[CubeJerry]

Problem and exact result

The public cached phase payload package (#6906) explicitly leaves complete
chirp convolution and payload-grid conversion open. A compression-only
precision ledger does not fund those operations or their actual intermediate
radius. This package connects that exact public consumer to canonical
Cyclic.run (#3763), retaining every original padded coordinate and the
same immutable rounded phase bank at all three chirp occurrences.

For D>=2, t_0=r>=4, t_i in {r/2,r}, odd original slopes, T=product t_i,
M=T/r=2^A and an actual q-grid input of maximum modulus at most one half,
the joined output satisfies

||yhat-Q_freq F_t x||_* < (8M+9D+18)2^-f + 2^(1-q).

Here the norm is maximum component-l1 and F_t is normalized. Choosing
q>=chi+2 and
f=max(q,16,A+13,chi+1+ceil(log2(8M+9D+18)),ceil(log2(6D)))
gives strict error below 2^-chi; f=O(p0) for the stated fine family.
The reached output modulus is below .501. The exact q-to-f lift and final
f-to-q projection introduce no normalizer or new Gaussian Gamma.

What changes and why it is useful

• Execute kernel-on-ones, both conjugate payload passes and the actual
  complete Cyclic child, consuming each protected return.
• Check input radius with actual exact scalar products; join the different
  native signed carrier widths and the public common-grid codec.
• Reuse the unchanged canonical cohort binder, which authenticates and
  compares every corrected ring export byte before enabling the APIs.
• Prove the actual-input defect from the shared rounded kernel A, and state
  the precise broad-expansion term remaining at the public compression
  caller. Keep all three full Fourier calls, complete padded-field scans,
  source setup, copying, cleanup and retained backing in the invoice.

Cached phase stability/travel (#4325/#5987/#6906), the cyclic algorithm and
Fourier identity (#3763), old expansion bounds (#5295/#5325), and the exact
binder are predecessor results. The distinct contribution is this executable
source/grid/convolution interface and its explicit joined precision/radius
budget. OVERLAP.md records the bounded audit.

Pins and reproduction

Canonical base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.
Cached public source: 3c0443b2aa721ce692199ff31abc18e8d917596a (#6906).
Public codec: 73d5adaba2b7dc662e86eb901e81369a3f73be9f.
Public Reference/ideal join: 4509310a9a58d4c41c74029f899cbf36be1af9a1.
Public caller audit: 96211247643059ae48af4c7d9dc960034798935d.
Exact file hashes are in SOURCES.json and DEPENDENCIES.json; public unmerged
heads are explicitly retained dependencies.

python3 -B scripts/research_package.py --package research/cached-chirp-cyclic-join
python3 -B scripts/check_research_commit.py --package research/cached-chirp-cyclic-join --check-modes

The registered normal and optimized runs emit
PASS_CACHED_CHIRP_CYCLIC_JOIN. The genuine T8 fixture has q10/f16/W19,
radices (4,2), slopes (3,1), 192 cached scalar products and 16 input-radius
squares, in addition to the actual Cyclic arithmetic. A separately written
integer oracle matches all output words; an exact dense Gaussian-rational
sum gives maximum component-l1 error 11/8192 below the 45/16384 bound.
Actual radius and pending-return failures plus conjugation, missing M and
grid controls detect the critical seams. All 75 finite allocation cases
pass; the proof, rather than that finite list, supplies the universal claim.

Limits, status and credit

No new kappa. The complete multiplication bound and selected certificate
are unchanged. The finite q2 native provider is not production GL64/O73.
The proof retains the source, scalar and Fourier contracts; it does not
establish their all-size acceptance by replay. Broad Gaussian expansion,
its reached radius/error, fine-family backend admission, opposite transform
and complete CRT/integer recovery remain separate obligations. The old
Tensor catalogue is neither invoked nor waived. Another session's broad
Core work is outside scope.

Prepared for @CubeJerry with substantial OpenAI assistance in research,
implementation, proof, independent exact oracles and packaging. Predecessor
credit is in NOTICE.md, and retained notices remain intact. Separate
model-assisted lead review is not human or proof-kernel acceptance.