[Clear1nsight]

Change and mathematical scope

Extend the standalone finite ordinary-module paid-history argument of #5425 in two scalar regimes. This is a documentation-only proof contribution with separate independent AI static reviews. It supplies no new paid-history principle, physical construction, admitted supplier or κ.

Both variants require ordinary initial history, distinct independent target input coordinates, invertible nonordinary target maps, fixed whole-field capacity and one actual new output row per paid call. Finite feedback and overwritten rows remain allowed.

Extension	Exact domain
Central normalizers	One fixed finite columnwise word with exterior c·2^(n f), c in Q(i) and integer n fixed per instruction. Exactness holds at every positive width or throughout a consecutive infinite tail.
Eventual pointwise scalars	Arbitrary exterior c(f), nonuniform finite words and finite N(f), with one fixed finite ordinary one-column group G and geometry. The bound holds at each f≥2p−1, where p=|PG| counts actual projective classes. No practical or arbitrary small-width bound is claimed.

In either domain, q qualifying target rows require at least q actual paid returns. The new note retains the self-contained proofs and distinguishes their width assumptions. Its q=120 example is a conditional LOCAL nomination of the ordered map T=A Jbar A^-1, with copied nonmonomial and full-current-input arguments. The physical BC ABI and implementation are not newly proved or admitted. The optional fixed P/T signed-profile consequence keeps the original optimistic target grants; it is not new funding.

Neither variant charges 120 additional calls at an actual parent cut containing paid/native/cached observations. Such correctly versioned observations can remove the additional charge. Shared conversions, changed ordinary groups or geometry, cross-column maps, changed capacities, several returns per call, dependent inputs, approximation and recursive compiler/precision/stock costs remain separate.

Task and overlap

User-assigned ceiling and verification support; scope key standalone-paid-history-scalar-extensions. Scope is changed from #5425's fixed-scalar 112-output application to the two stated scalar/width contracts and a separate conditional source nomination. This is written mathematics with static review, rather than a proof-assistant formalization.

Base: 32890bc5de034ab8fc93ac88ead7279a6a416d04. Proof/review snapshot: 56c52a79f7663bc59ad3a5013c242a113039a701. Publication head: 0d0bcf3fa261c9591e7e60ed5926c6e04d360f9e; the second commit changes only HANDOFF.md to pin that snapshot; the third escapes Markdown table delimiters in README.md without changing mathematics.

The direct predecessor is #5425 at 00005c8c5e81c7f27b5769f9309ff88758e26f51, including its feedback/overwrite history argument and ordinary-initial, fixed-capacity contract. The precise additions are central scalar normalizers, eventual pointwise arbitrary scalars/nonuniform finite words, and the separate conditional 120-coordinate application. Its 112-output source specialization is not transferred.

#5436 supplied the support-width prompt; its four-control 2705-source corollary is not adopted. Adjacent #5441 concerns fixed physical transfer-table endpoints and cut dimensions. #5442 states a finite-group lifting lemma, but its separate complete-body claim is challenged by review 6385984727; that closure is not a dependency here. A small-width affine result for a different operator family cannot replace the large fixed ordinary-group/coset argument. A final refresh through 2026-10-24 17:10:39 UTC records an author repair of #5442 at 4d5abe038ac106bd79acf548df4aeb66f32eff09; independent revalidation was not present. The challenged frozen head and subsequent repair are neither dependencies nor independently validated endpoint evidence for this note.

The supplied sole-monitor prepublication check covers 100 recent all-state PR-metadata/body records through 2026-10-24 16:50:42 UTC, with a later static locator addendum. Main/guidance/selected are unchanged. This bounded incremental check is not exhaustive novelty, Discussion or unpublished-work coverage.

Validation

The sixteen-file note contains twelve unchanged proof/acceptance/review/source payloads and four new navigation/provenance documents. The sealed scalar-extension lemmas received independent AI static mathematical reviews without a required correction. Their original PROPOSED headings are preserved; additive reports and decisions record acceptance.

Copied-byte hashes and lengths, JSON/navigation, exact path scope, seven handoff labels and Git whitespace/diff checks are recorded as passive assembly checks. They are not theorem or supplier certificates. Separate independent static reviews cover the mathematical variants and polished documentation scope. Committed-byte and exact-path checks are separate custody evidence; hosted outcomes remain tied to the PR head.

There is no verifier, VERIFY manifest, numerical RESULT or scientific reproduction command. No operator, native/phase/rank/calibration, Powers/price, Lean or registered replay was run for the note. Source-native facts remain conditional inherited premises, not a new executable closure. The checkout is shallow/shared; no complete history or file corpus is claimed. The CI-navigation repair is already canonical and has no duplicate delta.

Credit

Retain the #5425 ordinary-module proof and independent reviewer as the direct paid-history predecessor, and zxv's #5436 support-width prompt. Anthony Chefles supplies the classical multi-copy/projective tensor-independence precedent (DOI); the field-linear proof used here is stated directly. Peter Kropholler and Karl Lorensen's strong-rank theorem (DOI) corroborates the self-contained central argument.

Credit the scalar/source authors, root's localization and paid-history question, independent reviewers, sole literature worker and BC/native/Clifford/product/compiler/precision/analytic lineage. Preserve Ali Romman, Dillon Amburgey/dillona, Cao Yifan, Cinna Davis, zxv, Dugongue, Jacob Alan Sussman, Chris R/Clear1nsight and the retained notices. Substantial OpenAI Codex assistance. AI static review is not independent human or formal-kernel verification.

Open next questions

Identify the actual cut's available history module with source versions, owners, paid observations and lifetimes, then test containment of the required target rows and charge acquisition/retention once. That question is not decided by this standalone theorem. A smaller sufficient source-specific width guard needs a justified subgroup or span argument; no such cutoff is computed here. Recursive use also needs its own uniform cost, productive-target, precision and stock contract.