[petalas]

Summary

Prove the sharp retained-payload storage required by one forward parent-input pass that emits complete low-type child batches sequentially. For n repeated cycles, L low types and s symbols per chunk, the exact capacity is (n-1)(L-1)s alphabet-symbol equivalents. The proof permits arbitrary lossless encodings and charges payload-dependent controller state. A literal-symbol schedule attains the bound and empties its buffer.

input order:  (cycle, low type, chunk offset)
output order: (low type, cycle, chunk offset)
exact buffer: (n-1)(L-1)s

Claim type: scoped written proof with finite corroboration. This takes the physical-dispatcher question in Ali Romman's #5829 at f1e522fc26371617ed60dea3c4609d422d491d2e. Its declared whole-high address contract gives n_j=T*(L*H)^(D-j-1), s_j=H^(j+1)*R, and the new necessary term ((L-1)/L)*(1-1/n_j)*V_j. The sixteen-symbol cover fixture needs exactly six retained symbols.

The address contract and its open dispatcher question retain Ali Romman's credit, with Mason Hale and the BIT/compiler/cover predecessors credited there. Streaming permutation cuts and information counting are classical. The new scoped proof, attaining schedule and checker were produced by Nick Petalas. No new kappa or selected-frontier change is claimed.

Evidence

Before, the cited address map left the physical dispatcher and local-space bill open. After, this restricted one-pass interface has a sharp storage term with a matching schedule.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passes all 512 batch shapes, 5,914 permutations through length seven including the empty case, 31,990 binary payloads, 2,262 distinct cut states across 20 families, 11,520 algebraic cover substitutions, 392 subcritical-capacity rejections and 14 malformed-input controls. Independent prefix-cut and full read/emit minimax oracles agree with literal output execution. Internal review caught an omitted dominated read/store alternative in the scheduling oracle; that alternative is now included, and all checks pass with unchanged outputs.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/low-type-stream-buffer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/low-type-stream-buffer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/low-type-stream-buffer
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/low-type-stream-buffer --check-source-history

Any CPython 3.12 path can replace the local interpreter path. Each registered command was timed locally at 0.21 seconds, below its 120-second timeout. Outputs include PASS_LOW_TYPE_STREAM_BUFFER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Warning-strict AST compilation and function-size inspection pass. Committed bytes/modes and declared source history pass at 61dac6c54e2be2e486a61da40ffeb0d2db743773. The official export passes integrity validation and an optimized read-only Git-free replay with identical generated receipts. Generated local receipts bound all six package files; the verifier can regenerate them outside the checkout.

The predecessor contract is cited with exact head, paths and SHA256 values, separately from runtime inputs. The standalone verifier imports no pending predecessor code. The source-history check authenticates the declared main provenance, not external mathematical citations.

Limitations

One read stream, one append-only sequential child-output stream, arbitrary independent finite-alphabet symbols and deterministic exact dispatch are explicit hypotheses. Storage counts retained payload information, including compressed/controller representations; unbounded integer or exact-complex cells do not have unit capacity. The result is compatible with #5829's O(V_j) premise. A different physical traversal may use a different contract.

Physical head movement, fixed-tape implementation, numerical precision, returned ownership and the complete supplier invoice remain unpriced. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected scientific cold replay were not run; no local check of this scoped package was deferred to CI.

Related open work

• Bind delayed-cover batches to recursive allocation and return #5829 supplies the whole-high address/return contract and open dispatcher question; this adds the exact one-pass buffering term under its declared order.
• Bridge signed-query suspension and return on 23 fixed tapes #5814 implements signed-query suspension/return and leaves production dispatch open; this isolates one input/output-order storage cost.
• Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612 studies delayed-cover stock and atom accounting; the new term is retained payload during reordering rather than fresh-row reserve.
• Bind all 7,548 G20 route markers to conditional ROW cover words #5805 binds G20 route markers to conditional cover words; this theorem does not price those route operations.
• Lower all sparse G20 routes with exact whole-word q-adic carries #5841 implements fused q-adic carry identities; this theorem concerns sequential low-type batching, with the high fields retained.
• Complete runtime G20 quarter-carry rotation with tight physical bounds #5849 implements a runtime carry leaf with physical space bounds; those bounds and this dispatcher buffer refer to different operations and are not numerically compared.
• Prove exact contiguous helper storage: 5q versus peak 4q (no new κ) #5897 studies fixed contiguous placement of helper lifetimes; its storage gap is a different contract from this one-pass stream-reordering requirement.

Completion refresh inspected all nine other account PRs opened since the advance CLAIM: #5897, #5898, #5899, #5900, #5901, #5902, #5903, #5904 and #5906. The storage relationship with #5897 is recorded above. The others study clean bank exchange, tail gates, Newton precision, invoice frontiers, phase witnesses, COPY approximation, dyadic observation acquisition and a two-burst query schedule. None computes the same one-pass buffer quantity or the sixteen-symbol dispatcher fixture. No additional computations were run to validate their mathematical results.

Open next questions

Can multiple independently writable child streams reduce the buffer while preserving complete arbitrary-input return? Start with the sixteen-symbol fixture and price stream count, head placement and return ordering. What literal fixed-tape time and space bill realizes the attaining six-slot schedule? Start with every retained-chunk read and release; source binding and cleanup remain blockers.

CI notes

At commit 61dac6c54e2be2e486a61da40ffeb0d2db743773, CI run 40501344081 finished with plan failed; replay skipped. The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are missing follow_up_steps and catalog_first_step keys. Assertion failures concern frozen-card/current-route expectations, missing-reference handling and sparse omission lists. All ten failure/error headers and exception/assertion messages match PR #5904's plan log exactly. Maintainers will address this shared main failure; no shared source or tests were changed.

Hosted replay has not certified this package; only local scientific and publication checks have run. Local normal, optimized, registered, committed-byte/source-history and read-only export checks pass. The CI result does not establish mathematical acceptance.

Merge Danger

Door: two-way.

Blast Radius: research.

Adds one isolated six-file package and opt-in bounded replay. Shared code, frozen evidence and selected artifacts are unchanged.