# Continuation checkpoint: noncommutative formula construction and exact witnesses

9 October 2026. The previous goal turn made concrete progress through the edit-distance source review, exact-library controls and verified GitHub/Dokploy publication. This turn deepens family 116 and adds a bounded source-derived reference. The perpetual goal remains active after the minimum duration; facility-planning assurance remains the first product experiment.

## Findings that change the next action

The characteristic-zero formula paper gives explicit rational hitting matrices of dimension 2*n*s^2, with a special one-by-one case. Their integration action admits a simple exact coefficient recurrence. The implemented visible-tree evaluator produces nonzero free-polynomial witnesses without expanding every word. This is a concrete source-derived component, while its universal zero acceptance remains conditional on the unaccepted source proof and correct implementation.

Existing tiny matrix probes catch the same generated rewrite errors more cheaply. A sixteen-factor expansion takes about 14.8 seconds and emits 65,536 terms, but a source-derived 32-coefficient witness and a conventional two-by-two witness take about 3.5 ms and 0.16 ms respectively. Those are single local samples and different outputs; the full-expansion comparison supplies no competitive speedup. Full source-dimension attempts hit rational-bit caps. Keep a specialist rewrite-witness SDK at low commercial confidence. [Dossier](../opportunities/012-noncommutative-identity-checks/2026-10-09-v2.md).

The positive-characteristic companion has dense dimension 16,306 already at n=2,s=3. The inverse-formula list has dimension 32,768 and visits more than 9*10^25 grid triples at n=s=1. These formulas were evaluated without allocating the matrices or traversing the list. Defer those literal backends. Model boundaries include formula tree size, rational scalar field, ordered multiplication, inverse domains, and fixed-operator relations versus free identities.

## Evidence and current editions

- [726 new controls and exact inputs](../prototypes/noncommutative-hitting-comparison-2026-10-09-v1/report.md): 686 reference/word/matrix/library controls, 32 same-objective existing-probe controls and eight model/resource controls.
- [35 utilities v19](../PROTOTYPES-2026-10-09-v19.md), [commercial priorities v19](../COMMERCIAL-PRIORITIES-2026-10-09-v19.md), [feasibility v17](../FEASIBILITY-2026-10-09-v17.md).
- [41-dossier catalog v22](../opportunities/CATALOG-2026-10-09-v22.md), [372-family directory v17](../FAMILY-ASSESSMENTS-2026-10-09-v17.md), [coverage v22](coverage-2026-10-09-v22.json).
- [Manuscript overview v13](../MANUSCRIPT-REVIEW-2026-10-09-v13.md), [ledger v13](manuscript-ledger-2026-10-09-v13.json): 36 selected-section manuscript records, 683 earlier-notes-not-imported or individually pending, 86 source hashes. Source proof and full manuscript coverage remain incomplete.
- [New artifact manifest](../prototypes/noncommutative-hitting-artifact-manifest-2026-10-09-v1.json): 27 research artifacts, five inert tool copies and ten source hashes. Existing Python 3.10.19/SymPy 1.13.1 reused; no runtime byte archive or package installation.
- [Integrity v20](integrity-2026-10-09-v20.json): 20,617 passed checks in 52 reports, 1,558 checked local links, 129 current Python syntax checks and zero issues. This checkpoint follows that integrity record; the static-site checker checks its links before publication.

Reading extents: characteristic-zero PDF pages 1-12, positive-characteristic 1-4/10-13 and rational-formula 1-4/16-21; five construction pages visually inspected. Family scope, both linked Comparators/configurations and both solution entries read. Imported proof closure, kernel and source program execution remain unverified. The Python construction is independently written. No source proof/build code was executed.

## Publication and continuation

The [upstream HEAD check](upstream-check-2026-10-09-v1.json) remains unchanged at `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`; do not repeat before 01:53 UTC without new evidence. The runbook now explicitly reflects the user's later GitHub/Dokploy publication authorization. No outreach, paid compute, purchases or customer data were introduced.

Publish this batch normally to [strykesg/mathideas](https://github.com/strykesg/mathideas), verify the automatic deployment on [mathideas.stera.ventures](https://mathideas.stera.ventures), and save a publication record. Dokploy application `6yFD3JCmfp0uZcCuJGex2`. The site remains a static catalog; no hosted formula or facility-optimization service is claimed.

Next source queue: family 104's deterministic, randomized, stochastic and parity game variants. Scope/dossier metadata was read this turn, but none of its four PDFs received a new reading record. Build a variant and output-semantics matrix before any solver prototype. Continue the facility-planning review-workflow specification separately; its customer demand remains unvalidated. Preserve historical files, environments and deployments. The existing hourly heartbeat remains active while the app/host are available.