A common beginning
Open the fifth essai · All co-design essais
Question: what can we commit before a future requirement is known, and which choices can wait?
The earlier crossing essais compare complete configurations, construction paths, supplied programme strategies and a monitoring-power loop. This independent model adds an early choice shared by two possible futures, followed by choices restricted to the information actually available. None of its units or results transfers into the other four models.
What the example contributes
Two good future designs need not have the same beginning. The example makes that difference inspectable through component connections, complete choice rules, two separate resources and a small exhaustive calculation. It illustrates existing co-design and adaptive-decision ideas; it does not propose a new theorem. All quantities and constraints are invented, not ecological or construction data.
The two future briefs require a guiding route east (E) or west (W). An early component supplies connection ports. A later component consumes a set of ports and supplies a set of routes. Required ports must be a subset of supplied ports, and the future’s required route must be supplied. No numeric ordering is imposed on the distinct connection types.
Catalogue
Costs and land are illustrative units; both are minimised independently. The early and later footprints are disjoint by assumption, so land adds. Common crossing work is excluded equally from all alternatives. Later-component costs include their installation and mobilisation. The compact component is integrated: all its route work and land are in the early price, so its later use adds zero rather than receiving free construction.
| Early component | Ports supplied | Money | Land |
|---|---|---|---|
| East-only foundation | E | 2 | 0 |
| West-only foundation | W | 2 | 0 |
| Reserved junction | E, W | 4 + extra cost | 1 |
| Compact integrated crossing | I | 8 | 1 |
| Later component | Ports required | Routes supplied | Money | Land |
|---|---|---|---|---|
| Direct east route | E | E | 1 | 1 |
| Direct west route | W | W | 1 | 1 |
| East route by a west-side detour | W | E | 5 | 4 |
| West route by an east-side detour | E | W | 5 | 4 |
| Outer routes serving both sides | E, W | E, W | 3 | 3 |
| Both routes from an east-only foundation | E | E, W | 6 | 4 |
| Both routes from a west-only foundation | W | E, W | 6 | 4 |
| Use already integrated routes | I | E, W | 0 | 0 |
Removing long fence corridors removes both detours, the outer dual routes and the two single-foundation dual routes. Removing the compact site removes its early component and integrated completion. The default reserved-junction extra cost is 0; the default survey cost is 1. The interface permits each cost from 0 to 4 in steps of 0.5. The engine also accepts other finite non-negative costs within its documented numerical bound.
No monitoring interaction, construction schedule, uncertainty about animal behaviour, noisy probabilistic observation or repeated learning process is modeled here. The two futures are different requirements, not forecasts assigned invented probabilities.
The three calculations
Write a completed assembly as (e,l) for an early and a later component. Each future has a finite list of compatible assemblies that supplies its requirement.
- Fixed: one early component and one later component must serve both futures.
- Later choices allowed: one early component is shared. A rule selects the later component from the observation available at that decision point. A perfect timely survey distinguishes E from W. A late or uninformative survey puts them in the same observation class, so the later choice must be identical. Fixed choices are included.
- Hindsight: early and later components may both differ between futures. This is a relaxation requiring the answer before early construction, not a plan that can be used under the starting premise. It is always labelled as an unavailable benchmark in this interface.
The proposed rule is chosen before the outcome. If its two complete assemblies differ, it requires the survey and pays the survey price once in each possible realised future. Identical assemblies use no survey and pay none. Thus information is optional, but it is not charged opportunistically after seeing the outcome. The hindsight benchmark uses the same rule for its survey charge. The engine’s before timing exposes hindsight as available for other callers; the essai fixes the early-before-information premise.
This convention lets the fixed choice set remain a subset of the allowed-later-choice set with identical accounting. Allowing more choices cannot worsen the best feasible set; a particular flexible design, or paying for information, can still be a poor bargain.
For each rule and future s:
money(s) = early money(s) + selected later money(s) + survey charge
land(s) = early land(s) + selected later land(s)
budget = (max_s money(s), max_s land(s))
The pair covers either future. Mutually exclusive futures are not added. The two maxima can occur in different futures; the pair is not necessarily one realised branch, nor an expected value. The screen retains both branch totals. This is dominance of budget envelopes, not the stronger comparison of every resource in every future.
A rule is dominated when another feasible rule has no greater money budget and no greater land budget, with at least one strict improvement. Equal resource pairs retain every distinct implementation. There is no weighted objective or unique overall winner.
Order of the calculation
The plain JavaScript engine enumerates every compatible assembly for each future. It then enumerates whole pairs, enforces common early components and indistinguishable-observation rules, evaluates the complete budgets, and only then finds the resource frontier. It does not prune each future’s assemblies by its separate resource frontier first.
