# Thirty-three runnable research utilities

Forty-four saved component reports contain **12,696 passed checks**. The preceding edition had 11,637 checks; this continuation adds 857 numerical-range reference controls, 195 existing SciPy diagnostics and seven documentary contract controls. These are finite calculation, source reading and artifact evidence; source theorem/program, actual customer workflow and profitability remain unverified.

## Complete utility directory

| Utility | What it does | Material boundary |
| --- | --- | --- |
| [proof_preflight.py](tools/proof_preflight.py) | Inspect selected formal configuration/module metadata | Static presence/trust settings, no proof execution |
| [withdrawal_impact.py](tools/withdrawal_impact.py) | Record explicit withdrawals and dependencies | Three notices/two explicit edges; not a full theorem graph |
| [audit_embedding.py](tools/audit_embedding.py) | Check finite embedding quantities | Sampled/finite floating calculation, no universal guarantee |
| [gad_capacity.py](tools/gad_capacity.py) | Evaluate scalar generalized-amplitude-damping expression | Numerical convention/optimization, no hardware or finite decoder certificate |
| [audit_claim_contract.py](tools/audit_claim_contract.py) | Lint documentary source/model obligations | Twenty curated contracts; supplied statuses and evidence contents unverified |
| [periodic_interface_reference.py](tools/periodic_interface_reference.py) | Periodic interface formula reference | Specified finite geometry, no general optimizer |
| [audit_binary_waveform.py](tools/audit_binary_waveform.py) | Exact finite correlation and sampled spectrum | No full channel or physical waveform certification |
| [audit_ramanujan_graph.py](tools/audit_ramanujan_graph.py) | Rational contrast-space spectral acceptance | Supplied finite graph, not source constructor |
| [cyclic_chain_reference.py](tools/cyclic_chain_reference.py) | Tiny group cyclic-chain enumeration | At most sixteen elements, no effective spectrum witness |
| [evidence_bundle.py](tools/evidence_bundle.py) | Join source hashes/static metadata/claim audit | Archives six exact inputs; no source/proof execution |
| [contingency_reference.py](tools/contingency_reference.py) | Exact bounded table count/rank/unrank/law | Dimension six/total128; conventional DP, not source sampler |
| [audit_matching_certificate.py](tools/audit_matching_certificate.py) | Check matching and attaining cardinality bound | Exact supplied witness/bound, no extra allocation constraints |
| [solve_and_audit_matching.py](tools/solve_and_audit_matching.py) | Existing NetworkX proposal plus separate audit | Unit cardinality model, no source accelerated matcher |
| [perfect_matching_reference.py](tools/perfect_matching_reference.py) | Tiny exact counts/ranks/edge fragility | At most24 vertices, no FPRAS |
| [audit_switch_chain.py](tools/audit_switch_chain.py) | Exact tiny specified-kernel TV | At most6 vertices/512 states/64 steps; complete-host model |
| [audit_tree_thinness.py](tools/audit_tree_thinness.py) | Exact all-cut finite tree audit | At most16 vertices; not broad thin-tree construction |
| [kserver_reference.py](tools/kserver_reference.py) | Rational offline optimum and named-policy replay | At most8 points/4 servers/64 requests; no source online policy |
| [transport_sharpness_reference.py](tools/transport_sharpness_reference.py) | Exact three-atom rational geometry | Uniform-square example, no arbitrary transport solver |
| [audit_fourier_aliasing.py](tools/audit_fourier_aliasing.py) | Sparse exact odd-power convolution/grid folding | Typed support/work/bit caps; no PDE solution certificate |
| [chromatic_basis_reference.py](tools/chromatic_basis_reference.py) | Tiny exact chromatic/e-basis reconstruction | At most6 vertices; not broad packet/witness theorem implementation |
| [queryable_permutation_reference.py](tools/queryable_permutation_reference.py) | Shared-switch point/inverse replay and tiny laws | Explicit bits/dyadic domain, no universal mixing constant |
| [permutation_provider_reference.py](tools/permutation_provider_reference.py) | Decimal-wire seeded or locked local stored bits | Distinct randomness models; capped append-only JSON store |
| [palindrome_minorant_reference.py](tools/palindrome_minorant_reference.py) | Exact tiny palindrome law/operator calibration | Two/four slots; no universal numerical P |
| [audit_contingency_law.py](tools/audit_contingency_law.py) | Complete finite law and fixed-event comparison | Explicit conditioning, at most2,000 tables, unknown on exhaustion |
| [bounded_flow_reference.py](tools/bounded_flow_reference.py) | Signed-bound private-vertex flow reduction/counts | Tiny exact table engine; no unrestricted FPRAS/sampler |
| [three_machine_reference.py](tools/three_machine_reference.py) | Unit-job schedule audit/exact tiny optimum/deadline evidence | At most18 jobs/50,000 states/2m transitions; exponential |
| [solve_and_audit_three_machine.py](tools/solve_and_audit_three_machine.py) | Existing CP-SAT proposal and finite independent evidence | Same strict unit model; solver status distinct from independent optimum |
| [subset_sum_reference.py](tools/subset_sum_reference.py) | Exact disjoint-half support/multiplicity/witness reference | Two–64 positive identified items; 50k sum records/2m generation/probe work, not source backend |
| [subset_sum_schedule.py](tools/subset_sum_schedule.py) | Exact known guards and conditional decision-error/amplification algebra | Fixed main cutoffs unselected; source implementations and independence unvalidated |
| [solve_and_audit_subset_sum.py](tools/solve_and_audit_subset_sum.py) | Existing integer CP-SAT proposal and original-ID witness evidence | Deliberate total/target cap 2^62-1; solver satisfaction status does not give uniqueness/count |
| [superstring_packaging_reference.py](tools/superstring_packaging_reference.py) | Emit indexed immutable byte-view artifacts and compare complete compressed bytes | At most128 records/16KiB literal bytes; exact DP at most 12 reduced strings; no source factor-two backend |
| [superstring_counts_reference.py](tools/superstring_counts_reference.py) | Bounded paper forced-count recursion and balanced base graph | Reduced length 128/512 substrings/2m charges; no full layer/request/cycle construction or proof acceptance |
| [numerical_range_polynomial_reference.py](tools/numerical_range_polynomial_reference.py) | Exact finite norm threshold, PSD support halfspaces and covered polynomial upper bounds | n≤8/m≤4/degree≤6; new constant two conditional, exact inputs and capped arithmetic only |

