Prove source-bound prime supply for the many-axis tensor caller

[ToikY]

Result and boundary

This supplies the odd, pairwise-coprime source lengths explicitly left open by #4024, bound to #3935's actual TensorPlan.production_axis. The old source08/Cinna Davis selector uses log(t)>8d; that premise does not port to the newer D=Theta(p^.997), ell=Theta(p^.003) family. This is a changed parameter-regime implication, not a failure of the old lemma.

Write D for actual scalar axes and d=D+1. For ell>=2^32, 2<=d<=ell^334, and target_halo+2w<=2^(ell-3), each interior interval (t(1-1/(3d)),t(1-1/(4d))], for t equal to either target scale, contains at least D primes. Ordered distinct selections satisfy all three source axis guards and full T/S<3/2<2. The separate period premise is explicit; arbitrary halo is not admitted by prime abundance.

PROOF.md derives that deliberately conservative effective domain from Fiori–Kadiri–Swidinsky, arXiv2206.12557v2 Corollary22, an imported analytic theorem. It includes the two-scale ordering/product argument and a uniform fixed-tape trial-division/list construction with O(r² ell²+D ell) steps and O(D ell+ell) visited backing for this prime producer, r=2^ell, including scratch erasure and head returns. Under the retained caller laws its setup is 2^o(p)=o(V). Existing CRT metadata retains its own separately paid bill; an explicit p cutoff requires fixed growth/halo constants.

The executed program is a host-integer finite reference, encoded_execution=false. It authenticates five local source exports and extracts the complete source axis-method AST using a semantic Count adapter. It does not execute encoded predecessor arithmetic, giant primes, a production tensor or native supplier. The uniform machine construction is written, not executed.

Validation

Only research/many-axis-prime-supply/ is added:19 files, one canonical-parent commit 726c9a3a1dc58d2b1223dd681eb712255be407c1; public tree equals the locally tested tree.

python3.12 -B scripts/research_package.py --package research/many-axis-prime-supply

Five mixed-scale/nonmonotone finite cases, full source guards, products and encoded source/flag hashes pass. The separate checker imports no primary executable and uses Eratosthenes plus exact rational/integer predicates and symbolic threshold inequalities. The primary has8 internal controls; the separate suite checks50 expected rejections and5 positives, including strict equalities, actual D versus d, same/mixed-scale duplicate primes, composites, ownership, authenticated AST and typed/corrupted receipts. No giant target or axis list is constructed.

CPython3.12.14 registered replay, exact-public-commit byte/source-history check and official export/check pass. All three standalone commands pass normally and with -O from a fully0444/0555, Git-free20-file official export with empty PATH/PYTHONPATH. Every file retains bytes, modes and inode. Archive SHA256 ce3232e7644d18733b9013654c52f401947d1fe219c80e77074c6b2fcdd2b468. Hosted exact-head run39237273642 passes:265 onboarding tests (11skips), only this package selected, all three commands on CPython3.12.15 and unchanged committed inputs. Logs bind this head to test merge70c5c6f2 against canonical20ecf289. Finite execution does not reprove the imported analytic theorem or formalize the full multiplication theorem.

Selection, credit and remaining work

Handoff-first changed supplier-boundary-audit scope; original card, attempted baseline command and missing sparse inventory dependency are recorded in TASK.json. Canonical claim. Canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority6, catalogue and selected record refreshed unchanged before submission. Source #4024 e99297be6749518a1b5b85cd129ecaf204d26c63; #3935 240597ed313fc2854c0462b23c89dc3bae72aaf5. Entry overlap:729 completely paginated open PR bodies including drafts, repository helper and retained prime-supply predecessors; pre-submission latest100-body delta through #4028 and bounded live handoffs/source comments. Broader Discussion/closed/code coverage remains partial. Jose Manuel's prospective numerical recovery lane is separate.

Credit Jose Manuel/jmbarrancoml's tensor interface, Tekalign/tekuuu's CRT converter and explicit premise, Cinna Davis's old-domain selector transfer, Chafik Boukhalfa/#1786 and JosieElliston's prior fine-prime transfer audit, Fiori–Kadiri–Swidinsky's analytic theorem, Harvey–van der Hoeven/source08 trial-division construction and Agarwal–Cooley CRT algebra. Frozen READMEs retain the OpenAI manuscript, Douglas Colkitt, RaD/hipotures and deeper runtime lineage. No invention of trial division or the generic explicit-PNT method is claimed. ToikY with substantial OpenAI Codex assistance and separate AI proof/test contexts; Apache-2.0. No independent human or formal-kernel acceptance.

Open next steps: lower/admit the written prime producer at the actual encoded caller, then complete numerical tensor composition and integer recovery with all production_common policies, total gamma and both final S factors. Native/analytic, precision, physical reserve/stock and supplier obligations remain inherited or open. No new κ, merge, closure or selected-frontier change. Maintainer edits disabled.

Discussion handoff. Source claim released; no continuing reservation.