb39f31647fc6f651625353710624627348 and PR677 613c02b8b4411f991f108dcc1481b3f9100b996f, plus the original tape sections at 5996e33ecad6a0ccd578aec58831457829d75575. Seven original source/license/notice files are retained byte for byte with Git blob and SHA-256 pins.

OS-random assignment seed 54, recursive interfaces. No pinned coordination issue exists, so the claim is local. Final refresh covered 636 open PRs through #839, after a 646-PR snapshot through #833. Upstream advanced to PR799 (f567884aeb8851b6e794af1184119a3bfbe1e631) during the sprint; its selected record, review and actual X4 construction were read. The two consumed PR heads and tape sections remain unchanged. The selected frontier is not edited.

Concrete non-overlap: PR756 changes the historical balanced-Fano enumeration's performance; this adds no such checker or replay. PR805 builds real rational cyclic leaves and restores one dirty array plus a scalar; this instead concerns exact odd-ring ADDRESS permutations acting on F2 VALUE arrays with queries on the genuine DATA target and no full private row. Neither PR's branch/worktree, construction, restoration interface, code or verification is consumed or changed. Concurrent dirty/clean echo, Clifford valuation and newer supplier work are separate objects; inspected heads and attribution are retained.

Credit Karl Appel / applecrusher (PR521), commdata2338 (PR677), OpenAI's imported tape framework, Douglas Colkitt and the retained projector/group-algebra lineage. Prepared for Wentao Li (VictorLiwentao, @Is__Victor) with substantial OpenAI Codex assistance, under Apache-2.0. Original notices remain unchanged; no human peer-review, proof-assistant or literature-priority claim.

Validation

python3 -I -O -B research/dyadic-query-pair/verify.py

Expected: PASS_DYADIC_QUERY_PAIR: children 4->3; affine 24->18; XOR 4->2; no new kappa.

• 112 signed dyadic lowerings through index 55 (including the doubled seed-54 query), 1,792 rational columns and 16,128 modular columns, with literal inverses and ADDRESS spectators.
• 110 signed symbolic program pairs; separate outer-product/literal-matrix projector catalogues; all 91,320 declared balanced-power two-query products reject the centered target. The universal exclusion is the written proof, not an extrapolation from this census.
• Separate coordinate interpreter/direct push formula and sparse-set replay of the full six-field domains mod 3 and mod 5: 154,160 dirty four-role basis columns per program, 7,552 further prime-power orbit columns, complete baseline/fused/inverse checks and 92 simultaneous dirty-row cases.
• All 25 mathematical/admission controls pass. python3 -I -B research/dyadic-query-pair/check_package.py passes offline relocation in a path with spaces and rejects six deliberate source/certificate/program faults, including an unpaid child and integer-to-Boolean substitution.
• Final normal/isolated optimized checks pass on macOS arm64 CPython 3.13.2, with identical complete evidence on CPython 3.12.14 and 3.9.6. Every retained source also matches its exact fetched Git object. Selected-record integrity and staged whitespace checks pass.
• One make verify on an untouched exported base passes seven historical groups, then stops at gen4-bank-verify with missing SymPy (exit 2). No full-suite, selected-supplier cold-replay, Linux/remote CI or Lean pass is claimed.

PROOF.md, CERTIFICATE.json, PROGRAM.json, SOURCES.json, VALIDATION.md and the local claim provide the review map and exact limits. Finite replay supports the scoped argument; it does not formally verify the multiplication theorem.

[Mister-Meeseeks]

100-PR batch review at da2cd0dd2229e08b61b174a8835a2c2bb99d2230.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Three-query completed native DATA shear pair in the specified dyadic ADDRESS-query alphabet; paid lowering, not an ordinary interchange supplier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.