[petalas]

Summary

Prove that unchanged PR2697 Callback.weight stores at most h+1 RETURN records on its value stack. This sharpens PR2758's four-tape retained-backing allowance while keeping its source contracts and live-symbol proof:

- 2F(h) + T + B + 2
+ F(h) + (h+1)B + T + B + 2

Here h>=1, A=2I+4, B=R+3, T=max(A,B) and F(h)=(h+1)T+hB. The package also proves exact depth-zero backing, accounts for arbitrary warm history coordinatewise, and states the conditional A>=B binary-grammar sharpness boundary.

For the already published GL64 depth-two example, the certified allowance drops from 340 to 214 cells, matching its known observation. The new result is the all-depth source lemma and per-tape rejection checks, not a new measurement or runtime saving.

Scientific source: Jose Manuel/jmbarrancoml's #2697 at a53fdf4bcbe33c6732f0bec1badc8c023380f9d1, authenticated through Nick Petalas's unchanged #2758 exports. Exact pins, proof, receipt, limitations and next questions are in research/selected-continuation-backing/. The #2770 numerical live-word result, Ross Richardson's #2831 arena backing and #2923 native ADD join retain separate credit. Source notices remain unchanged. Substantial automated assistance was used for analysis, proof drafting, implementation and internal review.

Canonical scoped claim. Started at current main a7482490b; refreshed main 3bee87dc2 has unchanged priorities, selected record, consumed inputs and replay runner. Fresh overlap covered 495 open PRs, 2,122 closed/merged PRs and 1,601 Discussion records. Metadata coverage omits arbitrary diffs and unpublished work.

Evidence

• Before: conservative GL64 depth-two allowance 340, observed 214. After: proved allowance 214, literal lengths 145+44+19+6.
• Registered normal/optimized Python 3.12 replay passes 524 branch histories, 914240 observed tape writes, three actual GL64 fixtures, repeated/warm histories and three real quota failures.
• Nine behavioral tests pass, including one-cell-short, retained blank arrays, bad source hashes and observer restoration after failed context entry. Optimized Python 3.14 compatibility passes.
• Research workflow suite: 144 tests pass, seven platform skips. Scoped Ruff checks/formatting, local documentation links, source pins and committed-byte preflight pass.
• Separate internal source/proof review reproduced the receipt and all nine tests. No independent human, formal-kernel or hosted-CI pass is claimed.

Replay: python3.12 -I -Werror -B scripts/research_package.py --package research/selected-continuation-backing.

Merge danger

Door: two-way. Blast radius: research.

One opt-in package. The theorem assumes successful private weight execution; general B>A optimality is open. Numerical arenas, input programs/fields, other caller categories, physical reuse/movement, complete reserve and supplier admission remain separate. The selected frontier and frozen predecessors are unchanged; no new kappa or complete multiplication theorem is claimed.