Research: Restricted three-query optimum

research/p104-query-interval-optimum/CHECKPOINT.md

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+# Wave 8 lead checkpoint
+
+Canonical base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+Branch: `research/p104-query-interval-optimum-20261109`.
+Package: `research/p104-query-interval-optimum`.
+
+Complete: exact source rank-three certificate and universal query-count
+proof in the stated fixed linear/affine model. The pinned predecessor
+provides three-query attainment. No whole-caller or kappa bound is claimed.
+All four direct normal/optimized forward/reverse checks passed; receipts
+agree across modes. Independent reviewer artifacts are retained unchanged.
+
+Next: registered package and metadata checks, commit, byte/mode/export
+validation, inspect additions-only diff, then lead-controlled fork branch
+publication and remote ref/tree/parent verification. No upstream PR.
+
+Restart: read PROOF.md, DEPENDENCIES.json and review/REVIEW.md. Reproduce
+with README commands. The durable session manifest records exact frozen
+local and API publication commits; do not interpret this checkpoint as
+proof-kernel or human admission. Remaining work includes possible changed
+boundaries, actual reachable-input subspaces and full physical precision.

research/p104-query-interval-optimum/DEPENDENCIES.json

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+{
+  "schema_version": 1,
+  "files": [
+    {
+      "path": "research/p104-query-interval-optimum/data/caller_efree.json",
+      "git_blob": "5a71ca9f80ced1aeb8a68e97516cab52eb72a52a",
+      "sha256": "c5b6c5236462ae84e5431a791a2f8ce795774342f13e11c4af35689092a8af30",
+      "repository": "DreamingOfClouds/integer-mult-bounds",
+      "commit": "a98df73ddae1d1ea85a62b6bdf2714b28cb7b359",
+      "source_path": "research/p104-two-chain-findings/data/caller_efree.json"
+    },
+    {
+      "path": "research/p104-query-interval-optimum/QUERY-ONLY-CALLER.json",
+      "git_blob": "cf55f4271009a06ad192a070d330a4466b6d44fa",
+      "sha256": "3828d53e80ea97e8b866320f081af20b6b93bfebd58b95b56b85a7afe151af5e",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "commit": "617903497387a98f7ca929ad0b6928dd32abffec",
+      "source_path": "research/p104-query-only-interval/QUERY-ONLY-CALLER.json"
+    },
+    {
+      "path": "research/p104-query-interval-optimum/retained/ATTAINMENT-PROOF.md",
+      "git_blob": "2f8610ae455bb8c6b79161223cc948e136253add",
+      "sha256": "0cede285a98dd61b78020651202729c6d9cd16fb078ee5fe9a01aa2400df5432",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "commit": "617903497387a98f7ca929ad0b6928dd32abffec",
+      "source_path": "research/p104-query-only-interval/PROOF.md"
+    }
+  ]
+}

research/p104-query-interval-optimum/NOTICE.md

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+# Credit and retained inputs
+
+Prepared for @CubeJerry with substantial OpenAI assistance in mathematical
+reasoning, exact source checks, implementation, independent AI cross-review
+and packaging. No independent human review or proof-kernel verification
+is asserted. Apache-2.0, consistent with the repository.
+
+The original two-chain caller and its definitions are by the contributors
+to CrocSwap/integer-mult-bounds#6937, from the pinned DreamingOfClouds fork.
+They are retained byte-for-byte and are not new execution in this package.
+The public QUERY-only construction by @CubeJerry with OpenAI assistance
+is a separate prior contribution at `617903497387a98f7ca929ad0b6928dd32abffec`.
+Its complete formal proof and candidate caller are retained unchanged and
+credited as the attainment of the lower bound, not newly invented here.
+
+The classical rank and product-difference argument is used transparently.
+The canonical query-information and affine-tag packages and #6825/#6800
+retain credit for their different query/rank investigations. This package
+claims only the source-specific optimality theorem under its exact model.
+
+The reverse-covector evaluator and `review/REVIEW.md` were produced by a
+separate shared-context AI worker without importing the lead forward
+engine or predecessor arithmetic. Both approaches share the authenticated
+source words and the claimed minor. This is method diversity, not a claim
+of statistical, organizational or human independence.

