Research note: restored DATA-shear boundaries for PR #521's native BIT query (exact query costs, no new κ)

[commdata2338]

Change and mathematical scope

Status: research note with exact verification. No new κ. The selected certificate, its κ (51110230636731/31250000000000000) and upstream/ are unchanged. All files are new, under research/native-fusion-restored-shear/.

For the unipotent ADDRESS permutation h of #521 (Karl Appel / @applecrusher, head a190fdb39f31647fc6f651625353710624627348), this completes the DATA shear (x,y,z) -> (x XOR y XOR h(y), y, z) under three explicit workspace contracts:

Implementation	Recursive children	Persistent full private rows	Workspace endpoint
Query the DATA target	2	0	Every unrelated dirty row preserved
Query only private storage with triangular XOR charts	4	1	Arbitrary private contents restored
Owned disposable storage, with paid copy and clear	1	1	Private row returned to zero

• Query minimums.
  • The two-query minimum is proved in the fixed-h, arbitrary-dirty-row XOR model.
  • The four-query minimum holds under the narrower private-only restriction.
  • Neither lower bound covers arbitrary alternative ADDRESS primitives.
• The one-query implementation relies on ownership that permits overwriting the private input. It is not a dirty-restoration implementation.
• Costs. Complete typed cost recurrences (PROOF.md §6) charge every child, affine event, XOR, copy, clear, routing and inherited tape cost.

Predecessor work. Details and pins are in NOTICE.md and SOURCES.json.

• Native BIT ADDRESS fusion: two recursive queries to one #521 supplies the ADDRESS map, its seven-event lowering and the dirty-module representation. This PR does not review or approve the rest of Native BIT ADDRESS fusion: two recursive queries to one #521.
• The scalar-projector/group-algebra model comes from Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ) #288 (eumemic) and Prove nine calls are necessary for three abelian bit registers #468 (Ali Romman).
• Certify 64 depth-coarse profiles and an adaptive-recursion limit #83 (Joseph Demarest) scoped the fixed-menu recursion barrier, which this note does not reopen.
• The OpenAI Weighted dual-suffix carriers: conditional κ = 7.9961717358e-5 #109 framework interfaces remain premises.

Open obligations.

• Place a completed shear inside an actual supplier.
• Authorize its DATA query or disposable storage.
• Price the resulting whole profile.
• Handle arbitrary widths.
• Discharge the inherited all-size and tape interfaces.

No saving is inferred from the local counts.

Research note, proof, attribution, source pins, and completed local claim.

Overlap. No pinned coordination issue existed at either check. The pre-submission refresh covered 494 open PRs and found none on #521's restored shear. The nearest are #663, which audits #628's nonlinear dirty restoration, and #665, which prices a width-9 spectral split. Both study different objects.

Validation

From the repository root (Python 3.10+, standard library only):

python3 -I -B research/native-fusion-restored-shear/check.py --check

Output, re-run on base 66849e1 before submission:

PASS_RESTORED_SHEAR_INTERFACES
60880 signed projector pairs; 4681 native-chart words; 776 BIT basis columns per program; 17 negative controls
Independent full-domain replay: 729 addresses; 2916 basis columns per program; 17496 endpoint/inverse cases

• Finite checks.
  • Exhaustive enumerations: 60,880 signed projector pairs; all 4,681 native-chart words through length four; dirty BIT columns on full and shorter prime-power orbits.
  • An independently implemented sparse-set replay of all 729 addresses in (Z/3)^6.
  • 17 negative controls, all of which the checker rejects.
• Proof arguments. The universal claims, the two-query classification and the private-only restriction are written proofs in PROOF.md. They are not inferred from the finite replay.
• Repository checks. make selected-record-check passes: PASS selected record, exact kappa, original source inventory and proof supplement: 51110230636731/31250000000000000. git diff --check is clean, and no existing file or upstream/ byte changes. The replay takes about 1.2 seconds on Linux with Python 3.12.3, standard library only.
• Not claimed. This is not a formal verification of the multiplication theorem and not a cold replay of the selected supplier. It is also not a full repository suite, Linux CI or Lean pass.

Sprint: recursive-interfaces track, OS seed 57. The claim and its result update are recorded in CLAIM.md.

Submitted by @commdata2338 with substantial AI assistance, under Apache-2.0: OpenAI Codex (GPT-6-Astra) wrote the analysis, proof and checker, and Claude Code packaged the submission. No independent human peer review is claimed.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Completed fixed-h DATA shear: two target queries, four private-only dirty queries, or one owned-disposable query. Laurent determinant and projector classification reviewed, complete finite checks passed. Typed recurrences are not self-recursion.

The scoped package check passed at 613c02b8b4411f991f108dcc1481b3f9100b996f. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.