[petalas]

Summary

Prove the exact persistent stock for a fixed chronological scalar common-query stream, at the minimum acquisition count. The scoped proof and compiler were produced by Nick Petalas.

r = rank(all request rows)
minimum acquisitions = r
minimum persistent cache = max_i [rank(past_i) + rank(future_i) - r]

retain a basis of past_i intersect future_i
answer dependent requests from that cache
query each new independent request at its actual request time
convert the cache in place, deliver the response, clear the response port

The lower bound allows arbitrary linear cache encodings and legal prefetch. One causal schedule attains every cut bound simultaneously. Its emitted scalar instructions use the minimum persistent cache plus one response/work field. For e_1,...,e_n followed by their sum, n acquisitions and one persistent field suffice for every n>=1, while retaining each individually acquired answer until its final use would occupy n fields.

Claim type: scoped written proof with exact finite corroboration and a bounded constructive compiler. No new kappa, selected-frontier change or complete multiplication theorem is claimed.

Dugongue / David Ganz's merged #5803, scheduling S12-S13 and SQ05, supplies causal query-basis acquisition and sufficient one-cut compression. This adds the matching per-cut lower bound, the global minimum-capacity theorem and the instruction-emitting compiler. Classical dimension, represented-matroid connectivity and elimination ingredients retain their usual status. Predecessor mathematics is cited, not audited or replayed. Card joint-query-controller is explicitly changed to this interface side result. Base is canonical main e54523517d0f03e58352e647c403a5af3e525a57; selected conditional kappa remains 0.00363005817.

Evidence

Before, #5803 supplied sufficient one-cut compression without this matching minimum persistent-capacity statement. After, PROOF.md proves necessity and simultaneous attainment under an explicit observational acquisition contract.

The new independent finite-state oracle enumerates every vector subspace by additive closure, tries every linear cache state for each capacity and minimizes query count. All 17,523 request streams agree with the compiler: F2 dimension two through length six, F2 dimension three through length four, and F3 dimension two through length four. The oracle uses no compiler elimination or intersection routines.

Literal scalar instruction replay checks delivered formal answers, every retained basis, entry scrubbing, exit zero and two dirty spectators. Separate exhaustive tests cover 1,234 matrices, 40 running-sum family cases, six maximum-domain/zero-rank cases and 21 rejection controls. The direct marker is PASS_CAUSAL_QUERY_CACHE_STOCK. Generated receipts bind all seven package files by SHA256. Normal and optimized receipt bytes agree.

CPython 3.12.15, macOS arm64, standard library only. From the worktree root:

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

Another CPython 3.12 may replace the absolute interpreter path. Both registered commands pass locally; their exact direct commands took 2.03 and 1.59 seconds, respectively, under their 120-second caps. These timings establish timeout headroom, not a speedup. Metadata/pins, in-memory syntax/function-size checks, real valid/invalid compiler CLI cases, whitespace and seven-file committed-byte/source-history checks pass. The official export integrity check passes; normal and optimized verification also pass from a read-only Git-free export with an empty PATH, producing the same receipt bytes as the checkout. No generated output stays in the checkout.

Limitations

The theorem fixes request order and total acquisition count r. Sources are immutable, all requests share one scalar common-query ABI, queries take only ordinary source combinations, and delivered outputs cannot be read again. All persistent query-dependent state is counted. Query feedback, non-scalar ordinary operator algebras, source mutations, differing charts, consumer feedback and a fixed scalar Q have different contracts. Owned erasure is permitted and charged. The result concerns persistent state at delivery cuts; the extra response/work field is a total-storage upper bound, not a claimed microstep optimum.

The executable compiler is bounded to 24 requests, 1..8 source fields and primes 2,3,5,7,11. The written theorem has no such finite bounds. Installation must charge scalar recipes, assembly, scrubs, delivery, routing, descriptors, clears and physical precision/access. General field elimination does not establish a dyadic-only implementation.

No selected-source request region is installed. Native realization, ownership/erasure admission, compiler, precision/tape, stock and analytic recovery remain separate obligations. The original card's complete physical caller and signed invoice gate remains unmet. Selected cold replay, Lean, Docker, independent human review and formal proof-kernel acceptance were not run or claimed.

Open next questions

• Can one actual common-ABI source stream use this smaller persistent bank? Export its immutable coefficient rows and delivery cuts first, then price every emitted conversion, route and clear. The physical request stream is the missing installation input.
• What is the exact stock versus additional-acquisition tradeoff? Start with a small stream whose capacity one below the theorem's bound requires more than r queries. No general tradeoff formula is supplied.

Related open work

The refreshed overlap search covered all 495 open PR bodies, plus Discussion records, with targeted full-main and merged-work searches supplementing sparse local notes. No other open PR was found studying this fixed scalar request-stream cache quantity. This is bounded search coverage, not a novelty certificate.

• Bound retained backing across the selected complex and BIT backend stack with history-safe stock reserves #7151 bounds historical backing across the selected complex/BIT backend stack. It has a physical storage contract distinct from this observational scalar cache; no reduction of that backing follows automatically here.
• Bound lane-wise BIT IDLE regrouping (0% κ change) #7188 studies lane-wise BIT IDLE regrouping and its complete moment obstruction. Its native paid/body-mask contract is unchanged by this cache theorem.

The completion check read this account's nine other open PRs created since the CLAIM: #7189, #7191, #7192, #7193, #7194, #7195, #7196, #7197 and #7199. They concern reader coefficients, polynomial insertions, rounded-inverse residues, address-twist rank, three-shear COPY floors, cyclotomic precision, coupled Jordan maps and compact switches. None computes this request-stream persistent-cache quantity. No additional computations or numerical comparisons were run for this check.

CI notes

CI did not certify this package. At head 50c6467d445e46e130f7d8e21a6e7d48eb70b6b3, run 43001765438 failed in plan, Onboarding helper tests. I read the completed checks once and inspected the job log directly. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError is that `make frontier7000-verify` was not found in the CI verification document. The package replay job was skipped because plan failed.

This mismatch is already present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. I checked both base blobs and confirmed this PR changes neither file. No package replay or package failure is inferred from this repository-level failure.

I also inspected all six other cited plan-job logs. Each independently shows the same exact test, the missing make frontier7000-verify assertion, 760 tests, one failure and 24 skips:

• Run 42905094314.
• Run 42903313790.
• #7193, run 43000937833.
• #7194, run 43001222481.
• #7195, run 43001319803.
• #7196, run 43001498725.

Verification evidence for this package remains local, as listed above. No fix or retrigger commit, push, shared-file edit, job rerun or additional scientific computation was made during closeout.

Merge danger

Door: two-way. Only the seven files under research/causal-query-cache-stock/ are added.

Blast radius: package. Shared code, predecessor sources and selected-frontier records are unchanged.