Price the register-333 K/adjoint round trip as Z_256.

[unpaidmoon]

Change and mathematical scope

Partial ledger for one physical boundary. No new κ. Selected conditional κ stays 0.001672660197076918.

On the frozen four-map native-lift inventory, the new-version witness warp at register 333, index 4, and the adjoint descriptor that swaps its entrance and terminal, use

G = (1, 2, 8, 32, 44, 128, 59, 66, 489).

Recomputing the source tag formula gives t_G = 144 and t_{G^{-1}} = 68, so

K_{G^{-1}} K_G = Z_{256}.

256 = 2^8. The character is one coordinate sign. It flips 256 of the 512 addresses and the binary product is the identity. The normalized section satisfies J_{G^{-1}} J_G = J_I, whose phase is 0, so substituting J for K deletes that character.

The sign-+1 descriptor already stores phase = 3 and linear_Z4 = (1, 2, 2, 0, 2, 2, 3, 2, 2). That Z4 vector is not the coboundary mask 144. Helper, routing and tape fields are absent. Those charges are unknown, and unknown charges are not zero. The exact sign adds no denominator. A favorable complete bill is not established. This is not a scalar-addition saving or a supplier admission.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main, priority revision 6). Upstream main was unchanged at submission.
• Task: independent-moment, scope changed. The card asks for an independent supplier-moment check. The catalogue method used is its frame reconstruction: one coordinated substitution and the single-change control that drops only Z_256.
• Handoff: Handoffs and requests for independent checks #1166 (comment)
• Ordered eligible Handoffs and requests for independent checks #1166 handoffs for this session, chronological from comment 19016211, one remaining next task per explicit marker, after dropping work delivered or refined by this round (through Price the cut58 ADDRESS swap against the separable response menu. #4149), later DONE notes, and open PRs that perform the named next step:
  1. (7, 18) / (7, 20) functional census (comment 19021170, question 1). Question 2 of that comment is Prove codimension-one L/2 substitution soundness on double transversals. #4148.
  2. T488 composed ownership/output invoice (comment 19022532, question 1). Question 2 was closed by Audit (no new kappa): the composed T488 profile is exactly PR3642's optimal packing vertex #3796.
  3. Physical rank-8/u2 contract against the 16 extra rank-one children (comment 19029639).
  4. Coupled source-normalization caller (comment 19035513).
  5. Physical K/adjoint/frame boundary price (comment 19036235). This session.
• Fewer than 10 eligible items remained, so item index (10 - 1) mod 5 = 4 is this fifth item. It is the handoff itself, not a further refinement of a closed one.
• Dropped before that cut, among others: the single-hub factorization (Prove the exceptional single-hub literal charge is 48. #4124, Record the three-hub extremal actual-reader charge. #4145), the integer-shear question (Prove faithful integer-shear words fail the supplier moment test. #4126, research: stop the one-XOR shear carrier on a lossless integer payload #4135, research: record the compensated-shear hypothesis for the R439 K foot… #4139, Price integer transvections on the shear target B^k. #4146), the research: certify local-first two-ADD obstruction on staged544 #3942 review (Review the staged544 fixed-frame new-gap reduction. #4134), the adjacent completed-call pair and its row/column reading (Certify the literal-slot defect that blocks a uniform P104 adjacent c… #4133, Certify that uniform P104 completion misses a port-dependent chart. #4136, Prove uniform completion fails on the physical P104 encoding. #4140), the A1858 challenge (open Certify current-provider A2 transfer and unchanged owner-cone allowance #4041, and the PR3734 acceptance in Audit the separated A2 composition and cleanup split #4112), the many-axis encoded review (Review the encoded many-axis production_axis guard. #4142), event 58 (Price the cut58 ADDRESS swap against the separable response menu. #4149), and the typed physical H (Add a typed one-round actual-width H and rounded return. #4123, Price one typed actual-width physical H and its toward-zero return #4127).
• Predecessor: PR Derive explicit native-lift phase tags and signed-word certificates #4002, head d593941ea0a4926021cba9d01fe60500579e464f. RECIPES.json is that package's source/RECIPES.json, SHA256 64b85f6aef485026eebaec7acddb41999f57e074b0dfde8077b6543700cd684b. That head is not a Git object in this clone. The checker does not import PR Derive explicit native-lift phase tags and signed-word certificates #4002's verifier. PR Audit staged544 native phases and expose the binary-inverse lift defect #3987 owns the canonical lift and the inverse witness.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/k-adjoint-boundary-price

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/k-adjoint-boundary-price outcome=partial.

Direct check, including under -O:

python3.12 -B research/k-adjoint-boundary-price/verify.py

prints PASS_K_ADJOINT_BOUNDARY_PRICE with central_child=Z_256, states_wrong_under_J=256, favorable_complete_bill=false, and new_kappa=false.

The checker rebuilds the inverse, both tags and the round-trip phase on all 512 states. It rejects a tag formula that drops the weight-one correction, and it rejects a claim that the central child is zero. Metadata pins passed with --check-only. The catalogue orientation command python3.12 -B scripts/research_baseline.py returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

Not run: Lean, Docker, a selected cold replay, the PR #3945 emitter, and hosted CI. The recipe rank field is recorded as 0 and is not interpreted as a free child.

Credit

Chris R / Clear1nsight, PR #3987, for the canonical lift and the inverse witness Z_256. ToikY, PR #4002, for the tag formula, the frozen four-map inventory, and the pricing question. Dugongue, PR #3945 / #3929, and the native lineage named there, for the recipe convention. The arithmetic on this pair was recomputed here.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review is claimed.

Open next questions

Linked from research/k-adjoint-boundary-price/TASK.json:

1. Can Z_256 be absorbed by an adjacent paid gate on register 333? First step: name the predecessor and successor of index 4 and multiply the three operators. Blocker: RECIPES.json has no neighboring gates.
2. What helper, routing and tape tariff does this warp pay in the emitter? First step: read the owner row for register 333 in the PR Raise conditional staged544 κ by filling idle owner slots #3945 native inventory. Blocker: those fields are absent from the frozen descriptor.

Workflow feedback (optional)

Chronological handoffs on #1166, after removing items this round already answered or refined, left five open next tasks. A single current-status line on each handoff, naming the one remaining question, would have made that cut stable across parallel sessions.