[apoursh]

Change and mathematical scope

Resolves open question 1 of Nick Petalas's #6036: the exact minimum total readout cost is R(10)=72 and R(11)=66. An attaining construction at Q=12 and support-preserving padding prove R(Q)=62 for every Q>=12. Together with the retained Q=7,8,9 values 112,92,80 and the Q<7 rank obstruction, this completes the query-count frontier for five-source partitioned independent-dirt walks.

The new elementary proof shows that eight points of F2^5 have at most 25 distinct nonzero pair sums, and nine points at most 28. Affine relation-code weight counting and a three-parallel-pair decomposition establish those bounds without enumeration. The partition argument covers any number k>=2 of restored walks, repeated labels and interior zero revisits. Literal witnesses attain every minimum.

Package: research/partitioned-walk-exact-frontier/. This is a scoped proof with finite corroboration, no new κ. The tariff obstruction is retained from prior work; path cost is reported but not optimized. No physical supplier, shared-dirt/transfer-coupled interface or full multiplication theorem is supplied.

Task and overlap

Canonical base 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8; fetched again before publication with no changes. The handoff-first task explicitly changes the retained connector-alphabet-change card to #6036's released fixed-alphabet sharpness question. TASK.json preserves the original card, decision gate and stopping rule. Canonical claim.

Live all-state PR searches, recent handoffs and predecessor notes were checked. The overlap helper had no remote cache and partial local coverage; this is targeted overlap screening, not a historical novelty certificate. No shared code or frozen package is changed.

Predecessor source heads: #6036 1cf36dd5f4d484a6c764f2b73f4f1e21c3eb3108 and #2779 1f2a413f9e27ce9609924f7ab196548c4cbe43d8. SOURCES.json pins the retained bytes as provenance, including Q=8/9 notes. The new checker executes only package-local code.

Credit

Credit Nick Petalas #6036 for the question/reduction and repeated-label extension; Wentao Li #2779/#2851/#2883 for the model and earlier minima; Arham Tahir #1898 for the tariff; and the retained Ross Richardson #943 / TennisBowling #1524 lineage. New proof/checker by apoursh, with substantial OpenAI Codex assistance in derivation, implementation and separate subagent reviews. Apache-2.0; no independent human or proof-kernel acceptance claimed.

Validation

Exact publication head: fc3647b48d02bb8aab2d2f9460db39d276497ed1.

python3.12 -B scripts/research_package.py --package research/partitioned-walk-exact-frontier
python3.12 -B scripts/check_research_commit.py --package research/partitioned-walk-exact-frontier --check-source-history --check-modes
python3.12 -B scripts/export_research_package.py export --package research/partitioned-walk-exact-frontier --ref HEAD --output /tmp/partitioned-walk-exact-frontier.tar
python3.12 -B scripts/export_research_package.py check --check-inputs /tmp/partitioned-walk-exact-frontier.tar

Passed on CPython 3.12.14, standard library only:

• Both registered normal/optimized replays: 2,925 normalized supports, an independent kernel-code oracle with four tiny literal calibrations, all query partitions through Q=12, six literal even-subset witnesses, twenty padding witnesses and seven semantic rejection controls.
• Matching canonical result SHA256 b35ef32a08d40076719f57d88fb56863383667f65da7cddc01d4ec379f225b2d.
• Committed bytes, executable modes, source history and the official seven-file export/input checks.
• Both direct modes from a read-only, Git-free export with PATH=/nonexistent; identical reports and unchanged bytes/modes/inventory.
• Selected-source baseline bindings and whitespace checks. No full selected cold replay, native supplier or Lean run was warranted by this isolated change.

Shared CI caveat: this unchanged base has the two current-status/navigation test failures already addressed by pending #6050. Those unrelated fixes are not duplicated here; a planning failure can therefore prevent the hosted scientific replay despite successful local package checks.

Open next questions

The independent-dirt readout frontier is closed. Further work needs a specified transfer-coupled/shared-dirt producer-consumer contract with shared-prefix billing, restoration and path costs, then a comparison of its complete tariff against the retained six-query endpoint. This package does not supply that changed caller or its cost contract; a larger search in the same model is unnecessary.