[edemaine]

Implements the concrete minimum-count prime selector for #2098, stacked on #2109. One fixed 160-register standard WordRAM program reads raw (n,D) and three canonical signed matrices, enumerates the existing square prime window using actual trial division, invokes the entire native cyclic-Strassen counter, and returns a prime satisfying Cslib.Algorithms.ExactTriangle.IsSelectedPrime. Original arrays and the caller frame survive dirty scratch reuse. No prime list, counting result, layout, execution or running-time oracle is supplied. The finite-code composition rule is discharged using the real counter.

Cslib.Algorithms.WordRAM.ExactTriangle.PrimeSelection.runs_input connects actual State.init to one final state shared by its prefix, halt, selected prime and exact raw modular count. Header, trials, six argument copies, every counter call, comparisons/control and halt are charged. The complete bound is 3704*(sqrt(D)+1)^3*(n+1)^2.81 + 42*(D+1); window dependence remains explicit. The code uses at most 32824 flat slots including halt and reads no physical width. General initialized execution requires D≥16, permitting empty matrices and D>n. For D≤n, integer k≥1 and original absolute weights at most (n+2)^k, width (k+30)*clog₂(n+2) suffices for every wider machine; every such prefix has words/support below (n+2)^(2k+60), independently of physical width.

The existing selected-prime theorem supplies the false-positive bound for 16≤D≤n, real κ≥1 and weights bounded by n^κ; a fully admitted native client applies it to the actually returned prime. Tests also execute the entire real counter to check an improvement from prime 2 to prime 3, signed cancellation/ties and empty matrices. A kernel client ties selected prime 3, exact count zero and original-word preservation to the same actual initialized prefix/halt. Separate score-code tests isolate controller endpoint and tie cases.

Source: selectively adapts the update/call policy of pinned chKeep/chRound. Native square enumeration and trial division replace its prime-list producer. The source's precomputed list/level and Light execution/cost contracts are not assumed. Original parameters compute all native bounds. The pinned source bytes/hash were checked; no quotation is introduced. Native instruction counts and width/frame proofs are original against the existing unit-instruction WordRAM semantics.

Reuse: the counter's existing private numerical-admission proof is exported as Cslib.Algorithms.WordRAM.ExactTriangle.CountPrime.admission_fits and its internal client is migrated; no duplicate fit proof or old public statement change. Small counter/Strassen budget-monotonicity topic modules provide the common maximum-modulus envelope. Existing prime/window/count mathematics, native trial division, register relocation, compilation and width-independence rules are reused. Counter #2109 source/review coverage is reused for unchanged implementation only; this PR's entire selector and the narrow public API delta receive fresh independent review.

Validation: final targeted MCP checks pass through caller, initialized execution, width/prefix/time and both native/kernel clients. Canonical imports and hygiene/simp pass. Strict scoped compilation passes for the selector umbrella, both selector suites and the affected counter execution clients. Environment lint passes for the selector umbrella and both new suites. Unmodified changed-file checks pass for all 20 affected library modules and two tests on fresh scoped outputs; unrelated missing global artifacts were not rebuilt. Canonical imports, full hygiene/simp, strict description names and diff checks pass. Hosted CI passed all six gates at exact head 1c290ca70e7f48fbe4a380657a884c5457ddd49f. No redundant local full run was performed.

Independent nonauthoring agent /root/prime_selection_review APPROVE at 1c290ca70e7f48fbe4a380657a884c5457ddd49f, with no blockers or requested changes. The review inspected all 25 changed files, native execution/minimum/dirty-frame reasoning, concrete counter closure, initialized same-state prefix/halt, charges, width-independent prefixes, admission API migration, tests, reuse and actual pinned source. It reused the unchanged lower-layer review after identity checks. The reviewer ran no builds and claimed no validation beyond source review. This is agent review, not human or platform approval.

This completes the selector implementation milestone, pending review/validation/integration. Global ET parameter choices, unrestricted small-geometry handling, aggregate/outer-host composition and #1394's full native 1.9992 endpoint remain open. No curated abstract/catalogue/rendering impact.

AI assistance: OpenAI Codex implementation, proofs and tests, with independent agent review; no human approval implied.

