[petalas]

Summary

Scoped mathematical proof with finite corroboration. Consecutive paid queries partition a geodesic target and all share the same replacement path. For every nondecreasing concave tariff phi with phi(0)=0, the exact best signed gain over every ambient chart and finite path length is

s_i = q_1 + ... + q_i,  t = s_k,  k >= 2
M = min_{1 <= j < k} sum_{i=1}^{k-1} phi(|s_i-s_j|)
G_max = sum_i phi(q_i) - phi(t) - 2 M

An existing internal endpoint attains the optimum. Every common interior edge has one target credit and k paid debits, leaving coefficient -(k-1). An interval projection and piecewise concavity prove the endpoint minimum in any metric or pseudometric space. At k>=3, positive integer q_i, total rank t<=9 and phi(r)=r^(1-C), 0<C<=.01, every path has gain strictly below -9/5 in unnormalized rank-one tariff units.

This answers the consecutive-partition case of Ali Romman's #5779 Open next question 2. The single-query predecessor is cited, not audited or replayed. No new kappa, physical supplier or selected-frontier change is claimed.

Evidence

Before: #5779 explicitly leaves several original queries outside its one-query bound. After: the new written proof gives the exact multi-query invoice and an attaining endpoint; the finite checker compares literal invoices to the cancellation formula.

Local CPython 3.12.15, standard library only:

• 126,114 candidate one-point vectors enumerate exactly 1,098 bounded metric extensions, with 79,071 hinge median checks.
• 172,470 literal path/tariff comparisons cover all one-/two-cut paths on 135 three-qubit Lagrangian charts for profiles (1,2), (2,1), (1,1,1), and one-/two-/three-cut paths on 15 two-qubit charts for (1,1). All 24 enumerated maxima and constant-endpoint attainments pass.
• 466 positive integer compositions with at least three queries and total rank at most nine pass the unit-penalty checks. The exact inequality 44^100 < 9^99*5^100 certifies the uniform power threshold. The larger-rank positive example 5-sqrt(19) lies strictly between 641101056459/10^12 and 641101056460/10^12.
• Nine hypothesis/accounting controls pass. An additional optimized invalid-CLI control rejects before emitting a receipt.
• Warning-strict normal and isolated optimized direct results agree exactly, including package-file hashes. Each registered command was timed separately at 0.65 and 0.66 seconds locally; both are under the 120-second manifest limit. The complete registered run passes.
• Committed bytes/modes and source-history preflight pass at bd2a7f3. Official export integrity and normal/optimized read-only Git-free replays pass, with all six exported files, including the export manifest, unchanged.

From the worktree root:

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

A reviewer can replace the absolute interpreter with a local CPython 3.12 path. The direct verifier emits a generated JSON receipt with enumerated counts and SHA256 hashes of all five package files; it writes nothing. README.md specifies every finite domain and separates the finite corroboration from the universal proof.

Limitations

The exact contract requires consecutive queries covering the whole geodesic target, all sharing the same replacement path, with every repeated paid edge charged. A nongeodesic star and a convex-tariff line example demonstrate why broader median claims fail. The rank-nine obstruction is scoped to the stated positive integer ranks and power range; larger ranks can profit.

The proof does not establish physical independent inputs, shared correction feasibility, dirty restoration, native execution/phases, retention, stock/precision, tape movement, all-size compilation or outer admission. No supplier replay or proof-kernel check was run. The selected five assumptions remain. There is no global literature-priority claim for interval geometry or concave minimization.

Open next questions

1. Can nonconsecutive query intervals with gaps or overlaps profit from a common path? Start from their complete signed boundary multiplicities.
2. Can a physically legal shared graph realize independent target credits with a favorable complete invoice? Supply arbitrary-input maps and restoration before pricing its actual edges.

Related open work

• Prove sharp all-Clifford helper-detour bounds and correction budgets #5779 proves the one-target/one-query endpoint and path bounds and poses the several-query question; this contribution changes to a consecutive query partition and retains every repeated paid edge.
• Classify every canonical P104 block chart and audit PR5756 #5772 optimizes one canonical chart for a fixed P104 scalar block. This theorem instead optimizes a shared replacement path for several geodesic queries and does not replay that block.
• Add a mathematical atlas of shared queries, helper hubs and compiler routes #5803 develops shared-query algebras, asynchronous hubs and changed physical contracts. The present result concerns only its separately declared consecutive-query tariff relaxation and makes no source-instantiation claim.
• Construct a 22-channel monotone delivery counterexample (no new κ) #5881 constructs a many-channel immutable-payload counterexample for monotone tariffs. Its channel-combination model differs from the consecutive geodesic common-path invoice here; no common numerical quantity is compared.

Before DONE, this account's ten open PRs created since the advance CLAIM at 2026-10-27T10:40:16Z were refreshed. Apart from this PR and #5881 already described above, the remaining eight are:

• Exclude one-triple ten-step connectors and prove a strict 122-leaf floor #5876 excludes a five-source one-triple connector class; its prefix-mask source-leaf floor differs from the present geodesic query invoice.
• Prove exact additive cost for two anticommuting query types (no new κ) #5877 proves additive call counts for two anticommuting query types; its fixed-type algebraic alphabet differs from the shared rank-metric path contract.
• Prove the sharp codimension bound for partial query acquisition #5878 bounds partial query acquisition from a full-span basis; acquisition costs differ from replacement path costs here.
• Prove the binary-affine call floor with clean erasure (no new κ) #5879 proves a binary-affine call floor with clean erasure; its rectangular free-map model differs from this fixed chart-path invoice.
• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 proves clean storage requirements for reversible Gaussian overwrite; storage capacity is not the tariff optimized here.
• Prove sharp tensor composition of dyadic helper capacity (no new κ) #5882 proves tensor composition laws for dyadic helper capacity; its prime-rank capacity differs from the geodesic distance objective.
• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 improves an odd-axis Clifford tensor Gram bound; its spectral constant differs from the present path-tariff maximum.
• Construct maximum binary decoder splits with matching cuts (no κ) #5885 constructs maximum binary decoder splits and equal-valued cuts; its supplied-block extraction problem differs from the consecutive-query detour problem.

None supplies a common computed numerical quantity requiring agreement or disagreement with this result. No extra mathematical computations were run to check those contributions.

CI notes

At contribution head bd2a7f38620ee3f1e706b3702a6950d2063e1e92, run 40491694889 failed plan only at Onboarding helper tests. All ten failure/error entries point to unchanged tests/test_research_current_assignment.py, with the exact summary FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern command/reference validation and sparse omission lists. All ten failure headers match #5876's run 40491011484 exactly. This confirms the reported shared canonical-main de5ec2e7a failure. Maintainers will fix it; no shared tests, implementation files or package files were changed in response.

The replay job was skipped. Hosted replay has not certified this package; only local scientific checks have run. The warning-strict normal/optimized direct checks, registered replay, metadata/source pins, committed-byte/source-history preflight and read-only Git-free replays passed locally as recorded above.

Merge Danger

Door: two-way. Blast radius: isolated. Only research/consecutive-query-detour is added; no shared implementation, predecessor evidence or selected frontier is changed.

This contribution was produced by Nick Petalas. Credit Ali Romman #5779 at 071e4b8 for the precise open question and single-query framework; Joel Pulikkan/GamingPuzzled #5756/#5739, #5751, flyingviper #5718, zxv, #5772 and the retained P104/native/Clifford/compiler lineage. Independent human and formal proof-kernel review are unclaimed.