Optimization assurance workbench: facility planning

Edition 9 October 2026 v4. First product experiment, with buyer validation pending. Defer the source approximation backend. The immediate opportunity is a review workflow around existing solvers. Its checks and packaging are conventional engineering; neither algorithmic novelty nor profitable demand is established.

Research finding

Family 125 claims a deterministic polynomial-time approximation of 1+2/e+epsilon for every fixed positive epsilon, on finite binary-rational metrics with specified candidate facilities. The objective is an unweighted sum of nearest-facility distances; output is a nonempty candidate subset opening at most k sites. The claimed hardness infimum assumes P != NP, concerns this general candidate model and does not assert attainment of the endpoint or the same hardness for facilities equal to clients.

The threshold paper's preparation, graph rounding and physical-history expansion first produce a constant-additive facility surplus. A separate reduction removes it. PDF pages 1-5 and 27-35 were read, including the fixed-accuracy parameter ordering and bit-complexity arguments. The polynomial may depend on epsilon; grid, rounds and allowances may be constants of any size. Section 6.3 retains the entire adaptive history tree for T fixed rounds. The external reduction adds n^O(c/eta) overhead, with further n^O(c) threshold enumeration. Those are asymptotic descriptions, not measured runtime, memory or a uniform efficient dependence on inverse epsilon. No complete numerical parameter selection or executable source backend was produced here.

The companion recovery paper's pages 1-6 require a supplied anchor with sufficiently accurate distinct proxies for all but logarithmically many comparison clusters, on a normalized polynomially bounded integral metric. Its independent general-metric application is randomized below two. An arbitrary existing plan does not automatically satisfy the recovery promise, and original-metric proxy accuracy cannot simply be assumed after normalization.

Problem and buyer

Hypothesized initial buyer: a logistics or operations consultancy repeatedly preparing facility-location recommendations for clients. It needs to explain whether a proposed plan meets the declared model, reproduce its reported cost, and distinguish a feasible plan from a supported optimality or quality claim. Misstated distance, demand or constraint assumptions can make a technically correct solver answer the wrong question.

The proposed review targets an existing recurring deliverable. No consultant interviews, customer datasets, paid reviews or purchase commitments exist. Established solver diagnostics and in-house checking may already be sufficient; this must be tested.

What the finding could enable

If independently accepted and practically implemented, the new theorem could strengthen a future backend's worst-case guarantee for its narrow model. The product can begin now with solver-independent evidence: a frozen input snapshot, explicit model eligibility, checked selected sites and assignments, exact objective recomputation, bounded tiny optima and checked lower-bound witnesses.

The concrete first software is an Optimization Assurance Workbench with a facility-planning module: import a plan and distance data, explain model mismatches, compare plans with an established solver, then export a traceable client report. A stronger theorem is a potential later engine, not required for these existing capabilities. First product specification.

Technical and commercial limits

A complete distance matrix on the relevant locations is required by the current strict-metric reference. A rectangular client-site cost table alone cannot establish all metric axioms. Distinct indices must have positive distance, the diagonal is zero, symmetry and every triangle are checked exactly. Client and facility sets may overlap; repeated client identities and demand weights are not silently collapsed. Empty client sets are allowed, but a nonempty feasible facility selection is still required.

Directed travel times, uncertain or rounded measurements, weighted demand, capacities, fixed site costs, fairness, time windows and mandatory sites need separate model/implementation bridges. Such features are legitimate optimization inputs; failing theorem eligibility is not evidence that their general planning problem is invalid. Symmetrizing, rounding or replacing distances with a metric closure changes the input and may change the objective. Do not attach the source guarantee after that transformation without a transfer argument.

The new reference accepts at most 48 locations, canonical rational strings with 128-bit numerators/denominators, at most 100,000 examined k-subsets and five million enumeration distance lookups. Metric parsing/triangle checks, primal and dual audits, Fraction bit operations, allocation and runtime are outside the enumeration counter. These caps are not wall-time or byte bounds. Exact enumeration is exponential and is not the source polynomial algorithm.

Minimal architecture

Versioned import: stable site/client identities, original file hashes, declared units, distance provenance and explicit constraints.
Model review: separate a general planner's declared model from eligibility for any source theorem. Route unsupported features to a clear implementation status.
Proposal engine: established optimization software with recorded version, settings, termination status and model hash.
Independent checker: candidate membership/budget, assignments, original-data objective, bounded reference and exact rational lower-bound witnesses where available.
Report: reproducible data/tool hashes, checked facts, solver-reported facts, unresolved assumptions, alternatives and human approval fields.

The first local CSV review/export workflow now runs around the bounded checker and optional OR-Tools adapter. It preserves original bytes, stable ID mappings, exact submitted costs, solver-reported statuses, independent bounds and human-readable HTML/JSON reports. Eight generated examples and input contract. The published site hosts static examples and research; no customer upload service or production optimizer is deployed.

