[aliromman]

The selected 22,803-event word now has a complete same-grid stochastic numerical execution and a literal retained-state inverse. Every one of its 2,886 distinct native edges is checked on all 512 columns. The new absolute-error contract is supplied by executable primitives, including small lattice inputs where a uniform relative-error allowance fails. Result type: construction with a source-bound all-width numerical error proof. No new complete kappa, physical reversible circuit or selected assumption discharge is claimed.

Review the proof and complete direct bill, exact receipt, caller and inverse, signed transfer engines and source pins. Reviewed head: ff038070b8d122a43993f2b2524154323cc6b4d7. Base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR #6030, selected kappa 0.003125804.

The selected word has 770 arbitrary public fields and 29 owned labels. At tensor width f, each field has N=512^f complex coordinates on one grid h*Z[i]. Each forward Gaussian layer and noninteger scalar primitive receives fresh uniform randomness conditional on the entire past. A seed may serve all coordinates within its stage. The inverse retains and reuses that seed; later forward stages cannot reuse it under the theorem.

Every native recipe is compiled with the unchanged source compiler and checked with the separate coordinate oracle. The source join covers all 1,477,632 edge columns, including Gaussian global phases. The mathematical all-width extension is the explicit tensor lift of this fixed source word. It is not an identification of every recursive caller with that lift.

For every field at every completed source prefix,

E||error||^2 <= (N*h^2/2)*(23423*f + 161509/8),
||error|| <= h*sqrt(N/2)*(2957*f + 4917).

All four signed-transfer score maxima occur at cut 20644, field 558. Forward and transposed reverse sweeps agree on every final signed coefficient, with separate reverse-cut checks at the peaks. For each positive tolerance epsilon,

Pr(max_fields_and_prefixes ||error|| >= epsilon)
 <= 45606*exp(-epsilon^2/(N*h^2*(23423*f+161509/2))).

At width one, h=epsilon*2^-12 suffices for each individual RMS error below epsilon; h=epsilon*2^-16 gives simultaneous error below epsilon with failure probability below 2^-40. The positive rational Taylor certificate avoids floating-point acceptance. The proof gives simple all-width grid rules. These are sufficient absolute-error budgets, not least physical precisions or a direct numerical comparison with the input-relative guards in PR #6526.

The inverse pays the state that lossy instructions would discard. ADD keeps its source and seed; noninteger SCALE keeps exact signed division residues; COPY and CLEAR keep their old values. The uncompressed schedule retains 54 whole fields, 578*N residue bits and 5952*f+18592 seed bits, with 824 live-plus-retained fields at its peak. At width one, that is 295,936 residue bits and 24,544 seed bits. Thirty of 31 ideal CLEAR inputs are nonzero. This is an injective record on reached states, not a full-state permutation on identical finite registers or a joint information minimum. The checker additionally copies all 770 initial and all 770 output fields; these verification copies are priced separately.

One forward run pays 2976*f*N rounded pair butterflies, 6,906 tensor monomial passes, every scalar instruction, copies, residues and seed reads. The inverse pays its own work. The proof includes fixed route tables, sufficient signed widths and O(f*N*W) indexed bit work. Physical sampling, reversible gates, fixed tape routing, retained-data cleanup, accelerated recursive tariffs and outer recovery remain open. This direct evaluation cannot be substituted for a fast recursive child at its old price. All five selected framework assumptions remain.

The registered verifier uses CPython 3.12.13 and NumPy 2.3.4:

python3.12 -B scripts/research_package.py --package research/selected-absolute-stochastic-caller
python3.12 -B scripts/check_research_commit.py --package research/selected-absolute-stochastic-caller --check-source-history

Normal and optimized runs pass, with source assertions enabled in normal subprocesses. The complete 22,803-event dense caller reverses exactly, and the independent endpoint decoder checks all 770 public outputs. Other checks cover 2,401 pair inputs, 48 wide signed pair round trips, 2,015 scalar contexts, 5,140 signed division round trips, all 32 fresh seed trajectories of a mixed program and a 262,144-coordinate rank-eight tensor inverse with 16 active layers. A separate exact-grid tensor input is seed independent and matches the independent source oracle on all 262,144 coordinates. Seven semantic controls, two operator mutations and three actual source-corruption rejections pass. No probability claim is inferred from the deterministic full-caller test stream.

PASS_SELECTED_ABSOLUTE_STOCHASTIC_CALLER
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-absolute-stochastic-caller outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Fifteen committed package/input files, source ancestry and official export integrity pass. A read-only, Git-free optimized export replay passes in 208.89 seconds with all 16 files and modes unchanged. Hosted CI is pending at publication. No independent human or proof-kernel review, complete predecessor scientific suite or selected cold replay ran. No contribution was merged or closed, and no selected-frontier data was edited.

Credit Ali Romman's PR #6526 for the signed transfer method and retained concentration proof, and PR #6532 for the conserved-sum pair sampler and its actual-input relative-error boundary. PR #6509 supplies the prior deterministic center family; PR #6540 gives a distinct coarse-grid residue codec. Their local results are not presented as new here.

PR #6499 supplies a different exact scalar grid contract between independent native returns; PR #6506 gives its Boolean scalar lowering. Their native/error/reset premises and physical gate counts are not imported.

Nick Petalas / @petalas supplied PR #6438's reversible-ensemble construction and PR #6418's adaptive reuse bound. Credit zxv's PR #6066 for the scalar stochastic-shear ABI and earlier restored-randomness warning. DreamingOfClouds / @DreamingOfClouds supplied the selected word. Cinna Davis / @1337isnot1337 supplied the consumed PR #4088 and PR #2226 native representations. NOTICE retains the full native, lifting, numerical and caller lineage. Classical rounding, residue retention and martingale methods are credited, not claimed as new.

Advance claim. Canonical main and all public open/closed PR and Discussion metadata were refreshed at the round and before submission. A second exact-main README/TASK search found no same stochastic caller join. The concurrent fused deterministic two-axis scope remains separate. Arbitrary diffs and unpublished work remain overlap-coverage limits.

Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Twelve files are added only under research/selected-absolute-stochastic-caller/. The next unreserved question is an accelerated physical child implementing this numerical ABI with its complete retained-state and tape bill.