research/p104-query-interval-optimum/OVERLAP.md

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+# Bounded overlap and scope record
+
+Canonical main was rechecked on 2026-11-09 at 19:21 UTC:
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`. The latest relevant proposals
+#6936 and #6937 remained open and did not replace the selected certificate.
+No unclaimed global improvement is inferred from them.
+
+| Predecessor | Established scope | Relation to this package |
+| --- | --- | --- |
+| #6937, `p104-two-chain-findings` | Exact two-chain source and QSELF-pair caller; QUERY-only route discussed, complete replacement then unbuilt | Original word and question; no claim to discover that a QUERY-only route was conceivable |
+| Public `p104-query-only-interval`, commit `617903497387a98f7ca929ad0b6928dd32abffec` | Full formal 223-event three-query replacement and its complete conditional invoice | Supplies the upper bound unchanged; this package supplies the matching local minimum |
+| Canonical `actual-query-information-bounds` | Historical stage-cover, snapshot and residual-information bounds for another source/task model | Read as a query-information predecessor; not the same fixed order-four interval |
+| Canonical `affine-tag-query-minimum` | Normalized quadratic-phase membership-query oracle model | Different query access and target; not this linear register alphabet |
+| #6825 with #6800 antecedent | Characteristic-two target/rank factorization, square-zero paid calls and constant free bases | Rank/query proof perspective credited; different field, target and source boundary |
+
+The reconnaissance used current metadata, the named README files, relevant
+PR descriptions and the exact proposed caller/proof. It was bounded, not
+an exhaustive literature or archive novelty certification. The general
+rank-subadditivity and telescoping method is classical and is not claimed
+as new. No search over all possible replacement words is claimed.
+
+Concurrent precision work examines rounded scalar/native contracts for
+the same public replacement. It is complementary: this package prices
+only the necessary number of formal queries at fixed endpoints, and
+does not establish physical execution or precision admission.

research/p104-query-interval-optimum/PR-DESCRIPTION.md

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+## Result and why it matters
+
+The exact #6937 interval at zero-based events 457..575 needs **at least
+three QUERY operations** in the stated fixed q-even linear/affine rewrite
+model, even with arbitrary finite helpers, copies and erasures. The public
+three-query replacement already attains this count. This rules out reducing
+that specific interval to two queries while preserving its full independent
+input/output boundary and the charged-operation contract.
+
+The proof evaluates the actual source at q=+1 and q=-1, D9=1. A response
+minor on rows/columns 0,1,19 has determinant 8. The classical telescoping
+identity bounds response rank by the number of rank-one charged operations,
+without requiring invertible helpers or free maps. A forward full-state
+rational engine and separately coded reverse-covector engine recover the
+same exact certificate.
+
+## Sources, novelty and complete scope
+
+Canonical base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+Original #6937 source: `a98df73ddae1d1ea85a62b6bdf2714b28cb7b359`.
+The matching construction is the [published QUERY-only package](https://github.com/CubeJerry/integer-mult-bounds/tree/617903497387a98f7ca929ad0b6928dd32abffec/research/p104-query-only-interval),
+which is retained unchanged and credited. The lower bound is new for this
+pinned source/model; neither the three-query construction nor the general
+rank method is claimed as new. `OVERLAP.md` distinguishes earlier
+query-information, affine-tag and #6825/#6800 rank-factorization scopes.
+
+The theorem requires lawful evaluations, regular free denominators,
+independent inputs, identical free initialization/output extraction at both
+signs and rank-one full-state charged queries. It excludes free odd-q side
+information, nonlinear/adaptive coefficients, unpriced wider payloads and
+changed endpoints. No complete-caller, native-rank or multiplication lower
+bound follows. Existing helper, arithmetic, storage, source access,
+precision and compiler costs are not waived or repriced.
+
+## Reproduction and verification
+
+```sh
+python3 -I -B research/p104-query-interval-optimum/verify.py
+python3 -I -B research/p104-query-interval-optimum/review/check_covectors.py research/p104-query-interval-optimum
+python3 -B scripts/research_package.py --package research/p104-query-interval-optimum
+python3 -B scripts/check_research_commit.py --package research/p104-query-interval-optimum --check-modes
+```
+
+The registered commands include normal and optimized Python executions.
+Expected response rank is 3, determinant 8 and 55 nonzero response entries.
+Each of six deleted charged-site controls loses rank three. Both complete
+source words and the predecessor proof are byte authenticated. The finite
+two-point attainment check is not substituted for the predecessor's full
+formal identity proof. Universal candidate-rewrite exclusion is the written
+telescoping argument, not a finite circuit search.
+
+## Status, limitations and credit
+
+**No new kappa.** This is a rigorous scoped negative result and verification
+advance. The accepted bound remains conditional on the five framework
+assumptions; no selected artifact changes. Whole-caller rearrangements,
+proved reachable subspaces or additional operator relations remain open.
+
+Prepared for @CubeJerry with substantial OpenAI assistance in proof, source
+audit, exact checks, separate AI review and packaging. The #6937 authors,
+the prior public QUERY-only package and cited query/rank contributors keep
+their credit. No independent human or proof-kernel acceptance is claimed.

