[unpaidmoon]

Change and mathematical scope

Negative stop, no new κ. Selected κ stays 0.001672660197076918.

PR #3947 sends B = X^2 Y^2 to one XOR, b ^= c, on a three-bit quotient. Discussion reply 19032401 asks whether any injective payload encoder makes that XOR a legal consumer. It does not.

• B^2 - I = [[28, 12], [12, 4]] has integer kernel {0}, so the order-2 carrier does not intertwine with B on any nonzero invariant address set.
• Identity 4-bit packing takes (1, 0) to (5, 2) at Hamming distance 2.
• Mod 4 is the largest quotient with B^2 = I. Every wiring into the six-bit caller fails. The quotient map itself is two disjoint XORs and does not use the carrier.
• The exact m-bit address caller is four additions. Fixed full adders cost 8m+16 XOR, 8m+16 AND and 4m+8 OR. Restoring the consumed source adds 2m XOR and 2m stored bits.

This is not a supplier, a group-algebra register, or a multiplication theorem. The separate faithful-word moment test M(a) < 1 is not redone.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6. Card bit-new-register-model, scope changed: the card asks for a new nonabelian alphabet, and this package prices the payload encoder left open by the existing shear carrier instead of searching a new word.

Ordered open #1166 handoffs, after exclusions, had 12 entries. Item 17 wraps to index (17-1) mod 12 = 4, item 5 (comments 19031016 and 19032401). The piece taken is the payload stop, not the moment inequality.

Nearest work: PR #3905 (four-call formal minimum), PR #3937 (commutative obstruction), PR #3947 (one-XOR carrier and the open encoder question). Overlap with those PRs is the predecessor statement; the kernel, wiring exhaustion, two-XOR quotient circuit, and adder bill are new.

Validation

CPython 3.12.3, standard library only.

python3.12 -B scripts/research_package.py --package research/sl2z-payload-encoder-stop

Expected: PASS_SL2Z_PAYLOAD_ENCODER_STOP ... wiring_hits=0 mod4_direct_xor=2 ... no_new_kappa=true and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result.

Also passed: metadata pins, check_research_commit.py at 900d5b61df7988895c5db95a8c36008f2229c1e8, and python3.12 -I -B -O -Werror on verify.py (checks are raises, not stripped asserts).

The wiring search is exhaustive for bit placement and constant scratch. It is not a search over arbitrary circuits. The integer kernel argument is proved in the README and corroborated on [-12, 12]^2. No selected cold replay, Lean, or Docker run.

Positive control: the same wiring search accepts the carrier's own XOR as a target. Negative control: three shears do not send (1, 0) to (5, 2).

Credit

PR #3905, 1337isnot1337 / OpenAI Codex. PR #3937 and PR #3947, ToikY / OpenAI Codex, including the payload question in discussion reply 19032401. New checks by unpaidmoon with substantial Cursor Grok 4.7 assistance. Apache-2.0.

Open next questions

Can the low two bits of an exact m-bit address use the mod-4 two-XOR map while a carry into the high half replaces one of the four additions? First step: one shear's high-half recurrence on a single m = 4 witness. Blocked if that carry is itself an addition; the split then does not beat 8m+16 XORs plus the matching ANDs and ORs. See research/sl2z-payload-encoder-stop/README.md.

Workflow feedback (optional)

#1200 — no separate session report.