# Feasibility and implementation boundaries

There are 34 runnable utilities and 21 documentary contracts. All 372 families are first-pass assessed, while 32 of 719 manuscripts have precise selected-section reading records. Finite calculations and live-library agreement are separate from source proof, unrestricted algorithm, numerical model and business acceptance. [Prototype directory](PROTOTYPES-2026-10-09-v17.md), [current ledger](checkpoints/manuscript-ledger-2026-10-09-v11.json).

## Live fixed-margin law comparison

A fresh preserved Python 3.12.14 environment with SciPy 1.18.1 and NumPy 2.5.3 passes 586 controls. The comparison covers every per-table mass and probability-order Fisher event on 284 two-by-two margin pairs at totals one through eight, comprising 494 observed tables, plus five public/generated inputs and explicit conditioning. The rational reference is compared with finite SciPy floats at relative tolerance 2e-12/absolute tolerance 2e-14. [Saved integration evidence](prototypes/scipy-live-contingency-comparison-2026-10-09-v1.json).

Default SciPy 1.15.3 and a first fresh 1.15.3/NumPy 2.2.6 environment fail through the native _spropack zero-fill section loader error. The succeeding environment is separate; no existing installation was removed or overwritten. No R execution, RNG law certification, source sampler/FPRAS, large-support scaling or actual experiment-design validation follows. The own reference caps dimension six, total 128 and 2,000 tables. Bounds explicitly condition the usual independence law; real structural-zero mechanisms need an actual model. [Public report](prototypes/public-contingency-law-review-2026-10-09-v2.md).

## Scheduling source and existing-solver boundary

The seven principal scheduling TeX argument files total 1,698 read lines. The source formula family allows K=10000 leaves, O(N^(3K)) labeled formulas and sequential-scan accounting O(N^(15K+13)); it states a looser binary-input exponent 150020. The structural proof uses fewer than 500 leaves, lists of at most five entries and fewer than 40 leaves per entry. Smaller observed descriptions or an aggressively pruned search require their own completeness and cost evidence; these bounds do not promise a useful implementation.

The precise model is a nonempty explicitly listed DAG of unit nonpreemptive jobs on exactly three identical parallel machines, without release dates, communication delays, eligibility or extra resources. The 100-line Comparator interface and 90-line solution entry point were read, with the 14-line configuration/15-line scope note. The selected statement quantifies one fixed finite machine and positive C for all valid encoded inputs. Imported MachineBridge/proofs, kernel comparison, compiler/executable extraction and actual source-machine execution are unverified.

The implemented ideal-state breadth-first search is conventional and exponential, capped at 18 jobs, 50,000 states and two million transitions. 3,346 controls include an independent slot-variable oracle on all 1,099 topologically ordered DAG edge sets through five jobs. A nine-job priority schedule takes four slots versus optimum three. A six-job synchronization case has capacity/critical-path lower bound two but optimum three. These finite facts validate evidence distinctions, not the source algorithm.

A separate OR-Tools 9.15.6755 adapter uses one-unit intervals, cumulative capacity three, strict precedence and makespan minimization. It matches independent optima on eight generated cases, with 40 controls. An FEASIBLE six-slot proposal for 18 independent jobs attains the capacity bound and independently proves optimality. An OPTIMAL status on the six-job case still leaves independent optimality unknown when its reference budget is exhausted. Solver status, valid schedule, exact finite optimum and selected source-machine guarantee remain separate fields. [Saved baseline](prototypes/scheduling-existing-solver-comparison-2026-10-09-v1.json), [current dossier](opportunities/009-three-machine-scheduling/2026-10-09-v2.md).

## Subset-sum source guards and witness/count boundary

The fast source is a two-sided randomized YES/NO decision with .49n asymptotic time in fixed polynomial-bit regimes; the low-space companion is one-sided with polynomial-times-2^(n/2) time and polynomial-times-2^(.199n) writable words, simplified to O(2^(n/5)). Neither decision directly supplies an exact count or original-item witness. No family formal-scope document is listed, no source program ran, and full numerical/filtering/representation/survival proof closure remains unread/unaccepted.

For b64, b^100000 <= 2^n first passes at n600000. The low-space guard u=n+b+2 <= 2^floor(n/10^9) first passes at n36000000000. Additional fixed cutoffs are unevaluated. Guard/cutoff fallback is exact meet-in-the-middle in the fast source and all-indexed-subset enumeration in the companion. Main branch eligibility remains unknown even when the known guard passes. The low-space displayed trial-work schedule is 2*u^620*2^ceil(n/2), aside from reset/scanning constants; it is a cap, not a runtime lower bound. [Exact parameter plans](prototypes/subset-sum-source-schedules-2026-10-09-v1.json).

The conventional reference caps 64 items, 50,000 sum records and two million generation/probe operations, excluding native objects/sort scratch/comparisons/input. On 64 ones with target32, disjoint-half support compression uses 98 peak records and preserves C(64,32) original-index subsets; an occurrence method exhausts the same budget. This favorable repeated fixture proves no general memory improvement or novel algorithm. Equal-weight source records with distinct overlap masks cannot be merged by weight alone.

