Open the sixth essai · Project Co-design

Question: can an early action remove useful discrimination while leaving every later physical remedy available?

This independent finite model extends the enquiry in A common beginning. That model varies when a survey arrives. Here the first action changes the available observation map. All quantities, hypotheses and mechanisms are synthetic; no real ecological or engineering effectiveness is implied.

Construction

There are two exhaustive explanations, H = {E,W}, with supplied prior P(E)=p, 0.1 ≤ p ≤ 0.9. The same initial passage count, zero, occurs under either explanation. E and W are incomparable labels, not ordered quality scores.

The first implementation is either grade first, with incremental cost 0, or read tracks first, with extra cost c ∈ [0,8]. Reading first includes grading afterwards; both reach the same prepared site before the repair. Common works are excluded equally. The extra cost includes any delay; no separate calendar claim is made. Both supply exactly the same repair-connection set {E,W,I}.

The default deterministic observation map is:

First action E explanation W explanation
Grade first blank blank
Read first E trace W trace

The available pre-repair observations are the passage count and this trace. Repeating the trace re-reads the same object, without fresh noise. No other diagnostic exists inside this model. Repair outcomes or new experiments after this decision are outside its horizon.

Repair Required connections Success in default model Money Land
East guidance E E 2 1
West guidance W W 2 1
Both approaches E,W E,W 4 3
Compact dual guidance I E,W 6 0.5

Resources are invented units. Repair costs include fitting and mobilisation. Each early action retains all four repairs; this is a controlled observation-loss comparison, not a physical option-loss comparison.

Formal interfaces and algorithm

An implementation is (a,π): a first action a and a deterministic rule π from the signals obtainable after a to repairs. Let g_a : H → Y_a be the observation map. The selected repair in state h is π(g_a(h)).

The observation component supplies a partition of H. Order partitions from coarser to finer: P ≤ Q when Q refines P. A repair rule requires the partition induced by its state-dependent choices. Compatibility requires that the supplied observation partition refines this required partition. The code implements this by assigning a single remedy per signal, so merged states cannot receive different choices. The repair’s required connection set must be a subset of the early component’s supplied set, in the ordinary inclusion order on the powerset of {E,W,I}.

Complete implementations provide a success probability:

f(a,π) = p · success(E,π(g_a(E)))
       + (1−p) · success(W,π(g_a(W))).

The requested functionality is a lower bound τ ∈ [0.5,1]. The default is 1, requiring success in both states because both have positive prior mass. A smaller threshold explicitly accepts failure under some explanations. These probabilities are model inputs, not estimated from data.

The two resource coordinates are budgets sufficient in either state:

money(h) = first-action cost + repair money in state h
land(h)  = repair land in state h
r(a,π)   = (max_h money(h), max_h land(h)).

The two maxima need not occur in the same state. This is neither an expected-resource pair nor a sum over mutually exclusive futures. Resource order is coordinatewise ≤; functionality is ordered by ordinary ≤. An implementation witnesses feasibility of (τ,budget) exactly when τ ≤ f(a,π) and r(a,π) ≤ budget. The resulting feasibility relation is downward closed in required functionality and upward closed in resource allowance.

The engine enumerates every rule over the distinct obtainable signals (4 grade-first and 16 read-first rules by default), checks component connections, evaluates both branches, filters on τ, then finds the resource antichain. Equal resource pairs keep all distinct implementation witnesses. There is no scalar weighting of money against land. For antichains A and B the co-design order is A ≤ B ⇔ ↑A ⊇ ↑B; it is not the claim that every displayed resource coordinate must rise.

The additional perfect-information benchmark assigns repairs directly to E and W with the grade-first cost. It removes the observation constraint and has 16 rules. It is deliberately unavailable under an erased or indistinct channel; when the intact control makes the trace fully informative, it coincides with an available choice set. It is not presented as an implementable plan in the default case.

This is a finite catalogue with an explicit information interface, not a general traced-category, recursive MCDP or measure-theoretic solver. The screen’s decision-tree arrows represent observation and choice. They are distinct from the catalogue’s functionality/resource compatibility conditions.

A finite loss-of-learnability result

After grading, g_grade(E)=g_grade(W)=blank. Therefore any observable rule satisfies π(g_grade(E))=π(g_grade(W)). No such rule can select east repair only under E and west repair only under W. Both repairs still physically exist.

For any allowed prior p, both explanations give blank likelihood 1. Thus:

P(E | blank) = p / (p + (1−p)) = p.

Repeating the same deterministic observation n times still gives a constant map, for every positive integer n. It cannot split {E,W} or update the prior. Reading first produces the singleton partition { {E}, {W} }; its posterior is 1 after E trace and 0 after W trace. These are elementary finite derivations, not a claimed new identification theorem.

The conclusion is conditional: learning that distinguishes these two explanations is impossible with the listed observations before the repair decision after grading. It says nothing about additional sensors, interventions, hypotheses or eventual diagnosis after repair.

