Research: Remaining diagonal

research/source-remaining-diagonal/CHECKPOINT.md

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+# Wave 4: source remaining diagonal — reviewed and verified locally
+
+Base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.
+Branch: research/source-remaining-diagonal-20261109.
+Worker: analytic_join; worktree: /workspace/scratch/a365d332d434/worker-analytic-wave4.
+
+Completed: actual canonical Reference coefficients, finite sufficient normalizer
+admission and source p^100 implication; certified upper-floor one-ulp quantizer;
+unchanged original Full Diagonal event/FFT scalar service; two actual global
+routes; exact same-grid codec with no added normalization; full cost/backing
+invoice; substitution into public ledger equation (4).
+
+Verified: 272 exact coefficients, 1904 independent signed-grid trajectories,
+1998 boundary quantizers, 9 source/coefficient controls. Native 32-record run
+executes 2 routes, 64 squares, 32 complex products and 192 scalar calls; all
+16 native parameter positions and accumulator margin checked. Six physical
+alias/return controls; borrowed input/heads restored; protected output consumed.
+Registered normal and optimized replays PASS. Lead independently checked the
+universal quantizer proof, 7992 rational enclosures, source/codec contract,
+exact route exponent inequality and accumulator margin; receipt retained.
+
+Dependencies: public ledger76a84e1c4bfa4035e5ee85410dbfc4541f0da168,
+source codec73d5adaba2b7dc662e86eb901e81369a3f73be9f,
+complete Core6fb9ab776b7d0956b8bcc2d3cfde0f3034421630,
+Full FFT service2882d4414e95a6884f7b588d9da7ca893f800cbf.
+SOURCES/VERIFY declare all 51 canonical live inputs; DEPENDENCIES authenticates
+20 retained public files. The registered card/provenance input is also pinned.
+
+Remaining: committed-byte/mode gate and lead remote publication. No remote
+branch or completed multiplier is claimed by this local checkpoint. Production
+context delivery, H/ideal-Gaussian comparison, native/route/lifecycle premises,
+complete source prefix/suffix, extraction and final recovery stay open.
+No new kappa. Follow README reproduction and exact pins; old public branches
+are frozen. This scratch worktree alone is transient until lead publication.

research/source-remaining-diagonal/DEPENDENCIES.json

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+}

research/source-remaining-diagonal/NOTICE.md

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+# Attribution and scope
+
+Prepared for @CubeJerry with substantial OpenAI assistance in source audits,
+proofs, implementation, exact independent oracles and packaging. Apache-2.0.
+
+Jose Manuel (jmbarrancoml) authored the canonical rational Gaussian Reference,
+with substantial OpenAI Codex assistance. Its source credits Vlado Vojdanovski,
+Jeremy Kothe and Cinna Davis for geometry/reference work, and Mason Hale for
+the interval arithmetic. Their exact source files are external authenticated
+inputs, and their notices are unchanged.
+
+The exact Full/FFT service credits the original Full/fixed-word compiler,
+global caller, Count/ownership interfaces and prior scalar/native work, notably
+CrocSwap/integer-mult-bounds#3787, #3709, #3627, #4205, #4253, #4178 and #6898.
+The retained public files preserve all original author and assistance notices.
+The successful physical route/owner pattern follows the public complete Core
+join at `6fb9ab776b7d0956b8bcc2d3cfde0f3034421630`; this explicit new producer/
+owner does not claim to be that prior program or its original FullRoot theorem.
+
+The same-session published source codec `73d5adab` and group ledger `76a84e1c`
+are immediate predecessors. Their public commits, paths and hashes are in
+DEPENDENCIES.json. The source Gaussian diagonal cache and older C/D
+normalization in CrocSwap/integer-mult-bounds#3851 are prior work, as are the
+Karatsuba Full physical service #6894 and separate FFT recovery #6895.
+
+No upstream selection, unconditional multiplier or new complete kappa is
+claimed. Source and native premises are stated explicitly in PROOF.md.