research/p104-query-interval-optimum/PR-TITLE.txt

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+Prove the pinned QSELF-pair interval needs three queries (no new kappa)

research/p104-query-interval-optimum/PROOF.md

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+# A source-specific three-query minimum
+
+Prepared for @CubeJerry with substantial OpenAI assistance. Apache-2.0.
+
+## 1. Claim, sources and exact boundary
+
+The original #6937 caller is pinned at
+`a98df73ddae1d1ea85a62b6bdf2714b28cb7b359`, file
+`research/p104-two-chain-findings/data/caller_efree.json`, Git blob
+`5a71ca9f80ced1aeb8a68e97516cab52eb72a52a`. Consider exactly its zero-based
+events 457 through 575, inclusive. They contain 119 operations, begin with
+`QSELF(0,-1)`, end with `QSELF(19,-1)`, and contain one QUERY between them.
+The complete state has 76 independent formal input registers. Only the
+38 A registers are touched; the 38 B registers are unchanged.
+
+The public package `p104-query-only-interval`, commit
+`617903497387a98f7ca929ad0b6928dd32abffec`, replaces that same interval by
+223 events using exactly three QUERYs and no QSELF. It preserves the
+entire boundary map, with all independent inputs and formal D variables.
+We retain and authenticate its proof rather than claim its construction
+again. The original and replacement words are included byte-for-byte.
+
+**Theorem.** In the fixed linear/affine rewrite model in Section 2, every
+replacement of this exact interval's full independent-input map uses at
+least three charged QUERY operations. The cited three-query construction
+attains this bound in the model. Extra finite linear helpers, copies,
+erasures and affine constants do not reduce the minimum.
+
+This is a local formal-operator result. It does not prove optimality of
+the complete caller, a paid native-rank lower bound, a bit-complexity lower
+bound, or a limit on multiplication kappa.
+
+## 2. Rewrite model and legal evaluations
+
+The required identity is over
+
+    R = Q(i)[q,D1,...,D18]/(q^4-1),
+
+with no additional q/D relations or D idempotence. In particular q is a
+formal operator parameter, not an input register. Both evaluations
+`q=+1,D9=1` and `q=-1,D9=1` are lawful; other D variables may also be set
+to 1, since none occurs in this interval. We take matrix ranks only over
+the field Q(i) after evaluation, not over the quotient ring R.
+
+A candidate has a fixed finite instruction stream and finite register
+state. It may add any finite number of helper registers. Free steps are
+linear or affine maps whose coefficients, constants, initialization and
+final output extraction evaluate identically at the two chosen points.
+Arbitrary q-even coefficients are allowed when defined at both points;
+any allowed rational denominator must be regular there. Free maps may be
+singular or rectangular. Copying, erasure and arbitrary q-independent
+initial helper values are allowed without cost for this lower bound.
+
+A charged query must have full-state matrix difference of rank at most
+one between the two evaluations, after homogenizing affine state. The
+retained instruction `dest += q^k source`, for odd k and distinct roles,
+has difference `2 E_dest,source`, hence meets this condition. Surrounding
+it by any permitted free maps still has difference rank at most one.
+Even powers have zero response and cannot improve the bound by themselves.
+The original QSELF steps also have rank-one difference, although QSELF
+is not an allowed uncharged operation in the replacement alphabet.
+
+All query-dependent side information must be accounted for. In particular
+a prequeried helper bank cannot enter this interval for free, and odd-q
+linear coefficients are not free. Every independent boundary input is
+preserved as specified by the source map; restricting to a reachable
+subspace or discarding selected outputs changes this task. Data-dependent
+branches, nonlinear coefficient synthesis and a wider payload query whose
+full-state difference has rank greater than one are outside the model.
+
+These restrictions are essential. For example, freely applying q to three
+independent registers has response 2 I_3 and would supply the rank-three
+witness with no charged queries. Restricting the input to two dimensions
+also makes any response rank at most two. Neither is an admissible rewrite
+of the problem stated here.
+
+## 3. Classical product-difference bound, including helpers
+
+Append one constant coordinate to homogenize affine maps. At the two
+evaluations let the chronological step matrices be A_j^+ and A_j^-.
+Initialization E and final extraction P are the same at both evaluations.
+Variable register dimensions can be kept in the following compatible
+rectangular products, or padded into the finite union of all registers.
+
+The difference of whole maps is
+
+    M^+ - M^- = P sum_j [ A_m^+ ... A_(j+1)^+
+                         (A_j^+ - A_j^-)
+                         A_(j-1)^- ... A_1^- ] E.             (1)
+
+This is the usual telescoping identity, with no inversion. A free step
