Research: exact rank-48 packing cap for the selected fixed data route (no new kappa)

[JosieElliston]

Scope and result

This adds research/frontier1094-fixed-route-packing-cap/, a source-bound, standard-library verifier and proof note. No new κ. The selected #1094 data route has child ranks only 0, 8, 16, 32 and 48. Keep that fixed child inventory and the selected local word. Then no whole-helper owner reassignment into those children reaches complex primitive saving a ≥ 0.001682094870065. The exact rational interval root bracket is (0.001682094870064, 0.001682094870065). Under the retained balanced conversion, the planning cap is κ < 0.001679270178313, at most 0.395178% above the selected conditional κ. The selected κ reaches at least 99.606% of this favorable cap.

This narrows Douglas Colkitt's #1240 rank-51 halving cap (κ < 0.001687202645326) under an explicit fixed-child premise. It is not a new construction, owner map, selected frontier or exact seven-step grid result. Axis transfer across child boundaries, or a route redesign that creates rank-49 through rank-51 children, is outside it.

Task card and overlap

• selected-packing, priority revision 3, chooser nonce 51f534e584664a07443b4736c814deea, scope key ab5d8f0f0d19bbab8625, refined to the fixed selected route's residual mass relaxation.
• Package base and source pins 1f7a934975e5eacaf83511c9ac870263bf095e05. None of the pinned inputs changed on main up to 0b1cccf8, onto which this branch is rebased. Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Predecessor Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240 supplies the rank-51 coverage/halving model. The merged PR1276 bounds within-group idle-axis repartitions (ranks up to 51). The merged PR1387 sharpens that domain and gives a semantic counterexample. Their numerical caps are tighter but do not cover whole-helper reassignment among unchanged child bins. The open PR1685 shows that at a = 0.00169 at most 1,603 idle mass units may sit at ranks ≤ 48, and asks which residual groups are legal. This package answers that for the fixed route: all of them have rank ≤ 48. PR1252 (open) repartitions across child boundaries and is outside this scope. PR1343/Research: exact PR1094 neutral two-move census (no new kappa) #1617/Conditional T16 κ gain through neutral whole-helper hosting paths #1407 enumerate or construct concrete owner moves.
• scripts/find_overlap.py --refresh was run at 2026-10-14 05:50 EDT (446 open PRs, 418 discussions). It is not a novelty certificate.

Evidence

• Route ranks. The selected source manifest pins DATA-PHASES.json (SHA-256 c5916ff3…), and those bytes equal the selected replay archive member. The checker recomputes the GF(2) rank of every idle child's orthogonal_axes in all 112 recipes.
• Capacities are the route ranks (review fix). All 288 (owner replica, child) windows in the selected T16 PLAN.json have capacity equal to that child's route rank at every port. The two rank-0 children have no window.
• Hosting is a prefix (review fix). All 69,120 hosted helpers in the selected OWNERS.json.gz lie in their windows. The owner map is pinned by the selected manifest and by SPEC.json. In every one of the 32,256 groups, the offsets tile [0,U_g) with U_g ≤ C_g, and the remainders C_g−U_g reproduce the replay's idle profile {1:2276, 3:377, 7:7, 48:4866} exactly. This is the hosting proof's prefix and remainder rule, checked on the selected data.
• Mass split. The replay, SPEC.json and PLAN.json are bound by hash. The checker splits the complete profile into the per-call local word {1:2412,2:418,3:432,4:142,5:82,8:8} and idle mass 237,024, and checks PR1012's identity 1,041,408 = 72·14,336 + 64·144.
• Bound. For 0<a<1, an idle child of rank r ≤ 48 costs at least r·48^(-a)·73^(a-1). So the normalized complex moment is at least [144 Σ_r local_r (r/73)^(1-a) + 4938 (48/73)^(1-a)]/14336. A code-disjoint rational interval engine (reduced-argument atanh, Taylor remainder bounds) certifies both root endpoint signs. It independently reproduces Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's rank-51 bracket.
• Controls. Five mutations must each fail at their intended check: rank/axes mismatch, a rank-49 route child, mass-preserving local-word drift applied consistently to plan and replay, a window capacity of 49, and an overlapping hosted offset.

Reproduce and limits

python3.12 -I -B research/frontier1094-fixed-route-packing-cap/verify.py
python3.12 scripts/research_package.py --package research/frontier1094-fixed-route-packing-cap

Both pass on macOS with Python 3.12, and the standalone verifier passes on 3.10 and 3.14 (about 4 s). tests.test_research_onboarding and tests.test_research_replay_selection pass on the rebased branch. Not run: a cold selected replay, Linux CI and independent human review.

The calculation assumes fixed data-route child boundaries, PR1012's coverage law, the existing hosting operation and the retained balanced transfer. It relaxes timing, physical capacity and indivisible-helper constraints. All inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, analytic and bit H1–H5 obligations remain.

Open next questions

Can a capacity-aware relaxation within this fixed route rule out the hypothetical all-rank-48 allocation? Can an exact redesigned data route create rank-49 through rank-51 children without an offsetting controller or supplier charge? The README gives first steps and blockers.

Credit and assistance

Douglas Colkitt and Claude: #1240 packing ceiling. PR1012: coverage identity. Rohan Arun and Jeremy Kothe: selected #1094 word and the T16 owner-packing line. The selected review records further data-route, controller, supplier and composition predecessors.

Prepared by Josie Elliston with substantial AI assistance. OpenAI Codex drafted the package offline (session 14 of a donated overnight run). Claude (Anthropic) reviewed it, added the source-bound capacity and offset checks, fixed a control that was being rejected for the wrong reason, refreshed overlap and published it. A separate read-only Codex review flagged both fixes. No independent human review. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 691534353b8c5172f5c157e79496686d8c80f196.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Fixed selected data-route capacities all at most48, with every owner/window/prefix remainder bound checked. Relaxing all idle mass to rank48 gives certified a<0.001682094870065. Scoped to unchanged child boundaries/local word; not an attainable packing or general supplier ceiling.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.