Research: exact scratch capacity and a complete linear tape bill for PR #2915's ordered high-field ADD

[commdata2338]

Change and mathematical scope

PR #2982 leaves the ordered high-field ADD inside its single-cube tariff unpriced. This package proves a local reserve and physical-I/O bound for the unchanged retained PR #2915 kernels.added and its actual pooled stream.add_pass.

For odd Q>=3, canonical positive extents B,M,S and c in {-4,-2,-1,1,2,4}, let V=BQMQS and b=bit_length(Q)+1. Exactly 10+[S>1] additional scratch roles suffice and are attained in every call, sharpening PR #2809's conservative 32-role lexical allowance for a complete ADD. Six additional numerical words suffice, with an attaining fixture; outside the coefficient callback this is PR #2810's bound for the same stream.add_pass, composed here with the actual callback (at most five). The complete local physical read/write/unit-move bill, including encoded arithmetic and pool acquisitions, is at most 4096V, or 4102V including returned-output disposal. Borrowed payload, counts, numerical residents and occupied entry roles are preserved; the output is separately owned.

The exact non-arithmetic invoice extends the existing countdown and payload bills to the pooled call. PR #2877's all-width countdown identity (its zero case was also derived earlier in PR #2473), plus one acquisition read, gives the fully drained pooled l-digit Counter cost 15n+14l-5 popcount(n)+37+4[n=0]; PR #2473's exact payload bill 23V+29 gains three payload acquisitions; the one-bit-suffix path and the three marginal lifecycles are priced from source. Source-derived numerical bounds complete the bill; no finite timing ratio is extrapolated. An occupied-content corollary is stated separately from backing capacity, which PR #2905 and PR #2923 bound for the Counter and joint numerical/count backing of the same ADD.

Artifact: research/high-add-scratch-invoice/README.md, with PROOF.md, source pins, a typed receipt, verifier and controls. Status: scoped source-level proof and independent audit, with finite corroboration. No new construction, kappa, supplier admission or selected-frontier change. Full finite-control machine steps, marker/template provisioning, enclosing setup, reversed-field B1 swaps, physical backing and whole-stage stock remain separate.

Task and overlap

Task card: selected-stock-envelope, adversarial-audit, priority revision 6, scope key f5c32b23bcd01d2e356d; the full pinned original and actual card are in TASK.json. Handoff first: PR #2982's released question, then specific-card retrieval with seed 00000000000000000000000000000000, with no random draw or reroll. The claim is in discussion #2319.

Scope is changed from a whole-stage envelope to this named actual-source reserve/time implication. Covered conditional 8194/6748 coefficients, lcm-wrapper fits, cutoff comparisons and PR #3113's m=73 source gap are not repeated. The distinct active NODE/MERGE claim audits enclosing coordinate and ownership refinement; this package audits only ordered ADD. The family decision is narrow: no dependent construction search.

The concrete method is a chronological lifetime and literal-loop proof with an independent integer invoice, modular-address oracle and physical metering on complete bounded packets. The acceptance/stop rule was to include every constructor, numerical and disposal charge or report the missing term. The finite domain has Q in {3,5,9,17}, four spectator choices, six coefficients, four occupied-pool cases and nine complete basis columns at Q=3; isolated arithmetic checks reach 82-bit words. The session cap was 60 minutes. The declared local milestone is complete; full caller restoration/capacity and H2 composition remain obligations.

Scientific base and refreshed canonical main: 463113d675af7410ac91d53a4fed51ec6494eb9a. The submission checkpoint git fetch origin main at 2026-10-17 17:50:13 UTC found no commit or file change, including guidance, selected record and inputs, and a read-only recheck at 18:42:53 UTC found main and the PR #2982 and PR #2915 heads unchanged. Frozen inputs were preserved. Contribution commit: 803f3593e8f00ed8591d5ff6fc69ce2a036a3293.

Exact predecessor/dependency heads:

Source	Head
PR #2982 prefix wrapper / handoff	b9ac0d07eb9cdc31c4dca8dcc218943998edd3e8
PR #2968 high scaling / scratch	cf4fc903951cd3f034531f7089d0a178f0f67b05
PR #2915 high kernels	fecc5a87574df441ddc3dad44f99fca9bd7fdd92
PR #2915 main compatibility integration	d112c69dcfae71849feb8ffdb463b244c16db545
PR #2899 physical numerical arena	2f0911ad735643a069b9790302fbf179e49f74f8
PR #2809 actual scratch specialization	e7d93dd0de0bb81308bc3356f65eddeeb22fd712
PR #2753 retained logical caller	7dba5452a4071a856a0b11cebdcd2d3b6bcd7659
PR #2157 stream routes	eb1bf93950e9b14f7fadca7892866e9f29a26ad3
PR #2877 countdown identity	bbafe95af535004c7b605059e611cc8a569f88f6

SOURCES.json pins the actual main bytes and complete retained archives. The inherited selected-record compatibility adaptation is distinguished from original author bytes. No retained archive is rewritten.

