[unpaidmoon]

Change and mathematical scope

Partial negative result. No new κ. The selected conditional exponent stays 60403/20000000 (0.00302015, #4470).

On the source-entry identities in research/r437-carrier-integration, the full parent payload is exactly 9475 * 3^124 times the executed 972-bit child packet. A fixed-tape blank-return that charges at least one visit per payload symbol is not paid by that execution.

lcm(28416, 28425) = 269241600 = 28416 * 9475
E = 2 * 3 * 2 * 3^4 = 972
V = 9475 * 12 * 3^128
V / E = 9475 * 3^124

9475 is the unused-row multiplier. 3^124 is the atom coordinates in the parent gap and absent from the executed gap. Either factor alone is a strictly smaller comparison. Gather, routing, checkpointing, table setup and the reverse coefficient stay unresolved. This is not a global admission.

Task and overlap

Base ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7. Task card r437-carrier-integration. Scope status: refined.

This is item 5 of the eligible list for session r27. Order: 19 October handoff index (discussion comment 19029944), then priority-table rows still open. Dropped first: work this account already delivered, results on current main, open PRs, and fork cursor/* topics.

1. Downstream consumer of (w, q, v_N) (comment 19030045).
2. Fresh T385 full-wrapper completion (comment 19030047).
3. Six-tape head-move bill at the L546 F0 event (comment 19030049), distinct from Price the F0 adjacent ADD after the first rank-1 climb. #4162.
4. BIT.md weighted-leaf and fixed-tape induction.
5. This package: parent spectator blank-return, open question 1 of research/r437-carrier-integration.

Excluded before the cut: retiming (comment 19030053; #4122, #4128, open #4567); four-provider rebind (research/mixed-provider-admission); QQ16 park/restore (open #4167, #4172); P104 slot/chart/encoding/fiber/port-chart work; compact-switch #4321; rank-8/16-child producer (#4151, #4156, #4158 and cursor/rank8-*); F_2^6 difference basis on main.

Nearest overlap is the carrier package itself, PR #4067, unpaidmoon #4147 (reverse coefficient, not repeated) and #4160 (constructor traffic inside preparation, not this ratio).

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/r437-parent-spectator-blank --check-only
python3 -B scripts/research_package.py --package research/r437-parent-spectator-blank

verify.py prints PASS_R437_PARENT_SPECTATOR_BLANK with blank_return_discharged=false. test_verify.py recomputes the lcm by a subtraction gcd and rejects witnesses that drop either factor. Metadata pins passed. Not run: Lean, Docker, the selected cold replay, physical.py, and the 9475-row parent packet.

Credit

Extents, 269241600, 9475, the 972-bit tape and the unpaid blank-return are Chris R / Clear1nsight, research/r437-carrier-integration, from PR #4067, with the reverse-coefficient distinction from unpaidmoon #4147. Lineage named there remains theirs, including ai-nanachi / #3647, Ali Romman / #1043, and Sagar Thakrar / #3700 and #2423.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Exact fixed-tape visit count of one blank-return of the 9475-row parent, including cursor motion across the 124 omitted coordinates. The factor here is a payload lower bound. Blocker: the published child bridge does not allocate that parent.
2. Instantiate one of BILL.md's remaining terms G, C or F at the original root without setting the others to zero. A blank-return lower bound does not sign the whole inequality.