Existing alternatives and differentiation

PySAL spopt's p-median documentation, reviewed 9 October 2026, already describes cost-matrix/geographic workflows, weighted demand and a capacitated extension. OR-Tools CP-SAT documentation, reviewed the same day, distinguishes feasible and optimal statuses and requires integer models. Generic site selection is an established capability. Our proposed differentiation is useful independent checks and reproducible client reporting, which remains a demand hypothesis.

The existing OR-Tools 9.15.6755 environment was reused without package changes. Eight generated cases use a two-second, one-worker limit with exact rational-to-integer scaling and no rounding. Seven returned OPTIMAL and matched independent finite optima. The 24-location, 12-site case returned FEASIBLE with cost 12; a separate hand-constructed exact dual certificate proves 12 is optimal without enumeration. These small synthetic cases establish no broad runtime or source-algorithm advantage.

Monetization hypothesis

Start with an assisted review priced experimentally at AUD 5,000-10,000 for a clearly scoped facility study. At an illustrative AUD 7,500 fee and 25 hours at an assumed AUD 180/hour, labor is AUD 4,500 and the remainder is AUD 3,000 before overhead, sales, tax, liability and support. At 45 hours, labor alone is AUD 8,100. These figures are assumptions, not quotes, market prices or realized margins.

A subscription is justified only if repeated study revisions or client engagements create recurring value and bounded support cost. This module belongs within the shared assurance platform; do not count it as an additional independent customer or forecast revenue from the number of mathematical results.

Validation experiment

The reference passed 5,127 controls, including all 4,992 client/candidate/budget models over 52 three-point integer metrics with distances 1-4 against a separate all-subsets oracle. Additional controls cover exact rational duality, false certificates, client/site overlap, malformed inputs and exhaustion. The existing-solver comparison adds 34 controls, and the 24-site dual extension adds four. Comparison and exact inputs.

For any accepted dual, alpha_j <= d(j,f)+beta_jf, beta_jf >= 0, and sum_j beta_jf <= lambda with lambda >= 0. These inequalities give lower bound sum_j alpha_j - k*lambda by weak duality. The checker uses exact arithmetic. A matching checked primal proves the finite optimum independently of the repository theorem and of solver termination status. Formal correctness of the Python implementation remains unproved.

Next evaluate the report on actual, permissioned planning studies against the consultancy's current review process. Record material errors found, false alarms, reviewer time, decisions changed, support effort and willingness to pay. A successful mathematical control does not supply these business measurements.

The CSV/review workflow adds 104 controls across stable-ID ordering, exact decimals, unsupported constraints, malformed inputs, named duals, solver status separation, resource/scaling limits, HTML escaping and immutable exports. A six-location plan costs 24 versus exact optimum 17. A separate solver run reports OPTIMAL while independent optimality stays unknown; a 24-location named-ID dual proves cost 12 optimal without enumeration. Workflow report. These do not add customer evidence or mathematical source acceptance.

Conditions to reject or defer

Defer the source backend until full parameter selection, implementation correspondence, independent proof acceptance and representative performance exist. Reject the first product hypothesis if ordinary solver reports already answer the buyer's questions, no material mistakes are found, the narrow model excludes most useful work, or delivery cost exceeds attainable fees. Do not sell a broad guarantee through an unvalidated weighted/capacitated/road-time bridge.

Next concrete action

The local workbench and eight generated reports are ready for inspection. A public workflow comparison now prioritizes a separate weighted/rectangular plan audit before browser polish. Add required-site and whole-client assignment/capacity checks explicitly, preserving the strict metric module as its own model. Compare the export against an existing consultancy review process before expanding features. Seek real study access and paid pilot evidence only through authorized outreach; none has occurred. Expand weighted/capacitated checking as separate tested modules when buyer needs justify it.

Source and verification record

Pinned revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb:

The threshold Comparator/config, refined-recovery Comparator, solution entry and RationalAllParameters, RationalParameterizedHistory and ReductionClosure files were read. Selected PDF pages 1, 5, 27, 32 and 34 were also visually inspected. Whole manuscripts, imported proof closure, Lean kernel, source program execution and independent mathematical acceptance remain incomplete. Comparator challenge placeholders are not treated as solution failures. Current source/model contract, synthetic 16-gap control.

Public planning baseline gap

The documented workflow comparison identifies material input/constraint gaps in the initial release. PySAL already demonstrates weighted rectangular costs and required/capacitated sites. Its synthetic tutorial is not a customer study, and we did not execute it. Existing software sets the baseline; the candidate paid value remains useful independent review, with no verified demand.