[Clear1nsight]

Change and mathematical scope

Prove a scoped ceiling for two FULL invocations with two main owners and arbitrary distinct simultaneous productive helpers. Under invertible owner controls over the separated central-monomial/transverse-Clifford algebra B, the required two non-B main jobs cannot be completed while retaining every original helper job. For n=2+m1+m2+s, the subgroup-projected operator has rank at least (s+1)dim(B), while the requested target has rank s dim(B). The written proof permits noncommuting query/helper words, zero divisors, restored B-valued spectators, and width-dependent scalar coefficients. A reversible clean-work extension requires full cleanup.

This closes the retained-helper premise excluded by #4920's separated main-only argument. It does not combine with #4920's separate two-owner full-routing theorem. Arbitrary joint224-main-owner control, full central/fresh routing, prefix taps, changed helper ownership, extra interleaved nonmonomial operations, rectangular COPY/DROP, and discarded main-dependent garbage remain outside scope. The motivating 224H17+2H-2P invoice concerns112 independent replications, and its optimistic positive margin is not an endpoint certificate. No new kappa or selected-frontier change is claimed.

The package adds a self-contained written proof, eight vendored pinned sources, exact native normalization checks, frozen independent review, portable registered verification, and the required handoff. Original #4920 files are unchanged.

Task and overlap

Canonical question: Discussion #4902, retained-helper feasibility following #4920. Canonical main refreshed before submission: 624de9dc66858af0945e5f8a0942488d62280d43. Submission head: 6df6ed294614423836cbde182fb4368c9c346fe7.

Checked predecessor and overlap coverage includes #4610, #4694, #4698, #4700, #4703, #4711, #4712, #4856, #4867, #4888 and #4920, plus the all-author live PR feed. #4712's three-call dirty-helper construction has paid-minus-helper cost 2*r^t against one replaced r^t; it supplies a rejection example and predecessor credit. This result addresses a new retained-helper projection obligation rather than repeating that construction. The separate changed-routing construction remains open and is not supplied here.

Validation

Both registered Python3.12 standard-library commands passed locally:

python3 -I -W error -B research/two-full-retained-helper-ceiling/verify.py
python3 -I -O -W error -B research/two-full-retained-helper-ceiling/verify.py

The exact source checks cover112 actual fullH9 normalizers and114688 absolute-phase equalities, helper224's native empty-to-full contract, and four symbolic noncommutative compression fixtures. The backward ABI is entrywise conjugation in unchanged primitive order, generally not adjoint; the fullH9 exception is derived explicitly. Eight semantic mutation runs reject altered endpoints, source orthogonality, backward translation and row selection. An agent's independent source/accounting and zero-divisor review passed; its frozen report and receipt are included.

Committed-byte/source-history preflight and official26-file export/integrity checks passed. Exported normal/optimized replay passed without Git (PATH=/nonexistent) and left all exported bytes unchanged. All16 frozen science files and all27 original #4920 files were preserved byte-for-byte. The written proof supplies universal quantifiers; finite checks authenticate its source premises. Hosted run39882962847 passed at exact head 6df6ed294614423836cbde182fb4368c9c346fe7:346 helper tests/11 expected skips, both registered source-bound replays and the committed-input integrity gate. Registered outcome remains negative-result; CI corroborates the scoped package and establishes no new kappa. Native632-field replay, compiler/precision/reserve transfer, outer recursion, Lean validation and a complete multiplication theorem were not run or established.

Credit

Credit Cinna Davis /1337isnot1337 and retained upstream contributors for the native interfaces, #4920 for the projection method, and #4712 for the dirty-helper predecessor. Native source: #4610 at 20bb93cd990d2ff683e4d0fc56b87a42cd5d3296; #4712 source at b75a27bf07f1584b20babfddf887493981cd5949. Both are ancestors of canonical main; the package requires no fork-only Git history. Source, mathematical and review contributions are recorded in the package. Prepared for Chris R / Clear1nsight with substantial OpenAI Codex gpt-6.1-sol/high assistance. Separate agent checks do not imply independent human or proof-kernel acceptance. Apache-2.0.

Open next questions

Bind one literal central/fresh monomial route under the corrected native ABI, preserving every distinct helper job and charging actual entrance conversions, emitted subcircuits, saved intermediates and cleanup. X's rank-one entrance differs from the empty Y/helper entrance. A complete optimistic invoice must survive before a larger construction search. A physical witness, profitable caller moment and funded compiler/outer kappa improvement remain three separate gates. Required operational handoff: research/two-full-retained-helper-ceiling/HANDOFF.md; public exact-head handoff.