[aliromman]

Change and mathematical scope

This follow-up raises merged PR5599 from conditional κ = 0.00310045 to 0.00310440546, a gain of 0.00000395546. The result is 2.5474% above selected 0.003027288235. The selected frontier is unchanged.

Proof · Fixed boundary relations · Independent cuts · Preparation sites · Complete receipt · Source pins · Validation.

At a common-frame cut, a helper input functional and a helper output direction have unit pairing. Explicit input, cut and output bases let one independent target finish early and a fresh independent input use the suffix. Both complete boundary supports are checked. This contribution composes that changed helper interface with the useful released components:

• PR5601 supplies the frame word. PR5605 and PR5612 supply the shared and delayed relays. Their 13 combined source edits are disjoint from PR5601's 21 edits against the common PR5580 source. All input corrections and late cleanup remain.
• Five fixed PR5602 relations each remove one paid H5 and replace one H9 target by H4. Its sixth relation uses a removed contact and is omitted.
• Nine independent helper splits add targets without added Gaussian calls. Every prefix functional, suffix direction and intermediate interface is checked again on the new word.
• PR5606 supplies two preparation identities. The two H1 queries are removed, the H2 target is debited, and all three new owned lifetimes are charged.

Complete quantity	Result
Local independent inputs / helpers	743 / 535
Owned roles / lifetimes / peak	9 / 11 / 2
Paid calls	66,546
Productive fields	12,958
Rank-mass surplus	760
Tested saving	0.003104405465
Exact raw crossing	(0.00310440547057, 0.00310440547058)

PR5613, published during this run, has the same signed recursive moment with 72 more rank-five paid calls and 72 more rank-five targets. This construction removes those matching entries. Its last 0.00000000546 gain above PR5613's rounded 0.00310440 uses compiler headroom, not a new physical mechanism. The reserve is 10^-12; tensor and logarithm powers stay 1024. The finite threshold is N = 2^80511200000000000.

The p16/a7 caller charges every helper normalizer, main adapter, scalar operation, route, COPY/CLEAR path, thin query and whole idle. Two exact moment engines, all compiler conditions and 54 strict plus four functional outer conditions pass. The complete nonrecursive allowance is 184647552, below 1e9. A negative derivative bound gives the unique raw crossing for this fixed invoice on [0,0.0034]. The claimed saving is within 1.058e-11 of that raw ceiling.

The five framework assumptions and helper-endpoint contract remain. The huge uniform word and complete rounded multiplier are not materialized. Neither a global optimum nor independent scientific acceptance is claimed.

Task and overlap

The follow-up starts from refreshed canonical main 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7. Frozen scientific inputs retain base 5e0052ca0aa3036a3afa8b0438927377c8d97d53. A maintainer merged PR5599 at 09dd0965711ca7eeb169057a420abab1c9052137 while this later composition was in progress. This PR contains the later scoped delta.

Task: complex-new-helper-inventory, scope changed at the donor's request for a different mechanism and a complete gain. The independent input count and target contract change; a complete caller replaces the old frozen-provider screen. TASK.json preserves the original card and reason. No task was rerolled.

Canonical coordination records the claims, released components and final artifact. The parallel PR5602 claim was released before its fixed relations were composed. The relay/preparation work uses published constructions; it does not extend those authors' active searches. The shared rank-one method is credited without an exclusive discovery claim. Every combined interface and bill is replayed; headline gains are not added.

The last full overlap snapshot covered 340 open PRs and 4,874 Discussion records with complete remote pagination and incomplete sparse local coverage. The later bounded open-PR check found PR5612/5613 and led to the final compatibility check. The final title/head check found no newer compatible kappa component. Canonical main, priorities, selected result and shared inputs were unchanged at submission. These checks do not prove novelty.

Validation

python3 -B scripts/research_package.py --package research/p104-helper-subspace-splits

Direct isolated, registered normal and registered optimized replays pass. The sealed export also passes an isolated -I -O -B replay with no Git directory, an empty PATH, read-only files/directories and all 37 file hashes/modes unchanged. A one-byte change to the retained PR5612 constructor is rejected before import; exact bytes and mode are restored. All mutation controls pass.

Metadata/source pins and the 36-file committed-byte check pass at publication commit ed4dc88a09a7d201971f6603b8c2074e95dd9c32. All 35 other package/input files match the frozen scientific replay byte for byte. Official export and integrity checks pass. See the linked validation record for commands, environment and limits.

Scientific commit: 3e038c5da1b56e9c9e3ad76c0592e91914f7eb6d. The publication commit changes only the validation record after those scientific checks. The final finite split screen finds no further candidate in 46 boundary groups. It covers only the stated after-MOVE basis-pair class.

Full-history preflight is unavailable in this shallow checkout; all seven source-pin/base commits are verified branch ancestors. Registered replay needs no cached unmerged-branch objects. Hosted CI and independent scientific review are separate from these author-side checks.

Credit

Credit flyingviper for PR5605/5608/5612/5613; Scratch-net for PR5606; Pierre-Loic Doulcet / hexapode for PR5565; the separate Ali Romman PR5601/PR5602 constructions; Joel Pulikkan / GamingPuzzled, DreamingOfClouds, thu-nmrc, Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv, Cao Yifan and all retained geometry, BIT, compiler and analytic notices. New independent-helper construction and this composition are by Ali Romman with substantial OpenAI Codex assistance. The same assistant constructed and checked this package; no subagents. Apache-2.0.

Open next questions

Independent review should check the disjoint edits, first split, fixed boundary relations, both preparation identities and the H2 target debit before replaying the complete price. More general bases, cut orders and further preparations remain open; check current Discussion scopes first. Structured next questions.