2,294 finite/planner controls and 150 live existing-solver controls passed. OR-Tools already supplies checked original-ID witnesses for repeated and distinct-power inputs. Satisfaction OPTIMAL does not prove unique selection/count; solver negatives do not fill an exhausted independent reference. The deliberate CP-SAT adapter total/target cap is 2^62-1, while the reference accepts 256-bit values; unsupported ranges remain explicit and unrounded.

For n64 and conditional all-call error budget 1e-6, a planner reserves n+1 adaptive calls with 261 odd-majority repetitions per two-sided query. It requires fresh independent runs of a genuine base-error backend. Final exact witness checks can certify positive feasibility, while amplified NO stays probabilistic. Do not combine the two source time/space claims or treat amount equality as an intended reconciliation match. [Revised dossier](opportunities/017-subset-sum-reconciliation/2026-10-09-v2.md).

## Constructed superstring and actual artifact objective

All seven main superstring argument files total 627 read lines. The constructed count/layer/request/link/cycle/Euler algorithm claims factor two against unrestricted optimum in symbol length. Selected scope/config/43-line Comparator/65-line solution entry were read; imported proofs, source execution/extraction and independent acceptance remain open. Ordinary maximum-overlap greedy and exact tiny DP are different backends. The capped source count-stage alone does not output a factor-two superstring.

Source enumeration uses O(L^2) substring vertices and finite periodic/blocking rules; final graph size at most 4L does not bound that earlier dictionary/work to linear size. Compact substring-index substitutions require an equivalence/cost bridge. No practical full-construction runtime or memory result was demonstrated.

1,445 controls compare 469 tiny collections with an independent all-binary-word oracle through length 9 and verify finite source occurrence-count inequalities and artifact round trips. The count stage caps reduced input length 128/512 substrings/two million conservative charges; unknown on exhaustion includes no W/counts/graph. The packer caps 128 original records/16KiB input literals and conventional exact DP to 12 reduced strings. Its work budget excludes native preprocessing, per-character slicing/comparison and compression/memory cost.

The SSP1 artifact includes stable IDs, header and offset/length entries. It represents immutable byte views, including zeros/duplicates/empty values, with no C terminator, pointer identity or mutability guarantee. Compression uses whole artifacts and measures no access latency/resident memory. Seven generated/public/runtime comparisons emit 31 checked artifacts. The public 48-title case saves28 raw bytes, no gzip bytes and grows XZ by 32; a synthetic periodic case is favorable already to conventional greedy. [Saved report](prototypes/literal-packing-comparison-2026-10-09-v1/report.md), [current business decision](opportunities/023-superstring-packaging/2026-10-09-v2.md).

## Complete polynomial inequality and numerical enclosure

The direct manuscript's entire 623-line main TeX text, including appendix, was read. Selected scope/config/Comparator, a 41-line import entry, 119-line complete bound and 95-line sharpness texts were read separately. Full imports/kernel/source execution/independent proof and companion structural manuscript remain open. Positive dimensions do not mean A must be positive semidefinite. The finite polynomial model uses arbitrary complex A, ordered matrix coefficients and Euclidean operator norm.

The source constant two is a claim here, separate from the prior published complete 1+sqrt(2) bound, rounded to 483/200 in exact calculations. The source proof does not furnish a practical support/supremum implementation. The new reference uses exact Gram PSD for independent finite threshold evidence, twenty rational support halfspaces and closed-cell coverage with Frobenius/Lipschitz correction for an entire-domain upper. Zero pivots are handled exactly. A failed sufficient upper is inconclusive about the actual norm; a completed direct false PSD threshold disproves that exact finite threshold.

Caps are n≤8, m≤4, degree≤6, 128-bit canonical rational inputs, 4096-bit intermediates, grid≤32, support bisections≤16 and two million selected arithmetic charges. Charges exclude parsing, integer/gcd bit cost, native support setup/allocation and serialization. Exhaustion supplies no unfinished enclosure/supremum result; a prior completed direct threshold survives. Uncertain floating/physical inputs require their own bridge.

857 exact controls and 195 SciPy diagnostics pass. On the sharp nilpotent case, the conditional source upper about 2.189 fits tolerance 2.2 where prior about 2.644 does not, but direct norm-two evidence already suffices with 231 charges versus 84,404 full-reference charges. Normal/constant/cancellation cases expose conservatism. Existing direct norms are strong incumbents; repeated enclosure reuse and large sparse validated supports remain unimplemented. [Saved comparison](prototypes/numerical-range-comparison-2026-10-09-v1/report.md), [business decision](opportunities/014-numerical-range-certificates/2026-10-09-v2.md).

## Facility threshold and independent plan evidence

The threshold's fixed-epsilon polynomial classification permits arbitrary-size accuracy constants. Its deterministic physical history is retained for T rounds; surplus removal adds n^O(c/eta) and threshold enumeration n^O(c). No full numerical parameter selection or practical runtime was established. Recovery separately assumes accurate distinct proxies for all but logarithmically many comparison clusters in a normalized bounded integral model. It is not a generic warm-start guarantee.

