[CubeJerry]

Result

The current exact positive child can reject Gaussian halves on an arbitrary
grid input, while the precision contract needs a total primitive on its
declared actual-input grid/radius domain. This package connects a paid
dynamic lift and one final nearest-up rounding step to the unchanged current
source, descriptor and lifecycle interface.

For every actual positive source rank r in 1..8 and positive parent
f<=cutoff, put g=r*f. The provider physically lifts every component by
g bits and uses w=h+g+ceil((g+1)/2)+1.

Complete-fibre unitarity gives strict internal signed headroom for arbitrary
h-bit inputs, and g low zeros make every Gaussian half exact. A private
header physically changes the complete record/row lengths; the exact
public cutoff child then performs its own (f,t)->(r*f,r*t) transport.
Original f is not multiplied twice and the public cutoff/header are retained.

The dynamic-g return codec rounds once, with negative ties upward, tests
true final h-bit overflow, and stages the entire original-width result
before commit. On the declared radius domain
R+Delta/2<2^(h-1)Delta, it is total and its residual from U of the actual
input is at most eta_X=2^((X-1)/2)Delta for a containing complete X-fibre.
A late final overflow preserves the original field and retires all work.

Complete cost and novelty

The inventory is 521 fixed private ports. For
V'=V(w+1)/(h+1), the full direct bill is
F_template+O(g V'+(E+1)^4+521), including both codecs, source/header work,
staging, restoration and erasure. Historical backing is retained per role.
Only a separately proved g<=h permits V'<=3V; the exact precision W/D
mapping gives a conditional size corollary, not an established caller proof.

The generic lift/round and unitary error principle is prior. New work is
the literal variable-g codec, current widened descriptor/source join,
complete signed/overflow domain and paid return boundary. This builds on
cutoff transport 785000b8
and its retained all-source positive leaf, targeting the actual-input
contract in precision envelope 56bc9383.
The complementary rounded scalar tables 6476455a
handle fixed denominator debt and are not imported here.

Verification

From the repository root:

python3 -I -B research/current-rounded-positive-leaf/verify.py --suite interfaces
python3 -I -B research/current-rounded-positive-leaf/verify.py --suite numeric
python3 -B scripts/research_package.py --package research/current-rounded-positive-leaf

All four registered normal/optimized gates passed, each capped at 120 seconds.
Independent integer/bitstring checks cover 392 accepted drop values,
36 overflow controls, 180 complete signed lift values and 32 all-rank header
cases. Both deliberate rounding/range mutants fail. Larger-to-smaller
reuse checks preserve visited history without adding old-blank-tail scans.

Seven complete small-factor executions total 25,146,787 literal actions,
with an independent exact Gaussian matrix, absolute phase, both tie signs,
f=2, rank2, spectators, rows/coefficient multiplicities, source/header
restoration and late-overflow atomicity. One invokes actual PAID 9896 under
a small leaf-local header. No 4096-record selected producer packet,
enclosing compiler trace or complete native MOVE was executed or claimed.
Separate model-assisted lead checks and their limits are retained in the
package. Exact source pins, expected receipts and negative controls are
included for offline replay.

Scope and κ status

No new κ. This is one local total rounded positive-leaf supplier theorem,
relative to the stated valid-format/disjoint-segment and inherited current
source contracts. It does not flatten the whole compiler, establish the
global producer/radius mapping, admit monomial phases, prove a nonleaf rate,
or discharge any of the five global assumptions. Failure atomicity covers
the stated valid-provider boundary and final overflow, not physical
interruption or malformed late output. Canonical base is
7c49fe32c55de02c2c873655c804b16e4c3fef0a; no selected certificate changes.

Credit

Prepared for @CubeJerry with substantial OpenAI assistance in mathematics,
code, verification and prose. Credit Arham Tahir, Nick Petalas and
#2771 for rounded-native/block-scale predecessors;
Sricharan Vagvala for the finite rounded-native counterexample; the
#4743 normalized butterfly lineage; and the
#3091 physical Gaussian/arithmetic ancestry.
The exact current leaf, header, phase and lifecycle notices remain unchanged.
Separate AI review is not a claim of human or proof-kernel acceptance.