+contributes zero. A charged query contributes a term of rank at most one,
+because multiplication on either side cannot increase rank. Rank
+subadditivity therefore gives
+
+    rank(M^+ - M^-) <= number of charged queries.              (2)
+
+For affine maps, restrict the columns to independent original inputs to
+obtain the same bound on the linear response. Extra initial constants do
+not invalidate (2). Noninvertible helpers, copies, rectangular maps and
+erasures cause no problem. Correlated q-independent initial helpers are
+already part of E. The rank and telescoping facts are classical; no
+novelty is claimed for this general argument.
+
+## 4. Exact source witness
+
+Evaluate the original 119-event word at the two points in Section 2.
+LAMBDA_s is the actual even polynomial
+
+    ((1+s i) + (1-s i)q^2)/2,
+
+so it evaluates to 1 at either sign. The single IDLE_ONE in the interval
+is D9 and evaluates to 1. All ADD/SCALE coefficients remain their exact
+rationals; SWAP is unchanged. Thus direct rational arithmetic suffices.
+
+Let Delta=M^+-M^-. Its rows and columns numbered 0,1,19 give
+
+| | input 0 | input 1 | input 19 |
+| --- | ---: | ---: | ---: |
+| output 0 | -2 | -4 | 0 |
+| output 1 | 0 | -2 | 0 |
+| output 19 | 0 | 0 | 2 |
+
+The determinant is 8. Hence rank(Delta)>=3 over Q(i). The supplied forward
+evaluator also computes the complete 76 by 76 response and finds rank
+exactly 3, with 55 nonzero entries. Rank at least 3 is all the lower bound
+needs. Applying (2) proves that two or fewer queries are impossible in
+the stated model.
+
+There are two independently coded source checks. `verify.py` propagates
+all original input columns forward and performs exact rational elimination.
+`review/check_covectors.py` propagates three output covectors backwards
+through literal transposed instructions, retaining all 76 input coordinates.
+The latter was written by a separate shared-context AI worker without
+importing the lead evaluator or the predecessor arithmetic. It recovers
+the same minor and determinant. Its proof review is retained separately;
+this is not human or proof-kernel acceptance.
+
+Both checks authenticate source bytes and fixed interval endpoints. Each
+also checks the corresponding response of the public replacement. Deleting
+any one of the three charged source sites or any one of the three replacement
+queries destroys this rank-three witness; all six controls are included.
+These are exact certificates for a finite, explicitly given matrix. The
+universal claim over all candidate straight-line rewrites comes from
+Sections 2 and 3, not from enumeration of candidate circuits.
+
+## 5. Attainment, budget and limits
+
+The pinned predecessor proves the full R-valued identity for its 223-event
+replacement, including the untouched independent inputs, the formal D9
+service, the correct unit-valued LAMBDA factors and their conjugations.
+It uses only three odd-power QUERYs and permitted even/constant free maps.
+Its entire formal proof is retained as `retained/ATTAINMENT-PROOF.md`.
+The two sign checks in this package alone would not establish that full
+identity at q=+i,-i or all D9; they are supplementary source checks only.
+
+The exact three-query minimum prevents a one-query reduction of this
+particular interval while preserving the model and boundary. It does not
+reprice the predecessor. That package's additional free-operation costs,
+helper access, width, copying, restoration, finite compiler and physical
+supplier obligations remain in force. In particular no uncharged wider
+payload query or queried initialization is being hidden in this result.
+We do not multiply the formal rank by the number of physical ports and
+call it a new native or whole-caller lower bound.
+
+No runtime algorithm, selected certificate or accepted kappa changes here.
+The selected kappa remains 781451/250000000, conditional on all five
+retained framework assumptions. The proposed QUERY-only caller still has
+its separate precision, complete native admission and compiler obligations.
+Global reordering, changing the interval boundaries, exploiting a proved
+reachable-input subspace, or exploiting extra relations of a particular
+native operator remain possible research directions and are not excluded.
+
+## 6. Relation to prior work
+
+The #6937 two-chain source and the public three-query construction are
+the indispensable predecessors. The latter supplied the upper bound;
+this package supplies a matching source-specific lower bound.
+
+The bounded overlap review read the canonical actual-query information
+and affine-tag minimum packages, and #6825's characteristic-two rank
+factorization result with #6800 antecedent. Their source targets, operation
+models and coefficient fields differ from this order-four characteristic-zero
+interval. We credit their rank/query perspective and make no claim to
+inventing the rank method or exhaustively surveying all circuit complexity.
+`OVERLAP.md` records the distinctions and exact scope.