All dated validators/publishers and archived historical copies are supporting tooling, not additional utilities. The [preceding prototype edition](PROTOTYPES-2026-10-09-v15.md) preserves detailed earlier measurements and controls.

## Newly completed statistical integration

[586 live SciPy controls](snapshots/2026-10-08-baseline/scipy-live-law-validation-2026-10-09-v1.json) compare the exact reference with SciPy 1.18.1/NumPy 2.5.3 on 284 two-by-two margin pairs and 494 observed tables, plus five public/generated cases. A working fresh environment was created without modifying the default or first failed fresh 1.15.3 environment. R remains unexecuted. Floating agreement is finite and tolerance-bounded, not an exact library proof or design validation. [Runtime and source hashes](prototypes/scipy-live-contingency-comparison-2026-10-09-v1.json), [public law report](prototypes/public-contingency-law-review-2026-10-09-v2.md).

```sh
python3 research/tools/audit_contingency_law.py research/fixtures/contingency-scipy-fisher-threshold-2026-10-09-v1.json
```

The declared probability-order event is held fixed. Conditional-independence probability 5/143 versus uniform-table probability 3/7 crosses the declared 1/20 threshold because the law changes. Bounds use explicit renormalization of the original law; they do not automatically model real structural zeros. No large-support or source-backend performance was measured.

## New scheduling references

[3,346 finite controls](snapshots/2026-10-08-baseline/three-machine-reference-validation-2026-10-09-v1.json) compare all 1,099 topologically ordered DAG edge sets through five jobs with a separate backtracking slot-variable oracle. The reference validates explicit model fields, job identities and acyclicity; checks capacities/strict precedence; computes elementary lower/checked upper bounds; and completes ideal-state BFS when needed. Empty, oversized, cyclic and mismatched inputs are refused. Budget exhaustion grants no exact optimum. Independently sufficient deadline evidence can survive exhaustion.

```sh
python3 research/tools/three_machine_reference.py research/fixtures/three-machine-greedy-counterexample-2026-10-09-v1.json
python3 research/tools/three_machine_reference.py research/fixtures/three-machine-sync-bottleneck-2026-10-09-v1.json --state-budget 1
```

The generated nine-job case has a four-slot priority heuristic versus optimum three. The six-job case has elementary lower bound two versus optimum three. A valid schedule alone is not an optimality certificate; a valid schedule attaining the elementary lower bound can be one for the finite model. The source exponent 150020 and fixed-machine proof are not implemented by this exponential search.