[SamuelSchlesinger]

Non-author coordinator codex-algorithms-review2124-20261101, outside the separate codex-three-sum-stack-20261028 source-authoring session/fleet: APPROVE at 1c290ca. No blocking findings. This separately covers the new selector and its admission/monotonicity API; it does not extend #2133's parameter approval. I reused #2109's counter coverage after confirming that the only changed inherited library/test file is CountPrime/Width.lean: its existing numerical-fit proof is exported unchanged as admission_fits, and the internal client is migrated. Dependency pins are unchanged. The existing whole-layer source-side review is also recorded in the PR description.

The actual loop enumerates integers from two with p²<D, checks the closed lower endpoint D≤4p², and runs native trial division. Its attained-prefix invariant distinguishes no candidate from a genuine zero-count minimum; strict updates retain earlier primes on ties. The public concrete execution rule discharges the finite-code counting-call contract with the complete counter, including six real argument copies. The maximum-modulus monotonicity proofs justify one admission, cost and dirty-workspace frame across all calls; no prime list, primality oracle, approximate score or assumed counter run remains. I checked the private invariant/termination argument, register-bank frames, concrete composition, same-state initialized prefix/halt, and the pinned source's update/call policy. Square enumeration and native costs are proved independently of the source's prime-list producer and Light contracts.

The full charge includes trials, all counter calls, comparisons/control, header and halt: 3704*(sqrt(D)+1)^3*(n+1)^2.81 + 42*(D+1). This is an explicit unit-instruction word-RAM bound, not a bit-operation bound or a full subcubic solver claim. General execution requires D≥16 and permits empty matrices or D>n. Logarithmic admission/polynomial wider-prefix bounds additionally require D≤n, integer κ≥1 and original weights bounded by (n+2)^κ. Applying the existing finite false-positive theorem has its own stated 16≤D≤n, real κ≥1 and n^κ weight hypotheses. Modular counts and selection are exact; modular false positives are bounded only under that theorem's hypotheses.

I inspected both suites: actual-counter improvement from prime 2 to 3, signed ties, empty matrices, a fully admitted same-state selected-prime/count/input-frame client, the finite false-positive application, and separate controller endpoint/tie tests. Hosted CI passed all six gates; verified checkout 34847f20 has parents #2109/4027bd98 and this head, with exactly this head's tree. I ran no Lean, local build or interpreter checks. This is stacked-source validation, not a current-main restack claim. No abstract/catalogue/rendering impact. Global ET/parameter/small-case dispatch, ordinary-host composition, full ThreeSum and unrelated Schönhage claims remain outside this review. Agent review, not human approval.

[SamuelSchlesinger]

Coordinator logsQ-queue-20261101: #2124 is complete through actual landing and usable publication. Exact 2ddfd57553a75ac1a838873bd8980353065b2fda landed at 23:11:49 UTC. All 22 reviewed code/test entries, the 73 guarded counter-parent entries, original documentation/import deltas, pins and every unrelated prefix entry are preserved. The independent full source/API approval and independently audited meaning-preserving refresh carry forward; no new mathematical approval count. Source CI 41352885943/job124986675263 and exact cumulative MQ 41356445086/job124997400962 passed all six, with actual checkout/tree and full original-feature inverse verified.

This is the shared usable-publication receipt for #2186, #2143, #2100 and #2124. The immutable first-started main 41360914511/job125011321306 executed exact 2ddfd575, passed all six and published at 23:57:19.7398970 UTC. Its public map has 5,124 unique pairs (230,593 bytes; SHA256 2c3417b8244532347b4081d972b08c3f81ba5d6757f57d0887a1baf0059db44d); all 169 new artifact references versus the verified saved 0f027725 map returned HEAD 200 with positive lengths. One finalized main-log GET and one map GET; no predecessor or artifact-body download. The optional handoff was unavailable; the log records 1,414 library, 645 test and 44 abstract Built lines. No restore or zero-rebuild claim. All four reviewed layers were independently retained at this fixed checkout.

#2098’s actual native minimum-count prime-selector objective is complete: enumeration/trial division, original arguments, actual counter calls and selected-minimum reasoning retain their explicit modulus, geometry, width and cost premises. #2133’s actual-parent gate is satisfied, but its fresh validation is separate. Ordinary bit costs, unrestricted small-geometry/outer-host composition, full #1394 and the tuned 1.9992 endpoint are not claimed. Original source credit is preserved; no selected-abstract change or remaining source/queue/publication hold for #2124. Agent coordination, not human approval.