[CubeJerry]

Summary

The source precision proof leaves a completed one-rounded global LAMBDA
primitive open. This package supplies a literal Gaussian table and a
source-specific exact Q² monomial/header join, retaining all routing,
owned-copy, source, head-return and cleanup costs. No new kappa, complete
caller admission or physical tariff-cap claim.

For the actual source direction q and positive column multiplicity f,
Q_f² is simultaneous address translation by q in every central column,
with phase zero. The new fixed recipe is identity9/offset-q/phase0.
It is executed through the unchanged serial monomial provider, with a
physically encoded private view changing exactly remainder to 0 and
selected_axes to 9f. All other descriptor fields and the full row are retained.

Global Lambda_s(Q_f)=((1+si)I+(1-si)Q_f²)/2 stays outside tensor powers.
Two odd factor characters explicitly distinguish it from a product of
one-column Lambda kernels. The two Gaussian numerators are formed at W+2
with cancellation intact, then rounded once per real/imaginary component.
STAGE writes only an owned translated copy; a separate literal COMMIT
writes the public field after every range check passes.

Source and exact result

Canonical base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.
Source: DreamingOfClouds' #6936 at tested head
03626137d0bd57f09b0bf9597376183ce6fa9683; current head d360b118...
changes only the verification timeout. The QUERY-only caller is public
617903497387a98f7ca929ad0b6928dd32abffec, descended from
#6937 at a98df73....

The caller has 14,480 expanded LAMBDA occurrences at 104 actual directions.
Each sign table has 5,275 rows. The rank-nine provider has 1,965 fixed owned
roles, including all retained private inventory, one new full-field bank
and two numerator banks. No new recursive paid request or productive helper
credit is introduced. Every unchanged dependency has exact commit/path/
Git-blob/SHA-256 authentication.

For valid current geometry, canonical common-grid data and admitted signed
range, the output defect from Lambda applied to the actual input is at most
eta_X=2^((X-1)/2)Delta on each containing complete X-fibre. The fibre includes
all 9f translated axes. Named late range rejection preserves the public value,
retires the partial copy/work and restores public/context heads explicitly.
No arbitrary malformed/native-interruption rollback is claimed.

Full cost and remaining interfaces

The full serial bill remains

O_9(f*((g+1)*(Vraw+M1*A1^6)+V+(E+1)^4+P)),

where g=popcount(q). The new Gaussian tape/control bill is bounded by
4096(S+1), with additional positive copy, header, retirement, setup and old-
backing charges. Historical capacities of every role remain allocated.
Only explicit padding absorptions give O_9(fV). This is not converted into
the old 1000-unit LAMBDA tariff or a supplier raw cap.

The new outer token/header contract is explicit. The old compiler has not
been shown to emit it. The selected #6030 catalogue has 22,803 events/3,453
MOVEs; proposed #6936 has 71,144/6,323. This package derives its new Q² recipe
from #6936 and does not inherit selected-catalogue membership. Actual caller
width/radius substitution, full recursive stock and the full source invoice
remain open. All five foundational framework assumptions remain global
premises, and the selected certificate is unchanged.

Reproduction and verification

python3 -I -B research/current-rounded-lambda/verify.py --suite all
python3 -I -O -B research/current-rounded-lambda/verify.py --suite all

Expected: five PASS_CURRENT_ROUNDED_LAMBDA_* markers. Registered suites have
120-second caps; standard library only, with no runtime Git/network/solver.

Author checks cover 544 exhaustive four-input cases, 1,536 wider endpoints,
12 literal 23-field views, both-sign complete reduced-rank composition,
late rejection/retry, two arithmetic mutants, four strict typed-source
controls, and 53,248 source-character/compiled-flip equations. Separate lead
AI checks cover 48 wider cases through 129-bit components, a read-only-public
STAGE trap, dirty displaced work/context, retained backing after long-to-short
reuse, an independent rational oracle, and 1,020 global character cases.
Unchanged independent runner/receipt and reviewed hashes are supplied.

The complete actual nine-axis numerical case is explicitly unexecuted.
Its minimum 4,096 records was screened above the 120-second attempt cap from
the measured 16-record reduced run before extra flips. This is a feasibility
screen, not a formal wall-time lower bound. Reduced physical evidence is
not described as a whole source replay; the universal join uses the retained
provider proof and the source-specific recipe/header proof.

Credit

Prepared for @CubeJerry, with substantial OpenAI assistance. Credit
DreamingOfClouds' #6936/#6937 source lineage;
the retained monomial/phase work including #3087
and #4172; CubeJerry's lifecycle, scalar and
serial-provider packages; and the source precision package at
56bc9383594a3efc303148e00bb746851abd1ed5. Classical Gaussian/rounding
identities are not claimed new. This contribution is the literal physical
source join, signed/defect proof and complete stated bill. Separate AI review
is not human or proof-kernel acceptance.