[40 live CP-SAT controls](snapshots/2026-10-08-baseline/scheduling-existing-solver-validation-2026-10-09-v1.json) compare eight generated models with OR-Tools 9.15.6755, retaining separately checked schedules and tiny optima. Runs with one worker and a two-second limit already reach those optima, so no new solver advantage is shown. A FEASIBLE proposal attains the capacity bound in the independent-18 case. The exhausted independent six-job control does not inherit its solver's OPTIMAL status. [Saved comparison](prototypes/scheduling-existing-solver-comparison-2026-10-09-v1.json), [wheel installation report](prototypes/ortools-install-report-2026-10-09-v1.json).

```sh
/tmp/openai-math-scheduling-20261009-FKmLBr/bin/python3 research/tools/solve_and_audit_three_machine.py research/fixtures/three-machine-greedy-counterexample-2026-10-09-v1.json
```

That interpreter is the preserved environment for this run; the saved report records packages/runtime. No existing installation was altered. Times are diagnostic single runs, not a benchmark or deployment guarantee. [Current scheduling dossier](opportunities/009-three-machine-scheduling/2026-10-09-v2.md).

## Subset-sum reference and existing-engine evidence

[2,294 controls](snapshots/2026-10-08-baseline/subset-sum-reference-validation-2026-10-09-v1.json) compare 28,404 method/target queries across all 1,089 positive value words 1..3 through six items with a separate binary-choice count oracle. Both exact reference methods preserve original-index multiplicity. Additional controls check invalid models/IDs, 256-bit exact arithmetic, guards, budget states and conditional majority-error algebra.

```sh
python3 research/tools/subset_sum_reference.py research/fixtures/subset-sum-repeated-64-2026-10-09-v1.json
python3 research/tools/subset_sum_reference.py research/fixtures/subset-sum-repeated-64-2026-10-09-v1.json --method occurrence_MITM
```

The 64-one target-32 generated case uses 98 peak sum records under compressed support and retains C(64,32)=1,832,624,140,942,590,534 distinct indexed solutions. The occurrence method exhausts 50,000 records and returns unknown. These record/work caps omit runtime objects, input and native sorting scratch/comparisons, so they are not byte, time or word-RAM bounds. The [duplicate-value control](fixtures/subset-sum-duplicate-identities-2026-10-09-v1.json) has two valid selections; repeated IDs are refused.

[150 live solver controls](snapshots/2026-10-08-baseline/subset-sum-solver-validation-2026-10-09-v1.json) cover 1,080 tiny target queries and ten generated cases. Existing CP-SAT already proposes exactly checked selections for repeated64 and distinct-powers32. A directly checked positive witness can suffice when count/reference search is unknown. Satisfaction OPTIMAL is no uniqueness/count certificate; solver INFEASIBLE remains a solver report unless independent negative evidence exists. Larger Python-exact totals are refused by the conservative integer adapter without rounding. [Saved comparison](prototypes/subset-sum-existing-solver-comparison-2026-10-09-v1.json).

```sh
/tmp/openai-math-scheduling-20261009-FKmLBr/bin/python3 research/tools/solve_and_audit_subset_sum.py research/fixtures/subset-sum-duplicate-identities-2026-10-09-v1.json
```

The preserved isolated OR-Tools 9.15.6755 environment was reused without installation changes. These are single-run generated controls, not real accounting data or speed comparisons. [Updated dossier](opportunities/017-subset-sum-reconciliation/2026-10-09-v2.md).

The [source-parameter plans](prototypes/subset-sum-source-schedules-2026-10-09-v1.json) expose known 64-bit main guards at 600,000 and 36 billion items, with further fixed cutoffs unevaluated. At n64 and conditional all-call error budget 1/1,000,000, 261 odd-majority repetitions are assigned per two-sided adaptive decision query. This requires fresh independent genuine base-error calls, not implemented source programs. A verified positive witness and probabilistic NO remain distinct. Fast time and companion low-space claims cannot be combined.

