# Essai 02 — changing method after commitment

A fictional bridge connects two peaks for walkers and an emergency utility service. The question is whether a cable method remains reachable after partial prefabricated construction and the observed withdrawal of the lifting service. This is a finite witness for the wider enquiry, **Keeping construction methods open**. It does not establish real engineering feasibility or complete that enquiry.

## Exact result

The complete declared graph has **37 reached states and 39 transitions**. There are **14 action-sequence completion routes from the initial state**. Routes count orderings, not independent engineering designs, quality or probability.

Two reached histories have **1 of 2 bridge units completed at elapsed time 4**, access at both peaks, a closed interface and a withdrawn lifting service:

| History | Retained difference | Completion routes after withdrawal |
| --- | --- | --- |
| Prepare → install provision → close → partial lift → withdrawal | Conversion provision installed | 1: convert → cable-complete → connect service |
| Prepare → record lift-only plan → close → partial lift → withdrawal | No conversion provision | 0 |

The planning record deliberately consumes one illustrative time unit without changing physical options. It supplies the matched elapsed time; it is not asserted to be equivalent in value to installing a provision. The progress measure counts bridge units only. It omits provision, closure, lifting availability, conversion and the utility service.

The successful continuation takes four more illustrative units and reaches completion at time 8. The stranded state has no outgoing action. The failure is exhaustive within the declared rules, not a timeout.

On the second history, the transition **close the interface** removes the last route to cable completion. Immediately before it, **install conversion provision** is the latest single extra action that would retain a cable route while permitting the same closure. Adding it to that history takes an additional time unit: the resulting post-withdrawal state is reached at time 5, not 4. The interface cannot be reopened here. Converting a retained interface costs one further unit after partial work; no provision can be installed after closure.

This boundary is indexed by the inspected history and the cable-completion target. Before withdrawal, an unprovisioned closed state can still finish by lift. Losing the cable option is therefore earlier than losing every completion route. Selecting a different history can change the boundary. Finishing by lift intentionally abandons cable completion; that also crosses the cable-target boundary and is reported as such.

## Declared state and target

A state records the set of already used actions, elapsed illustrative time, peak access, provision installed, interface closed, bridge units built, lifting availability, conversion done, completion method and utility service connection.

The target requires **both bridge units built, utility service connected and access at both peaks**. Finishing the second unit alone is not completion. Access is opened by preparation and remains available thereafter. Closing the interface is not closing peak access.

Each state ID is the integer bitmask of used actions, in the action order below. This intentionally retains whether the optional planning record occurred. Elapsed time and all other state fields can be reconstructed from that mask. Histories that differ only by the permitted ordering of independent preparation actions may reach the same state; all such action-sequence routes are still counted.

## Complete action rules

Every action can occur at most once. Every listed effect is irreversible within this model. The interface, provision and lifting rules are authored toy rules, not structural-engineering assertions.

| Action / bit | Duration | Demand | Preconditions, in addition to not already used | Effect |
| --- | ---: | --- | --- | --- |
| Prepare / 1 | 1 | One crew | Access not prepared | Open access at both peaks |
| Record / 2 | 1 | One planner | Access prepared; interface open | Record the lift-only plan; no capability added |
| Provision / 4 | 1 | One crew | Access prepared; interface open | Install conversion provision |
| Close / 8 | 1 | One crew | Access prepared; interface open | Close interface irreversibly |
| Partial / 16 | 1 | One crew and lifting service | Interface closed; no unit built; lifting available | Build first unit |
| Withdraw / 32 | 0 | External event | Exactly one unit built; lifting available | Permanently withdraw lifting service |
| Convert / 64 | 1 | One crew | Exactly one unit built; provision present; both peaks accessible | Enable cable assembly and exclude subsequent lift completion |
| Lift finish / 128 | 2 | One crew and lifting service | Exactly one unit built; lifting available; not converted | Build second unit by lift |
| Cable finish / 256 | 2 | One crew | Exactly one unit built; converted; both peaks accessible | Build second unit by cable |
| Service / 512 | 1 | One crew | Two units built; both peaks accessible | Connect and check utility service |

The model stops at the completion target. There is no action to remove completed work or restore withdrawn lifting. The explorer's Back button changes which history is inspected; it is not a model action.

