[petalas]

Summary

Construct one fixed seven-tape inventory for typed actual-width H rounds whose selected-bit counts change. This answers Open next question 1 of unpaidmoon's PR6342.

Claim type: scoped tape-model construction and written simulation proof, with exact finite evidence. For ell=3, flags=(True,False), k=1, p=4, the g=1 then g=2 witness uses 21,504 cells. The achieved general allocation is 6 M r (w + W_max). The intervening output reset adds 6,144 writes and 12,284 moves. No new kappa or selected-frontier change is claimed.

allocate seven maximum-width bit arrays once
for each externally presented round
  reset previous root and consumer
  expose the required work prefixes
  reuse low/high/paired after each blank butterfly return
  retain the latest root and consumer

Only research/variable-g-cube-stock/ is added. The one-round arithmetic is a cited premise, not an audit target. The construction, proof and verifier were produced by Nick Petalas. Credit unpaidmoon for PR6342 and PR4127, Chris R / Clear1nsight for PR3623, Sagar Thakrar for PR3617, and their retained PR3493/3593/3599/3611 lineage. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before, PR6342's exact-length allocation did not cover different selected-bit counts. After, checked prefixes and per-bit buffer reuse execute five schedules comprising fourteen rounds with exact equality to fresh-call outputs and complete read/write/move/gate ledgers. Actual source, root and consumer bit packets are checked. Eleven rejection controls cover prefix/capacity bounds, inactive dirt, allocation state, input bounds and restoration after failure.

Local CPython 3.12.15 on macOS arm64 passes both warning-strict normal and optimized commands, each about 0.42 seconds, below its 60-second registered limit. Registered package replay and committed bytes/modes/source history pass at 809e6a47b4b7ab34cfaa36a93b118e5e1a06bc66. Official export/input integrity passes for eight files. Normal and optimized read-only, Git-free, empty-PATH export replays produce identical generated file-hash receipts.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/variable-g-cube-stock/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/variable-g-cube-stock/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/variable-g-cube-stock
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/variable-g-cube-stock --check-source-history

Any CPython 3.12 executable may replace this installation path. Direct execution prints PASS_VARIABLE_G_CUBE_STOCK; registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE. The base is canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. SOURCES.json authenticates the engine, question and scheduler sources. The advance CLAIM matches the delivered scope.

Limitations and unrun obligations

The stock counts physical bit arrays, excluding Python control objects, diagnostic scans and verification snapshots. The executable bounds layouts, precision, addresses, capacity and schedule length as recorded in PROOF.md. Its private engine-module binding is used serially and restored on every exit.

Each round has an external input presentation. Only the latest physical output remains live. This is not an output-to-input transfer, a framed diagonal, an optimal stock theorem or a complete machine lowering. Production widths, retained-output banks, physical control costs and all five framework assumptions remain open. No selected cold replay, Lean, Docker or predecessor scientific suite was run.

Open next questions

• Compose a fully charged framed diagonal and the next round on this stock, starting with the same ell=3 witness. Initial framing and input transfer remain unpriced.
• Give explicit output banks and lifetimes when both round outputs must remain live.

The first verified improvement is delivered without further optimization.

Related open work

The completion refresh inspected all fourteen open account PRs created since the advance CLAIM, including this PR. The other thirteen are #6380 through #6384 and #6386 through #6393. Their scopes concern query-tree topology, helper respanning, coded-job substitution, lattice descent, mirror capacity, carrier merges/routes, ternary denominators, G20 captures, moment splitting, signed idle completion, nonunit handoffs and dirty-input queries. None studies this changing-g typed-H inventory or computes its stock/reset witness. No shared-instance numerical disagreement was identified, and no extra computation checked another contribution.

PR6342 and PR4127 are merged predecessors and are cited above. The pre-publication refresh searched the latest 100 returned open PR titles/bodies, relevant merged PR bodies and canonical-main READMEs. The cached overlap helper does not cover every omitted local note or arbitrary code diff.

CI notes

Both hosted checks, plan and replay (research/variable-g-cube-stock), passed at head 809e6a47b4b7ab34cfaa36a93b118e5e1a06bc66 in run 41511516476. The completed checks were read once. Separate run metadata confirms the same head and successful completed conclusion.

Hosted replay certifies the registered normal/optimized finite checks and source bindings. The general simulation claim rests on the written proof; independent human and formal proof-kernel review remain unclaimed. Local publication/export checks are listed above.

Merge Danger

Door: two-way

Blast Radius: isolated

Adds only the scoped research package. Shared code, predecessor files and the selected record retain their bytes.