research/source-remaining-diagonal/PR-DESCRIPTION.md

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+## Why this matters
+
+The public group scale ledger leaves an explicit missing service: after the
+actual Core/halo routine returns the inverse divided by 2^U, apply the source's
+remaining `2^(U-K)Dtilde` on the same original padded field, actual-state radius
+and common q grid. A norm bound or generic cached diagonal pass does not
+establish those source, scale and physical interfaces.
+
+## Exact result
+
+This package binds the actual canonical Gaussian Reference coordinates,
+including identity holes, to one original Full Diagonal event and the unchanged
+exact FFT scalar owner. Two real whole-field routes restore original order;
+the existing external codec has zero grid drop and zero added normalization.
+For all admitted q-grid inputs of component-l1 radius 3R0, the error is at most
+`(3R0+2)2^-q`. This fits the ledger's existing q budget, so that particular
+remaining-diagonal error obligation is supplied without changing gamma.
+
+The finite guard `K-U>=2 sum alpha_i^2` is explicit and follows from the
+conservative source p^100 budget and its existing alpha/gap admissions.
+Certified reciprocal exponential intervals and an upper-endpoint floor with
+clipping give one-ulp coefficient error without a transcendental separation
+assumption. The complete full-period stock, spectator code, both routes,
+all radius squares/products, format conversions and retained backing are paid.
+
+## Novelty and predecessors
+
+This is the specific source/remaining-scale/common-grid/physical join, not a
+new scalar multiplier, Gaussian cache or generic perturbation inequality.
+CrocSwap/integer-mult-bounds#3851 already has C/D normalization, #6894 the
+original Full physical service, #6898 its exact FFT scalar owner and #3787 the
+scalar theorem. The source GAUSSIAN.md already gives cached prefix products.
+#6895 addresses a different whole FFT recovery boundary.
+
+Immediate public dependencies:
+
+* [Group ledger 76a84e1c](https://github.com/CubeJerry/integer-mult-bounds/tree/76a84e1c4bfa4035e5ee85410dbfc4541f0da168/research/core-group-scale-ledger).
+* [Source codec 73d5adab](https://github.com/CubeJerry/integer-mult-bounds/tree/73d5adaba2b7dc662e86eb901e81369a3f73be9f/research/core-halo-source-bridge).
+* [Full FFT owner 2882d441](https://github.com/CubeJerry/integer-mult-bounds/tree/2882d4414e95a6884f7b588d9da7ca893f800cbf/research/full-fft-scalar-join).
+* [Complete Core family 6fb9ab77](https://github.com/CubeJerry/integer-mult-bounds/tree/6fb9ab776b7d0956b8bcc2d3cfde0f3034421630/research/complete-core-physical-join).
+
+Canonical base is `7c49fe32c55de02c2c873655c804b16e4c3fef0a`; all source paths,
+individual hashes and dependencies are recorded. Retained source bytes are
+unchanged, and large canonical archives remain external inputs.
+
+## Reproduction and checks
+
+```sh
+python3 -B research/source-remaining-diagonal/verify.py --physical
+python3 -O -B research/source-remaining-diagonal/verify.py --physical
+python3 scripts/research_package.py --package research/source-remaining-diagonal --check-only
+python3 scripts/research_package.py --package research/source-remaining-diagonal
+python3 scripts/check_research_commit.py --package research/source-remaining-diagonal --check-modes
+```
+
+Expected status is `PASS_SOURCE_REMAINING_DIAGONAL`. Exact independent
+positive-exponential and forward-coordinate oracles check 272 coefficients,
+1,904 signed-grid trajectories and 1,998 adversarial quantizer cases. There
+are 9 coefficient/source controls and 6 physical controls. The actual 32-record
+native witness runs 2 global routes, 64 exact radius squares and 32 complex
+products (192 scalar calls), checks all 16 parameter positions, matches every
+original-coordinate result, restores input and borrowed heads, and consumes
+its protected return. Its scalar widths use the fixed cutoff's base branch;
+the universal FFT bound is credited separately.
+
+## Limitations and kappa status
+
+**No new kappa.** This is one source-specific conditional service theorem.
+Whole production-context delivery, rational H/ideal-Gaussian comparison,
+native/Count/route/lifecycle premises, the rest of the source transform,
+final extraction and complete integer recovery remain necessary. A finite
+fixture does not prove those claims. The integration is supplied at the
+service boundary; a complete Core/halo-plus-diagonal physical fixture is not
+asserted here.
+
+Prepared for **@CubeJerry**. Predecessor authors and exact retained notices are
+credited in NOTICE.md. Substantial OpenAI assistance was used in proofs,
+source analysis, implementation, checks and packaging. Independent lead
+review is a separate recorded review, not proof-kernel or human acceptance.

research/source-remaining-diagonal/PR-TITLE.txt

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+Supply the actual remaining Gaussian diagonal on the common grid (no new kappa)