## Bound, algorithm and quantifier

All actions are sequential. One crew and one planner are always available; lifting is the only resource lost. No concurrency, calendar, costs, deadline, load calculation, weather or peak-access closure is included. The durations are illustrative. The graph includes choosing the withdrawal event after partial work as well as continuing while lifting is available. The matched result conditions on that observed event, not on a probability distribution or a policy against all future events.

Ten one-use actions give at most ten transitions in a history. Every transition sets a fresh bit, so state masks strictly increase and the graph is acyclic. No waiting, loop or expiry changes state or reachability under these rules. Introducing such effects requires a different model and a new completeness argument.

The runtime generates every reached state and permitted edge. It starts with target states and repeatedly adds every state with a successor already included. This backward fixed point answers the existential question: **does at least one permitted route remain?** A separate recursion counts every route in the acyclic graph; the viewer lists every completion route from the inspected state. Already completed states have one empty continuation. The complete adjacency list and JSON export expose every state, edge, count and completion route.

Independent tests use a separate Boolean-mask forward enumerator, without calling the runtime's eligibility, transition, target or graph functions. It checks the entire state set and edge set, reconstructs every state variable, and enumerates completion counts from every reached state. Additional tests replay every exported route and check the matched witness, the failure, the last preserving choice and unavailable actions.

Backward reachability is a planning construction, not permission to reverse irreversible work. The starting reading is [LaValle, Planning Algorithms, Chapter 2](https://lavalle.pl/planning/ch2.pdf). The resulting completion set is not automatically a viability kernel or a robust winning set. A robust policy would require a declared uncertainty model and the appropriate “all allowed successors” condition; this essai does not supply one.

## Three separate explorations

1. **Finite switching graph:** the new declared states and actions above.
2. **Drawing interpolation:** three four-vertex polylines, with every ordered comparison. Corresponding vertices are blended linearly. With equal segment parameterisation, `H(s,t)=(1−t)A(s)+tB(s)` is a continuous endpoint-preserving homotopy in the unrestricted plane. No admissible construction space, obstacle or intermediate feasibility check is assigned to these drawing coordinates. The third pictured approach, advancing from both peaks, is not an additional method in the finite graph.
3. **Task-affinity field:** sixteen task records, four types and supplied priorities mapped to a bridge setting. Seed 190 fixes initial positions; bounded, damped simultaneous updates produce a reproducible illustration. The 600-step cap is not convergence. Affinity is not a prerequisite, schedule, duration, resource demand or construction phase.

The three engines are independent. Moving a slider or selecting a task changes no finite construction state. A smooth line or a spatial cluster cannot certify a switch.

## Preserved provenance and boundaries

The drawing and task field derive from Gimmer 28, *Homotopic Cliff Shed Construction*, including its repaired six-file implementation. The [pinned earlier source](https://github.com/lawrencerowland/gimmer-crag-project-mountain-refuge/blob/c9481a81a731682906837b500602948407d5f630/input/homotopic-cliff-shed-construction.tsx) records their lineage. Three path geometries, all sixteen typed/prioritised records, six ordered comparisons, slider/play/reset behaviour, milestone correspondence, field inspection, strength control, deterministic reset and bounded playback are preserved. Names and scenery are adapted to the two-peak bridge.

Earlier invariant ticks and claims about optimal construction phases or self-organisation remain withdrawn. Neither retained illustration establishes weather resilience, structural/access safety, a seasonal deadline or budget compliance. The new graph supplies a precise fictional contrast, with no claim of observed reader benefit or engineering applicability.

Return to [the enquiry](../) or [Essai 01: resource feasibility](../essai-01/).