A separate strict-metric reference uses exact rational arithmetic, checks every triangle, audits unweighted nonempty candidate plans and verifies LP-dual inequalities. Bounded exactly-k subset enumeration also solves the at-most-k objective because augmenting a facility set cannot increase nearest-distance cost. Caps are 48 locations/128-bit rationals/100,000subsets/five million enumeration distance lookups. Parsing, metric/primal/dual checks, rational bit arithmetic, allocations and time are outside enumeration counters.

5,127 reference and 34 live existing-solver controls pass, with four additional exact-dual controls. On the generated 24-location case, a cost 12 solver FEASIBLE plan meets an independently checked dual lower 12 with no enumeration. This finite certificate is separate from source theorem acceptance, physical input fidelity and general runtime. Weighted, capacitated, directed and other constrained models require separate adapters and do not inherit this theorem. [Comparison](prototypes/metric-facility-comparison-2026-10-09-v1/report.md), [first product](plans/optimization-assurance-first-product-2026-10-09-v1.md).

## Other operational limits retained

| Research connection | Concrete boundary | Current decision |
| --- | --- | --- |
| Family 115 general table sampler | Literal outer T=d^200(k+b)^2; saved two-by-two schedule has a 260-digit transition count; rare exhaustive correction separates expected cost from bounded latency | Defer unrestricted source backend; use bounded exact law/reference diagnostics |
| Families 120/113 matching algorithms | Extreme schedules/prerequisites; exact cardinality certificate and tiny perfect-matching counts use separate conventional engines | Existing-solver assurance before direct speedup claims |
| Family 131 switch chain | Specified half-hold/complete-host kernel and all attempted steps; restricted hosts may be disconnected | Audit the precise protocol; no blanket restricted-host or exact-sampler badge |
| Families 174/178 graph constructions | Unselected constructor constants/thresholds or enormous parameter sweeps; exact cuts and finite LDL acceptance are separate references | Defer broad construction backend |
| Family 110 k-server | Uniform implementation permits unrestricted finite additive cost and exponential activation | Exact finite replay/horizon comparison, no production competitive-policy claim |
| Family 103 logspace | Language decision equality does not preserve sampled outputs; paper compiler bounds are separate and enormous | Feasibility/semantics audit, no practical compiler speedup |
| Family 374 transport | Continuous uniform source/fixed compact targets and unique quadratic optimum; entropic/discrete solver bridge absent | Exact tiny sharpness geometry and assumption review |
| Family 371 Fourier/supercritical PDE | Odd-power grid folding can introduce a phantom retained coefficient; blowup claim has selected powers/dimension/quantifiers | Exact sparse aliasing audit; no broad numerical solver certificate |
| Family 169 positive algebra basis | Tiny exact reconstruction; broad witness/cone algorithm and all-r proof not implemented | Specialist low-confidence reference module |
| Family 238 queryable permutations | Short seed support cannot inherit full fair-bit statistical law; universal P unselected; cycle restriction can require n-m+1 calls | Separate provider models; no blanket data-loader mixing/performance promise |
| Family 115 bounded flows | Private-vertex reduction gives q=|V|+|E|; public model q14/total3422 exceeds exact dimension6/total128; cost/commodity/path-count bridges absent | Tiny scenario/model diagnostics; defer unrestricted FPRAS/sampler |

For a 2^30-slot domain and 256-bit seed, the conservative statistical TV lower bound is 1-2^(-15,569,256,192), obtained symbolically without allocating that integer. This does not assess cryptographic security or invalidate seeded replay goals. The measured 256-slot JSON provider uses 35,972 bytes after 16 points and 293,486 bytes after full traversal, versus a 2,048-byte packed int64 representation; the current backend cannot justify a generic memory-saving pitch. Finite palindrome calibration gives exact four-slot centered norm 1/4 and source-oriented controls, but does not evaluate the universal classically chosen threshold. [Permutation specification](plans/queryable-permutation-sdk-2026-10-09-v3.md), [contraction evidence](plans/coordinate-contraction-review-2026-10-09-v2.md).

## Evidence and commercial acceptance

The [current registry](prototypes/curated-contracts-2026-10-09-v11.json) has 21 source/model contracts with exact file hashes. A complete manifest still does not verify evidence contents or mathematical applicability. The [new metric-facility report](prototypes/metric-facility-evidence-bundle-2026-10-09-v1/report.md) flags 16 gaps in a self-authored claim and archives six exact input files. Earlier bundle/packet editions remain preserved; no historical tool bytes are executed.

All 65 declared entries in the finite ten-interface semantic queue have selected source observations, with no detected intended-model mismatch in that selection. Whole imports, proof/kernel and program verification remain open. The ten-case expert comparison and actual buyer workflow are prepared but unperformed. New finite tests and package execution do not fill those gaps. All prices/margins remain assumptions; source novelty, engineering feasibility and willingness to pay are distinct.

The [preceding feasibility edition](FEASIBILITY-2026-10-09-v13.md) preserves earlier parameter derivations and source-reading context. This edition gives current decisions and the newly completed integrations; no prior artifact was discarded.