Research note: restoration deferral is mass-neutral in the hosted T37 recurrence (no new kappa)

[thakrarsagar]

Status: research note / scoped negative result. No new κ; the selected result is unchanged.

Track 50–59 (recursive interfaces), assignment seed 57 (OS CSPRNG via Python secrets). Analysis base bbc600db2fc15007b7c914f4f2b2ed55fa909c87; rebased onto 2921874911956718fcaaaf9396fe8267b3cfa9f0 and re-verified there (selected PR942 T37, κ = 0.001661960547713947, unchanged; all pinned inputs byte-identical). Overlap re-checked against open PRs at submission (#974, #987, #991, #992, #1000, #1009, #1010): none prices restoration deferral or the coverage identity.

What was learned

This note prices one explicit recursive-interface mechanism, restoration deferral, and counts every child the change creates. A U8 helper ends at the frame it holds before its final restoration climb. The host coordinates that climb advanced go back to the same hosting window, and that window's idle child advances them (hosting lemma, research/idle-axis-hosting/PROOF.md §2).

1. Coverage identity (exact, every live family). On the selected T37 payload, all 33,152 live families satisfy 8·k_f + 1 + hosted_f + idle_f = 73, and all 18,816 T21 families do too. So rank mass = 72W + 64N and the deficit is exactly D = W − 64N (11,840 and 6,720). This is the per-family form of the per-replica 8v − 9C in research/larger-gain/geometry. Outside the eight centre paths, which run on private work families, paid mass equals advanced live coordinates, family by family.
2. The full-frame inverse cleanup consumes every U8 restoration climb. 378 helpers (636 mass) and 112 sources (230 mass) have positive final climbs, and every one is touched by a full-frame cleanup gate. The other 126 helpers are already zero-restoration low ends. Index by index, the helper restoration equals the canonical provider word bound by the hosted template. In both emitted programs, the 490 final climbs have exactly the replay's rank multiset and run before full_frame_cleanup. The data contract fixes the sources' full endpoints.
3. Deferral is mass-neutral; its whole value is consolidation inside shared windows. Exact ceilings use the pinned engines and the selected outer witness:

Scenario	κ	Gain
selected baseline (reproduced)	0.001661960547713947	—
defer all helper restoration, selected owner map	0.001708640379500299	+2.809%
same, ideal re-hosting into capacity-48 windows	0.001730427469823602 (PR589 bit side binds)	+4.120%
best single helper index (9), all calls	0.001662126446464193	+0.00998%
relaxation: one rank-one climb per call merged into a 48-child (exceeds every window capacity, so not realizable)	0.001662339932746932	+0.0228%
PR974-style zero-cost deletion of one rank-one child per call (leaves 333 live coordinates unadvanced)	0.001708820303553035	+2.820%

The value is strongly superadditive. Single-helper moment drops sum to 53.9·(mW)⁻¹, while joint deferral drops the moment by 325.0·(mW)⁻¹. Source restoration has no idle-window target in the fixed nine-stage template, so it is not priced.

Relation to concurrent and prior work

• Research: T21/T37 interface budget and bit-binding headroom (no new kappa) #974 (avirazza, open, same track) prices the same boundary classes with zero parent cost. This note reproduces its T37 one-unit value independently (last row). It shows that the restoration rows are coverage-productive: their coverage-preserving value is the consolidation above, not deleted mass. Of Research: T21/T37 interface budget and bit-binding headroom (no new kappa) #974's classes, only the centre path (64 per call) does not advance live coordinates.
• Conditional κ = 0.001614202339036103: 46 hyperplane-ended helpers per call via commuting-prefix cleanup #601 retimed cleanup with commuting prefixes to make 46 helpers per call hyperplane-ended. Research: full-center U8 composition and bounded cleanup retiming #935's retiming census found no local gain, but it held the final caps fixed, which excludes this mechanism.
• Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 supplies the hosting lemma, Independent verification: PR869's U8 word and κ with a code-disjoint checker (no new kappa) #881 the U8 frame semantics, and Dominik Scholz's research/larger-gain the 8v − 9C relation and the centre obstructions.

Reproduce

make t37-restoration-deferral-verify

The expected last line of verify.py is PASS_T37_RESTORATION_DEFERRAL_REPLAY, followed by 13 unit tests, OK. It runs in about 8 s on standard-library Python 3.10+ with no network.

• verify.py reads pinned members of the committed T37/T21 archives in memory, without extraction. The template payload equals the selected ASSEMBLY.json template_sha256.
• It recomputes CENSUS.json and PRICES.json and requires byte-equal JSON.
• Both pinned engines must accept b and reject b + 10⁻²⁴. The outer witness must accept κ and reject κ + 10⁻¹⁸.
• A double-precision bisection with no shared code must agree with every row.
• price.py's transcription of outer_witness is bound to the source digest and its literal lines, and the bit endpoint is cross-checked against ASSEMBLY.json.
• It runs nine negative controls: coverage mutations, cleanup removal, zero-cost deletion posing as a deferral, a pin corruption, the halving guard, the next endpoint and a perturbed baseline.

Scope and remaining obligations

These are exact identities on the frozen payload and exact model prices. They are not a construction and not a new κ, and neither ceiling is a lower bound for any mechanism.

Realizing any part of the ceiling needs all of the following, none attempted here:

• full-frame cleanup retiming;
• low-end cone and compensation validity;
• exact phases and adapters;
• hosting and owner admission;
• the finite invoice and assembly.

All inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, analytic-transfer and bit H1–H5 assumptions remain. Scalar maps, compensation and cones are not rechecked. The checked files are standard-library Python, and upstream/ is unchanged.

Credit

Selected lineage: Jonathan Capalbo (#942), Chafik Boukhalfa (#890), Jeremy Kothe and NewBArch (#869/#870), Hao Yan (#858), DreamingOfClouds (#589) on eumemic (#555).

Building blocks: avirazza (#974), Jeremy Kothe (#601), goatliamia (#935), DaysSky (#442), SethBurkart123 (#881), Dominik Scholz (research/larger-gain).

Prepared by Sagar Thakrar with substantial AI assistance from Claude (Anthropic) via Claude Code. New material is Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 4ceaf380ed41b28dadcc41d2f974f025ac4d8c03.

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

Reviewed scope: T37 per-family coverage identity and fully priced restoration deferral; moved work is mass-neutral, not free deletion.

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.