That order preserves a distinction the question needs. A reserved junction can lose to the east-only specialist in E and the west-only specialist in W, while still being a useful common start. The two specialists cannot be substituted for one shared early installation. This is not a failure of valid Pareto reduction or Censi’s composition theorem: it is a warning against forgetting a decision identity before imposing the constraint that uses it.
This is a finite two-component compatibility model, not an implementation of a general traced-category or fixed-point co-design solver. The timeline shows when choices occur. The separate connection checks show which component supplies each required port. Co-design arrows must not silently be read as information or temporal flow.
Results that can be checked by hand
With default assumptions there are 6 fixed rules, 34 rules permitting later choices and 100 hindsight rules. Counts include resource-equal implementations.
| Comparison | Frontier money / land | Explanation |
|---|---|---|
| Fixed | 7 / 4; 8 / 1 | Reserved junction + outer dual routes; compact integrated crossing |
| Timely survey | 6 / 2; 8 / 1 | Reserved junction + matching direct route; compact integrated crossing |
| Hindsight | 4 / 1 | Matching single foundation + direct route + survey |
The adaptive reserved rule costs 4 + 1 + 1 = 6, uses 1 + 1 = 2 land, and has the same totals in both futures. Its 6/2 is incomparable with compact’s 8/1. The fixed reserved rule uses 4 + 3 = 7 money and 1 + 3 = 4 land. Hindsight’s 4/1 comes from 2 + 1 + 1, but requires different foundations in E and W.
- Make information late or uninformative: the allowed-later-choice set equals the fixed set, including component identities.
- Add 2 to the reserved junction’s early cost, with the default survey fee and compact site available: compact’s 8/1 dominates reserved’s 8/2. Every frontier choice is fixed.
- Remove long corridors and the compact site: no fixed design works. A timely survey still allows the reserved junction and matching direct route at 6/2. Late or uninformative information then leaves no feasible common plan, although each separate future remains feasible.
- Make the survey cost 0: default hindsight falls to 3/1 and reserved adaptation to 5/2; fixed choices keep their original costs. Information is still required when choices differ, even if its price is zero.
These are exact results within the declared catalogue, not evidence of a real project’s saving or feasibility.
Reading the originals
- Censi, A Mathematical Theory of Co-Design, §III, Definitions 10–12. Implementations have functionality/resource maps; composition keeps compatible implementation tuples. We use whole decision rules as named implementations. The MCDP manual, tutorial §§2.9–2.10, also makes implementation identities and equal-resource alternatives explicit.
- Zardini, Co-Design of Complex Systems, §§3.1–3.5, printed pp39–50. Its component interfaces and compatible tuples motivate our port checks. Its much broader recursive examples are not implemented here.
- Categories and Compositionality with a View to Applications, 4 November 2024 draft, §§30.1–30.4 and §36.2. Reducing implementations to feasibility forgets identity; co-design composition checks requirements; finite solutions can be represented by antichains. Our model retains identities until the early-choice condition is imposed. We keep the timing picture separate from compatibility connections.
- Censi, Uncertainty in Monotone Co-Design Problems, introduction and §II. Its interval bounds on design problems should not be mistaken for a fixed shared design across outcomes.
- Huang, Furter and Zardini, On Composable and Parametric Uncertainty in Systems Co-Design, §III-D. Choice before uncertainty differs from free choice afterwards; shared parameters preserve decisions that must remain common.
- Huang and Zardini, Distributional Uncertainty and Adaptive Decision-Making in System Co-design, August 2026 v3, §§IV-B/V-D. The paper already formalizes observations and adaptive choices and compares degrees of adaptation. Our exhaustive finite example omits its distributional and measure-theoretic machinery. It is a teaching construction, not a competing general method.
Only the relevant sections were revisited for this essai; no whole-book implementation or reading claim is implied.
Reproduce and inspect
The dependency-free model is common-beginning-model.js. From a repository checkout, run:
node --test tests/co-design-common-beginning.test.cjs
Tests include hand-derived frontiers, equal-information constraints, duplicate witnesses, infeasible combinations and independent exhaustive enumeration over 320 parameter cases. The test oracle constructs complete quadruples from its own numeric fixture, then checks the rule sets and resource results. Passing those checks validates the implemented finite calculation; ordinary browser checks separately exercise the controls, selection, address restoration and recovery.
The page uses its address for settings, view and selected choice. Refresh restores them; malformed or obsolete parameters are replaced visibly. It writes no server record, local storage or file and has no upload, account, project history or deletion operation. These are intentionally outside this temporary calculation. Reset restores the opening state and address. The four earlier apps remain separate and unchanged.