Research: Production Reference gap

research/production-reference-gap/CHECKPOINT.md

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+# Wave 5: literal production Reference gap
+
+Status: lead-reviewed source proof and registered normal/optimized checks
+PASS. Committed-byte/mode gate and lead remote publication remain next.
+No remote publication is asserted by this local checkpoint.
+Base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.
+Branch: research/production-reference-gap-20261109.
+Worker: analytic_join; worktree: worker-analytic-wave5.
+
+Result: the actual Reference.for_cap allocation already forces
+E>=47+ceil(log2 p^2), from its own normalized qbar and gauge allowance.
+Combined with the credited ideal gap and exact outward endpoint budget,
+its literal narrow_gap exceeds (1/8-2^-46)/p^2 >1/(16p^2) whenever the
+fine-theta/lower-alpha domain and existing constructor cuts are legal.
+No larger target or precision is introduced. This is not an assertion for
+arbitrary manually specified References or all theta/alpha values.
+
+Prior result #4648 supplies the ideal/half-gap budget; RAW-GAUSSIAN-PROOF
+supplies the actual certificate and for_cap error bound. Neither predecessor
+ties that certificate's positive production threshold to for_cap's existing
+normalized conditioning quantity. forward-fine-theta-conditioning concerns
+the broad kernel and explicitly excludes this narrow claim.
+
+Executed: 37 actual for_cap constructors, separate rational log/pi/positive
+exponential checks, 505 integer budget cases and 6 failure controls.
+The largest original target period is 2^37, with lazy source coordinates;
+no full period or original weight bank was materialized.
+Proof, receipts, source hashes and bounded reproduction are local files.
+
+Complete: source pins, package metadata, independent lead review and
+registered normal/-O replays. Next: committed-byte/mode verification and
+lead publication. Old
+published diagonal at b00d64b6204e83d468dde2db7480a2822e6940aa is frozen.
+This independent package uses canonical inputs only. No new kappa.

research/production-reference-gap/NOTICE.md

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+# Attribution
+
+Prepared for @CubeJerry with substantial OpenAI assistance in source analysis,
+proofs, exact independent calculations, implementation and packaging.
+Apache-2.0.
+
+The unchanged canonical Reference, normalized Gaussian and rational circuit
+source is Jose Manuel/jmbarrancoml's work with substantial OpenAI Codex
+assistance. Its source credits Vlado Vojdanovski, Jeremy Kothe, Cinna Davis,
+Chafik Boukhalfa, Mason Hale and the retained mathematical/compiler lineage.
+The source files are authenticated external inputs with unchanged notices.
+
+The ideal majorant/half-gap bound in CrocSwap/integer-mult-bounds#4648 is
+unpaidmoon's result with Cursor Grok assistance, crediting LOG-A/PR1673,
+the PR1786 fine-prime audit and Cinna Davis's PR4100 integration. Its generic
+rounding budget is a premise, not a new result here. The actual cap error,
+interval width and positive conditioning enclosure are likewise credited
+canonical source theorems.
+
+The new contribution is the implication from the literal cap allocator's
+normalized qbar to its already-sufficient global narrow certificate budget.
+The proof does not assume that narrow inverse in order to certify it.
+The broad `forward-fine-theta-conditioning` work is distinct and credited
+as prior overlap context, not imported as a narrow proof.
+
+Independent direct positive-series checks use classical rational series;
+they import no retained interval backend. Finite evidence and AI lead review
+do not imply human or proof-kernel acceptance. No new kappa is claimed.

research/production-reference-gap/PR-DESCRIPTION.md

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+## Why this matters
+
+The actual canonical `Reference.for_cap` computes a narrow row-gap certificate
+but leaves a positive value conditional. The Core/halo production family then
+requires that same literal certificate to reach `1/(16p^2)`. A generic ideal
+gap or unspecified high-precision rounding budget does not establish the
+constructor's actual admission.
+
+## Exact result
+
+For every integer p>=2, radius R>=1, target t>=1 and legal cap/cut request,
+under the source domain `theta in [1/(4p^2),1/p^2]` and
+`pi alpha^2>=4 log(16p)`, the unchanged constructor returns
+
+`reference.narrow_gap > (1/8-2^-46)/p^2 > 1/(16p^2)`.
+
+No extra precision, larger target, changed Reference or different rounding
+is introduced. The source's own normalized cap quantity has `qbar>=16pB`,
+forcing `Q*>=2^21 alpha^2 pB` and `E>=47+ceil(log2 p^2)`. That existing
+allocation funds the full reference error and both outward endpoint excesses.
+The global narrow inverse is not assumed to derive its own admission.
+
+## Novelty and predecessors
+
+CrocSwap/integer-mult-bounds#4648 supplies the ideal/half-gap budget and leaves
+actual W/precision binding open. The canonical raw Gaussian proof supplies
+the literal certificate and cap error, while leaving positivity conditional.
+This package supplies the missing allocation-to-certificate implication.
+It does not reclaim those predecessor estimates or prove a new general
+Gaussian theorem. `forward-fine-theta-conditioning` concerns the broad inverse
+and explicitly excludes this narrow statement.
+
+All 14 scientific inputs and card provenance are pinned to canonical base
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`. The original constructor/interval
+bodies execute unchanged; no source body is projected by this adapter.
+
+## Reproduction and checks
+
+```sh
+python3 -B research/production-reference-gap/verify.py
+python3 -O -B research/production-reference-gap/verify.py
+python3 scripts/research_package.py --package research/production-reference-gap --check-only
+python3 scripts/research_package.py --package research/production-reference-gap
+python3 scripts/check_research_commit.py --package research/production-reference-gap --check-modes
+```
+
+Expected status is `PASS_PRODUCTION_REFERENCE_GAP`. Exact source-bound evidence
+covers 37 actual cap constructions, 505 integer interval-budget cases and 6
+failure controls. A separate rational log/pi/direct-positive-Taylor oracle
+checks alpha ceilings and actual certificates without importing the retained
+interval backend. The largest period 2^37 and cap 2^15 remain lazy metadata;
+no huge field, weight bank or native production run is claimed.
+
+## Costs, limitations and kappa status
+
+**No new kappa.** This removes one source-specific numerical admission,
+conditional on the explicitly credited ideal, scalar enclosure and constructor
+theorems. It does not collectively discharge the five selected assumptions.
+Legal period cuts, complete shared-context delivery, source H/ideal N and
+full transform integration, physical/native contracts and final recovery
+remain required. Arbitrary manually specified References can still fail;
+the controls demonstrate that scope boundary.
+
+The proof records an explicit sufficient onset for the unchanged O(p)
+allocation bounds and keeps source descriptor costs, O(p^2 log p) offline
+weights, subsequent page/Schur work and backing separate. No constructor-only
+bill is passed off as the cost of the complete Gaussian transform.
+
+Prepared for **@CubeJerry** with substantial OpenAI assistance. Jose Manuel/
+jmbarrancoml, Mason Hale, unpaidmoon/#4648 and the retained PR1673/PR1786/
+PR4100 contributors retain their credit in NOTICE.md. Independent lead review
+is separate evidence, not human or formal proof-kernel acceptance.

research/production-reference-gap/PR-TITLE.txt

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+Prove the actual cap constructor funds the production narrow gap (no new kappa)