## Literal packaging and source-derived count stage

[1,445 controls](snapshots/2026-10-08-baseline/superstring-reference-validation-2026-10-09-v1.json) compare 469 tiny collections with every binary output word through length nine. Exact overlap DP matches their unrestricted optimum within this finite alphabet/domain. Paper forced counts do not exceed actual occurrences in every common word in that domain. Artifact round trips, original IDs, empty/duplicate/zero/UTF8 byte values, invalid models and budgets are checked.

```sh
python3 research/tools/superstring_packaging_reference.py research/prototypes/literal-packing-comparison-2026-10-09-v1/synthetic_greedy_gap-input.json
python3 research/tools/superstring_counts_reference.py research/prototypes/literal-packing-comparison-2026-10-09-v1/synthetic_periodic_windows-input.json
```

[Seven saved comparisons and 31 emitted .ssp files](prototypes/literal-packing-comparison-2026-10-09-v1/report.md) include stable IDs/header/offset-length data. The public 48-title sample shrinks raw SSP1 by 28 bytes, leaves gzip unchanged at 1,685 and increases XZ from 1,616 to 1,648. A synthetic periodic sample shrinks SSP1 from 308 to 228 and gzip from 163 to 138, already with ordinary greedy. The generated greedy-gap control takes 10 raw bytes versus exact 9. Complete artifact objectives and API constraints matter; no customer, linker or access-performance result is inferred.

The count-stage reference implements source section02 only. For a^16 it recovers W=16 and m(a^k)=17-k. Source W validity in general depends on its unverified proof; finite comparisons do not certify that proof. The full forced-count/periodic-layer/request/link/cycle/Euler factor-two construction is unimplemented. The package model uses immutable length-aware bytes, not drop-in C strings or preserved pointer identity. [Revised dossier](opportunities/023-superstring-packaging/2026-10-09-v2.md).

## Exact polynomial and numerical-range evidence

[857 controls](snapshots/2026-10-08-baseline/numerical-range-reference-validation-2026-10-09-v1.json) include 625 independent complex Hermitian 2-by-2 PSD oracle cases, zero pivots, exact rounding, eight generated matrices/polynomials, tensor order, finite Rayleigh probes, thresholds and budgets. An exact complete Gram PSD check supplies finite norm-threshold evidence independently of Crouzeix. Formal Python correctness was not proved.

```sh
python3 research/tools/numerical_range_polynomial_reference.py research/prototypes/numerical-range-comparison-2026-10-09-v1/nilpotent_sharp-input.json --grid 32
```

The separate twenty-direction PSD support enclosure and covered-cell Lipschitz/Frobenius calculation gives an entire-domain upper. The source constant-two result is conditional; the prior published constant is rounded upward to 483/200. On the sharp nilpotent example, the source-conditional upper about 2.189 fits tolerance 2.2 while the prior upper about 2.644 does not. A direct exact norm-two check already suffices with 231 charges, versus 84,404 for the full reference. No standalone library advantage is established. [Eight-case report](prototypes/numerical-range-comparison-2026-10-09-v1/report.md).

[195 SciPy diagnostic controls](snapshots/2026-10-08-baseline/numerical-range-scipy-validation-2026-10-09-v1.json) reuse the existing isolated SciPy 1.18.1/NumPy 2.5.3 environment without installation changes. SVD/eigvalsh agreement is finite and tolerance bounded, not a rigorous library rounding guarantee. [Revised numerical dossier](opportunities/014-numerical-range-certificates/2026-10-09-v2.md).

## Current documentary evidence

The [twenty-contract registry](prototypes/curated-contracts-2026-10-09-v10.json) adds Euclidean tensor evaluation, entire numerical-range coverage, supremum and arithmetic obligations. [Seven controls](snapshots/2026-10-08-baseline/numerical-range-contract-validation-2026-10-09-v1.json) check registry/linter behavior. A self-authored eigenvalue/sampled/nonlinear-stability claim has 17 gaps in the [new joined report](prototypes/numerical-range-evidence-bundle-2026-10-09-v1/report.md), which preserves six exact input files and 2,933 source hashes. A complete manifest verifies no evidence contents or applicability.

The frozen ten-case expert packet and older bundles preserve historical tools/contracts. Expert comparison, source kernel/program acceptance, actual workflow advantage and buyer value remain pending. [Feasibility](FEASIBILITY-2026-10-09-v14.md), [pilot](plans/evidence-platform-pilot-2026-10-09-v11.md).