Results and controls

At c=1, p=0.5, τ=1:

Query Resource frontier: money / land
Grade first 4 / 3; 6 / 0.5
Read first 3 / 1; 7 / 0.5
Either first action 3 / 1; 6 / 0.5
Unavailable perfect information 2 / 1; 6 / 0.5

The informative matching policy needs 1 + 2 = 3 money and 1 land in either state. Grade-first matching is disallowed by the merged observation, so a universally successful grade-first rule must use one of the two dual repairs. Compact’s 6 / 0.5 remains incomparable with reading-first’s 3 / 1. A mixed informative policy using a single repair in one state and compact in the other needs 7 / 1 and is dominated.

  1. Expensive preservation: at c=4, the combined frontier returns to 4 / 3 and 6 / 0.5. Preserved information is still informative, but every retained policy grades first.
  2. Decision-irrelevant distinction: change only repair success so either single repair works for both explanations. Grading still destroys identification, but the combined frontier equals the grade-first frontier, 2 / 1 and 6 / 0.5. Two resource-equal single-repair witnesses are retained.
  3. Evidence survives grading: change only the grade-first observation map to distinguish E and W. Grading then attains 2 / 1 and 6 / 0.5. The apparent advantage of delaying grading disappears.
  4. Assay cannot discriminate: both first actions yield the same trace under E and W. Reading first adds cost without knowledge; the combined frontier is 4 / 3 and 6 / 0.5.
  5. Accepted probabilistic failure: at p=0.9, τ=0.9, always using the east repair is feasible at 2 / 1 without learning. At τ=1 that policy is infeasible. This is an explicit tolerance change, not improved evidence.

These controls hold the physical remedy catalogue fixed. “More learning is always better” is not the result.

A failed interpretation worth keeping

An initial expected-money / expected-land projection yielded a misleading same-remedy control. An informative signal can serve as a randomizer, mixing cheap and compact repairs even when the explanation has no bearing on success. At a fair prior and cost 1, a single/compact mixture has expected pair 5 / 0.75; it is incomparable with grade-first’s 2 / 1 and 6 / 0.5. That would not establish useful identification. A fair independent coin available to grade-first produces 4 / 0.75 and dominates the supposed information benefit.

The maintained comparison uses state-covering resource budgets, which remove that confound. Any later expected-resource extension must supply matched independent randomization and keep risk/mean/budget semantics separate. The failure changed the construction before publication.

Source contributions and boundaries

  • Andrea Censi, A Mathematical Theory of Co-Design, §II (antichain order), §III (implementations and composition). Supplies the functionality/resource/implementation distinction and compatibility-based construction. We retain explicit policy witnesses and derive finite antichains. No fixed-point or feedback theorem is exercised.
  • Censi, Formal Tools for Co-Design manual, §§2.2–2.10. Provides/requires interfaces, catalogues and incomparable resource choices inform the visible catalogue. No MCDPL interpreter is used.
  • Gioele Zardini, Co-Design of Complex Systems, §§3.1–3.5, printed pp39–50. Motivates the explicit interface constraints and compatible implementation tuples. Its warning that co-design diagrams are not signal-flow diagrams governs the separate decision-tree explanation.
  • Categories and Compositionality with a View to Applications, 4 November 2024 draft, §§29.3–29.4, printed pp405–408. Supplies the finite minimal-resource query and implementation-tuple composition. Our information-partition interface is a proposed finite construction; it is not taken as a theorem from that text.
  • Yujun Huang and Gioele Zardini, Distributional Uncertainty and Adaptive Decision-Making in System Co-design, August 2026 version, §IV-B. Explicitly models observation and policy stages depending on previous choices. This supports the placement of action-dependent observations in a co-design enquiry. The present two-state deterministic calculation omits their general distributional and quasi-universal-space machinery.

Relevant source sections were read for this trial. No Churchman or Ackoff technical result is attributed here. The synthetic catalogue, trace-erasure mechanism, elementary Bayesian derivation and controls are the essai’s own construction, not empirical findings borrowed from these sources.

Reproduction and interface boundary

The dependency-free engine is learnability-model.js. Run:

node --test tests/co-design-learnability.test.cjs

The suite checks hand-derived frontiers, observation equivalence, repeated readings, the four mechanism controls, accepted-failure changes, duplicate witnesses, resource-envelope semantics and invalid inputs. A separate integer-scaled oracle enumerates state assignments, rejects unobservable rules, and compares complete frontier policy IDs over 900 parameter combinations. It does not reuse the engine’s catalogues or enumeration helpers. This is bounded calculation evidence, separate from ordinary browser review.

The page is an ephemeral calculation. Its URL holds settings, selection and repeat count. Reload or the “Open this exact comparison” link restores that state. Changing assumptions can invalidate the selected policy; a visible message explains the replacement. Malformed settings visibly return to valid defaults. Reset replaces the address with the opening case. There is no server write, local storage, upload, generated file, project receipt, app History, correction record or deletion action; those loops are deliberately inapplicable. The persistent object is a browser address, which the browser may retain. None of the other five numerical models is changed or coupled to this one.