Overlap searches used the repository helper offline for the task plus lease/scratch, complete-route and ordered-ADD terms, and inspected merged main packages and the disposition ledger. Preclaim snapshots: open PRs 17:26:19.594925 UTC, Discussions 17:27:03.117721 UTC. Final snapshots: 253 open PRs at 17:49:11.608413 UTC, 2015 Discussion records at 17:39:04.580509 UTC, complete pagination. A further add_pass query against the 18:20 UTC snapshot (267 open PRs, 2,034 Discussion records, complete pagination) found the merged ADD audits named below and no competing open PR or claim on this object. The expected offline warnings remain; there is no usable closed-PR snapshot. Cached metadata, local notes and ledger do not cover arbitrary diffs or unpublished work and are not a novelty certificate.

Nearest work: PR #2982 proves the prefix wrapper's 265V/277V overhead; PR #2968 proves separate scaling/pool charges; PR #2877 supplies the all-width countdown identity, including zero. PR #2473 audits the unpooled encoded ADD with exact countdown and payload bills and a 947V non-arithmetic bound, leaving arithmetic as O(B+UML^2); PR #2810 bounds stream.add_pass numerical Words outside callbacks; PR #2905 and PR #2923 bound its Counter and joint backing; PR #2809 gives a conservative 32-role lexical allowance. This contribution adds the exact pooled scratch inventory, the pooled and marginal invoice terms and an explicit complete local I/O constant including arithmetic. PR #3041's residual/CALL and the enclosing-node work remain distinct.

Validation

CPython 3.12, standard library; all commands run locally and sequentially:

python3 -I -B scripts/research_package.py --package research/high-add-scratch-invoice
python3 -B scripts/check_research_commit.py --package research/high-add-scratch-invoice

Expected primary output:

PASS_HIGH_ADD_SCRATCH_INVOICE cases=109 payload_bits=37241 scratch=10,11 numerical_extra<=6 io<=4096V

Registered metadata/source checks, the primary verifier and all ten optimized behavioral groups pass. Evidence: 109 complete packets / 37,241 payload bits; 51 padded/zero countdown cases; 36 isolated numerical checks through 82-bit words. The saved case digest is 9f0fe7257cc2f18a7040e42c2cbf713535a389e8f0cfeaabbaaa38e3f4ffe79a.

Controls include one fewer scratch role, one fewer numerical slot, extra dirty entry roles, pool reuse, wrong coefficient, an endpoint-preserving extra read, noncanonical count padding, omitted zero charges, three source-pin corruptions and Boolean/integer receipt substitution. Raw physical method interception agrees with summed unique-tape meters; alias meters are not counted twice. Publication preflight matches 34 files. A committed export of precisely those files passes transport-integrity checking and the direct verifier/control suite from a read-only directory without Git metadata. git diff --cached --check passed before committing.

The proof, rather than finite tests, supplies the all-width claim. Not run or claimed: full selected GL64 payload, selected supplier cold replay, whole caller/bootstrap verification, formal proof, hosted CI, independent human review or general interrupted-call recovery. Failure fixtures are explicitly cleaned by the harness. One initial invocation of the inherited publication helper with -I failed its local-module import; the documented command above passes without modifying that helper.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. Jose Manuel/jmbarrancoml supplied PR #2915's kernels, PR #2899's arena, PR #2809's scratch specialization and PR #2157's factor words; David Arias supplied the retained PR #2753 logical caller. Nick Petalas supplied the PR #2982 released handoff/wrapper and PR #2968 scaling invoice. Arham Tahir's PR #2877 all-width countdown identity, including its zero case, and ywz65's earlier PR #2473 countdown and payload bills retain credit; the pool, one-bit-suffix and marginal terms and the ADD composition are new here. Ross Richardson's PR #2810 numerical bound and Nick Petalas's PR #2905/#2923 backing bounds for the same ADD retain credit.

The original numerical arithmetic, ywz65's parking work, this account's PR #2399 substitution algebra, and DreamingOfClouds' PR #589 on eumemic's PR #555/source527 lineage retain their authorship and notices. Existing NODE/MERGE work is complementary. Source archives and licenses are unchanged; new work is Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

See the structured questions in research/high-add-scratch-invoice/TASK.json and the README.

1. Audit high_route.extents/execute's remaining encoded setup, then compose sign/scaling/translation and complete native B1 charges, retaining owned intermediates and both swaps on reversed ADDs. This would advance PR Certify sharp linear wrapper costs for complete high routes #2982's single-cube tariff; native contracts and those separate ledgers remain blockers.
2. Join local role/content bounds to actual physical backing and caller stock at the same chronological cuts, retaining entry residents, markers/templates, blank backing, native inventories and recursive lifetimes. No whole-stage reserve follows from occupied content alone.

Workflow feedback (optional)

The released PR #2982 handoff made a useful bounded task possible after the card's old numerical-envelope territory was covered. A compact list of unclosed local implementation obligations beside the broad stock card would reduce time spent distinguishing coefficient arithmetic from actual implementation work. An index of merged audits by the source routine they bound (for example every stream.add_pass audit) would also surface earlier countdown, numerical and backing results sooner.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG