Appendix: illustrative corridor studies

The study slate is expressly fictional. It uses named serial functions to make the mechanism easy to follow, not to imply an actual mine, railway or port. Complete inputs were authored before freezing and executed through the Corridor lab compute function. The output objects, scalar bindings and claim ledger are retained with the paper. The manuscript does not duplicate the numerical method in its authoring code. This appendix describes what changes, what remains constant and how each case challenges the service decision.

The common base has producer release, rail movement, port stockyard and vessel loading. All stages use the same illustrative cargo quantity and incur their stated variable and fixed operating costs on the common flow. Demand is a chosen annual bound. The stage capacities and availability assumptions are chosen to create a rail constraint close to the loading limit. That structure makes the next-constraint effect visible without assigning a measured frequency or claiming that a particular industrial geography behaves this way.

Every intervention case preserves the declared serial topology and homogeneous cargo basis. Operating costs remain unchanged across the physical interventions to isolate capacity and denominator effects. This is a deliberate analytical restriction. It does not imply costless expansion. A capital appraisal would need the incremental investment, implementation period, additional operating obligations and customer value. The slate should therefore be used to understand the physical method and its applicability limits, not to select an actual supplier or investment programme.

The additional cases challenge the original interpretation rather than simply extend a preferred expansion story. Lower availability tests readiness sensitivity; lower demand removes the annual capacity bottleneck; tied capacities challenge isolated relief; zero loading availability tests the undefined-cost boundary. Their purpose is to give an independent reader admissible counterexamples within the implemented method. Other important challenges, such as routing, queues and quality, remain outside the engine and are handled conceptually in the main analysis.

The base chain calculates 11.90 Mt/year and 23.73 USD/t, with Rail corridor binding. The capacities, availability and cost inputs are printed below on their declared units. The result is a common annual flow, not the sum of local outputs. Its main implication is that every stage carries the same modelled quantity while the smallest effective annual bound prevents the higher-capacity stages from increasing that quantity on their own.

Illustrative. The loading-only case changes the vessel-loading nominal capacity and nothing else. It calculates 11.90 Mt/year and 23.73 USD/t, with Rail corridor still binding. The unchanged common result falsifies an unqualified claim that enlarging that visible asset must raise annual corridor flow. It does not falsify a separate marine-window or resilience benefit, because those outcomes are absent from the method. The reader should preserve that narrow scope when using the comparison.

Illustrative. The rail-only case changes the original binding nominal capacity and calculates 12.00 Mt/year and 23.67 USD/t. The next annual constraint is Vessel loading. The implication is not that rail relief is pointless. It is that the represented gain is limited by another unchanged stage. A design brief should therefore include that stage's readiness and any relevant service mechanism before the sponsor interprets the nominal rail increment as added delivered quantity.

The built-in single-stage expansion table offers a related controlled test for the base fixture. Its rows change one stage according to the implemented method, preserving all other inputs. The manuscript publishes the resulting scalar readings and keeps the complete table in the frozen output. This is useful for checking the logic of isolated relief, but it still omits feasibility and investment costs. A positive tested gain identifies a physical opportunity under the model, while a zero gain identifies an annual limitation rather than a universal absence of value.

The coordinated case changes rail and vessel-loading nominal capacities together. It calculates 14.40 Mt/year and 22.39 USD/t, with Port stockyard binding. The result demonstrates that joint relief can reach a different annual bound from either isolated intervention. It does not establish that simultaneous procurement is superior to a staged operating and investment sequence. That choice depends on evidence outside the fictional operating-cost comparison.

Illustrative. The lower-availability case preserves nominal capacities and costs but changes the rail readiness assumption. It calculates 7.00 Mt/year and 29.14 USD/t, with Rail corridor binding. This case isolates the distinction between physical nameplate and assumed usable annual capacity. Its sensitivity is structural: less usable capability at the limiting stage reduces common flow and raises the fixed-cost allocation per delivered tonne under the unchanged cost basis.

Illustrative. The engine's deterministic stress produces a further result for each case. The base stressed flow is 10.50 Mt/year; the coordinated stressed flow is 12.80 Mt/year; the lower-availability stressed flow is 5.60 Mt/year. These quantities are scenario outputs, with the stress rule retained in the canonical methods metadata. They do not have likelihoods and should not be presented as expected outcomes or confidence bounds.

The comparison also clarifies which evidence a practitioner would need. Coordinated physical relief needs technical and operating feasibility at several stages. An availability improvement needs a defined readiness mechanism and proof that the changed calendar or capability is usable. Both need demand and acceptance evidence before becoming a service promise. The frozen cases cannot supply those properties, but they prevent the committee from confusing a nominal expansion, a readiness change and a demonstrated increase in accepted service.

The lower-demand case changes only the annual demand bound. It calculates 8.00 Mt/year and 27.50 USD/t, with no capacity stage listed as binding. The result reverses the capacity-expansion interpretation because the customer-side bound is below every effective stage capacity. A practitioner should then examine eligible demand and the service proposition rather than claim that physical additions create more current flow. Spare capability can still serve a separate reserve objective if its duty is defined.

Illustrative. The tied case changes rail's nominal and availability basis so that its effective annual capacity matches loading. It calculates 12.00 Mt/year with Rail corridor and Vessel loading binding. The annual model warns that isolated relief may produce no common-flow gain. This is an admissible counterexample to a simplistic single-asset recommendation. A real application should investigate whether equal limits are genuine or merely an artefact of rounded or poorly defined capacity inputs.

Illustrative. The zero-service case sets loading availability to 0.00 percent and calculates 0.00 Mt/year. Whole-chain unit cost is not defined. The essential serial function is unavailable, so upstream spare capacity cannot substitute for it within this topology. Fixed operating obligations remain present. The output should be interpreted as an unavailable per-tonne ratio, not as a finite estimate of the financial loss or as evidence that the continuing obligations vanish.

These cases should be retained in review even if an investment narrative prefers the expansion cases. They demonstrate why the method needs a demand definition, correct handling of tied constraints and a clear unavailable state. They also show the limits of the slate: an alternate loading route would change topology and require another method. The zero-service result cannot establish that no real workaround exists; it establishes that the declared essential function has no substitute in this particular serial representation.

A replication should begin with the exact frozen input, not with numbers copied from a rounded PDF table. The table supports reading and audit, while the JSON preserves full input and output precision. Execute the canonical compute function and compare the complete stable result object with the frozen output. A difference can reflect a fixture change, an engine change or an error. The reviewer should identify which occurred before interpreting the revised conclusion.

Scalar bindings use explicit input or output paths. A printed token resolves to one scalar in the canonical object and to one ledger claim. The claim records illustrative status, method, fixture and sensitivity, with reciprocal manuscript references. String readings such as binding-stage names are included, so directional conclusions are tied to the same reproduced object as the numerical values. This prevents a narrative from retaining an outdated bottleneck label after a quantity or method changes.

The complete output also retains tables, warnings, methods and limitations. Those fields are part of the interpretive record, even when the PDF prints only the decision-relevant scalars. A reviewer should inspect them alongside the stage results and intervention tests. Warnings about demand, shared constraints or zero flow should not be discarded because they make the headline less simple. A faithful export should preserve the method and limits as well as the attractive result.

The study slate also supports a contrast-selection instrument. Use loading-only against base to test an intervention at a nonbinding stage; use rail-only against base to test relief of the original limit; use coordinated against rail-only to expose the next receiving duty. Use lower availability to separate readiness from nominal rating, lower demand to separate use from capability, tied limits to test a joint constraint and zero service to test an undefined denominator. Selecting the pair by the disputed mechanism gives replication a practical purpose. Combining all changed cases into an apparent trend would lose that purpose because the fixtures alter different causes. These remain fictional controlled contrasts, with no field validation supplied.

Binding membership is a property of the supplied capacity ordering, not a convenience tolerance measured in one chosen unit. The small-distinct-capacities study gives stage A nominal capacity 1e-09 and stage B 2e-09, each at 10% availability, against demand 1e-09. Its effective capacities are 1e-10 and 2e-10 Mt/year. Only A binds. Its tested single-stage expansion leaves A as the next constraint. This artificial scale has no operating significance; it makes an incorrect absolute comparison unmistakable.

The revised engine uses rational products and comparisons for the canonical decimal spelling of its validated inputs. That keeps capacity membership, demand caps and expansion comparisons invariant under a supported common rescaling. The full output records the exact fractions and their display approximations. Annual operating cost remains a numerical calculation, and the availability stress remains a declared deterministic change. Exact arithmetic cannot establish achievable capacity, accepted cargo, cost reliability or intervention feasibility.