[CubeJerry]

Change and mathematical scope

The public Core/halo and group-scale packages compare actual rounded
compression with a fixed rational Reference H. This package joins their
explicitly open H-versus-ideal-periodic-N boundary on the same original q
image/holes, through the actual remaining D diagonal, final extraction C and
unchanged restoration gamma. Assumption progress; no new kappa.

The literal certificate delta=1-adjacent_upper-5*distant_upper-epsilon also
proves ||N^-1||<=1/(delta+epsilon). Including the actual D factors and group
normalization yields xi_j=mu_j sum epsilon_i/(delta_i+epsilon_i). The actual
for_cap allocation and actual remaining-diagonal error leave enough room for
this term at the unchanged public Core t and external q allocation under
the exact finite gate mu_j |G_j| 2^U_j<=2^42. The source's p^100 budgets satisfy
that gate. Source targets/radii must cover the real caller; a source-first
target rule removes circular dependence on later Reference normalizers.

Sources, novelty and checks

Canonical base 7c49fe32c55de02c2c873655c804b16e4c3fef0a. Exact public
dependencies and copied proof/helper blobs are recorded in DEPENDENCIES.json:
scale ledger 76a84e1,
remaining diagonal b00d64b,
production gap 9a07c08.

#3851/#3935 already perform a resolvent/Tensor join for a different
ProductionPlan domain, coefficient cohort, gap and normalization. Their method
is credited. RAW-GAUSSIAN-PROOF and #4648 supply analytic/allocator premises;
#6880 and later public Core/halo work supply the same-reference splice. The new
work is the present caller binding and unused actual precision-budget slack,
not a generic perturbation identity or complete old Tensor repackaging.

Reproduce from the repository root:

python -B scripts/research_package.py --package research/reference-ideal-compression-join
python -B scripts/check_research_commit.py --package research/reference-ideal-compression-join --check-modes

Expected PASS_REFERENCE_IDEAL_COMPRESSION_JOIN and registered/commit markers.
Normal and optimized modes check the unchanged complete for_cap constructor,
all 4096 entries of a T64/S61/p3 Reference, 549 independent original-offset
Gaussian terms plus an infinite alias tail, and two 128-step fixed-grid
Neumann diagnostics with exact residual certificates. Both solves use one
named 64-component complex input x. On that input, the exact inverse-output
difference is <=2^-158 and the selected normalized post-D difference
<=2^-169. These two bounds are input-specific, not inverse operator norms.
Also included are 216 unchanged-budget fixtures, 30 normalization gates and
10 rejecting/source-mathematical controls. Separate lead AI review checked
144 rational tensor/resolvent and 162 allocation cases without importing the
submitted checker. Complete gates and receipts are in the package.

Limitations and credit

The small fixture admits neither complete Core geometry nor its dense native
radius, and performs no physical Core or diagonal call. Its source radius2
tests the additional bridge-radius binding only. The evaluated combined
2^-12 budget is a conditional invoice, not an observed physical output error.
Production source generation/context delivery, native/ownership/compiler
contracts, retained infinite analytic identities, actual expansion/Fourier
errors and full CRT/product/carry recovery remain explicit obligations. All
five foundational assumptions retain their global status.

@CubeJerry with substantial OpenAI assistance; separate source-aware AI
mathematical review, no independent human or formal-kernel acceptance claim.
Credit Jose Manuel/jmbarrancoml, Mason Hale, Jeremy Kothe, Vlado, Cinna Davis,
unpaidmoon, the retained public Core/halo/ledger/diagonal authors, and the
Harvey--van der Hoeven/classical analytic lineage. Full notices retained.