[CubeJerry]

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.
#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.
• Source codec 73d5adab.
• Full FFT owner 2882d441.
• Complete Core family 6fb9ab77.

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

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.