Record that the rank-8/u2 16-child slack depends on the family split.

[unpaidmoon]

Change and mathematical scope

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

PR #3889's tight nine-call cell pays local mass as {8: 698, 2: 1} on every call, with centre 64 and two unpaid coordinates. Its pooled idle histogram still passes 16 additive rank-1 children at a = 1/98 and fails at 17. The same 16/17 signs hold for five calls of 100 families and four of 99. Transferring one family from one 100-call to another makes 16 fail and leaves 15 as the last pass.

This is not a rank-8 producer, a consumer, or a supplier admission. Scope is refined from complex-new-helper-inventory: the open question asks for a physical contract charged against that slack, and no such word is on this main. The note charges the integer family splits the coverage identity already allows.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6. Card complex-new-helper-inventory. Package research/rank8-u2-split-boundary.

Item taken: #3 of the ordered eligible list below. Nearest work is open PR #3889, which states the pooled 16/17 screen and leaves legal pooling open, and discussion comment 19029639, which asks for the rank-8/u2 contract. Open PR #4066 is a different rank-16 surcharge against the five-percent gate. A cached overlap scan at 2026-10-20T03:21Z covered 842 open PR bodies and 2670 discussion records; no match prices this family split.

Ordered eligible list. Chronological open #1166 next tasks, dropping a task when a later DONE, an open PR, or this account's delivered round through #4149 performs it:

1. Physical child-rank bound on one support alphabet (comment 19021222).
2. Invoice-level audit of the composed T488 ownership and outputs (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. This rank-8/u2 slack. The producer named by comment 19029639 is still absent; the piece charged here is the legal per-call family split inside that stop test.

Dropped before that cut, among others: the #3605 replay (open #3681), typed actual-width H (#4123, #4127), the single-hub factorization (#4124), the L22 census (#4117–#4120, #4148), the residual consumer (open #3943), the T385 replay (open #3936), the #3942 review and moved-ADD invoices (#4134, #4122, #4144), the shear words (#4126, #4135, #4146), the tensor bridge (#4125, #4131, #4143), event 58 (#4149), and the stock rational (#4130, #4141).

Validation

python3.12 -B scripts/research_package.py --package research/rank8-u2-split-boundary

Local CPython result: PASS_RANK8_U2_SPLIT_BOUNDARY pooled=LT/GT balanced=LT/GT shifted=LT/GT no_new_kappa and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result. The moment test uses pinned ceiling.child_cost enclosures. Controls reject a 77-family call and check that replacing one rank-51 child by 51 rank-1 children raises the moment.

Not run: a census of every integer split, a rank-8 word search, selected cold replay, Lean, and hosted CI.

Credit

Douglas Colkitt, with Claude assistance, for research/complex-local-word-ceiling. ToikY for PR #3889 and Nick Petalas for PR #3874. Coverage identity PR #1012. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

See the package README. The useful next step is to classify which nine-call splits of 896 families, each at least 78, still pass 16 additive rank-1 children. A literal rank-8 word with this paid profile is still missing, so the screen does not reject a construction.