Two miniature brick rings. One shared hoist pool. Derive the states and possible futures—not an animation supplied in advance.
First try · Follow one shared-hoist decision
Leave the defaults and wait for the model to load, then press Next tick → in Build & paths. Ring A starts course 1, brick 1; the sole hoist is reserved and 33 ticks remain on this completing path. From start rewinds the replay.
Now change Hoists available to 2 and press Construct model. With the other defaults unchanged, the minimum completion time changes from 34 to 22 ticks. This is a consequence of the supplied rules, not a forecast for a real site.
For the state question, open What state is enough? and press Test and refine. The two witness states explain why the same placed-brick picture can permit different next actions. Formal construction contains the source mechanisms; End · way · result states the boundaries. The method note connects the rules and proof obligations.
Want the smaller first step? Start with choices and policies in #12. The continuous tab is a separate smooth toy, not a proved limit of this brick construction.
Constructing the finite model…
Settings live in this URL. No project record, file, booking or approval is created. Replays are temporary.
—reachable states
—sufficient state classes
—minimum completion ticks
—shortest labelled paths
FROM THE ORIGINAL FOUR LINES · A MYERS-INSPIRED CONSTRUCTION
A space of states. A way of moving through it.
First the manifold. Then the tangent spaces, their bundle, and a section choosing the dynamics. Only then do we obtain a trajectory.
Imagine replacing discrete brick placements with a continuously moving workfront. Track its angle θ around the ring and the fraction w of the specified work achieved. This deliberately simplified continuous toy omits bricks, curing and shared hoists; it does not change the finite experiment above.
Choose the way, keeping the end fixed
The open system becomes a section through feedback
State x ∈ Xangle + completed work
r = id →
Output o ∈ Ofull state exposed here
κ →
Input (ω, q)chosen rates, not a next state
u(x, ·) →
vκ(x) ∈ TₓXa direction at this same x
The open update has a family of possible rates. Wiring its readout into a fixed smooth policy closes that choice: vκ(x) = u(x, κ(r(x))). The initial angle chooses where a trajectory starts, not the rule of motion.
The full lens, and what this picture leaves out
Use Myers’s convention, with change/input above state/output: (u,r): (TX → X) ⇆ (I → O). Here O = X, U = ℝ², I = O × U, and p(o,i) = o. Readout r = idₓ. The pullback r*I is X × U, and u(x,(ω,q)) = ω∂θ + q∂w lies in TₓX, so π(u(x,i)) = x. The input arena permits signed rates mathematically; the three displayed ways have nonnegative work rates along their shown trajectories.
The policy κ : O → U lifts to the bundle section κ̂ : O → I, with κ̂(o) = (o, κ(o)). The feedback lens (κ̂,id): (I → O) ⇆ (O → O) removes external input choice. Composing it with the system gives the vector-field section vκ. Projection π ∘ vκ = idₓ checks where every chosen vector belongs; the differential equation γ̇ = vκ ∘ γ checks whether a history actually follows the chosen way. Those are different equations.
The visible strip 0 ≤ w ≤ 1 is only a window in X = S¹ × ℝ, not a claim that a closed strip is a boundaryless manifold. Each tangent plane contains all mathematical directions, including decreasing w. TX is four-dimensional, so these attached planes are samples, not a literal complete embedding of TX in this image. The angle cut at 0/2π is a drawing convention: the state and tangent direction are continuous across it.
The end is an observed first-hit condition, not a force supplied by the manifold. Steady work is displayed only up to its first hit; we have not made a smooth field stop there. The taper trajectory from w(0)=0 exists for every future time and never reaches the end at finite time. We make no global-completeness claim for that field from every point in X. A whole brick project still needs resource, support, curing and terminal rules, as the finite tabs construct.
Begin with the prompt, then follow the sources
The original intuition names X, each TₓX, their bundle TX, and a vector field selecting a vector at every x. The correction is small but decisive: a section is a way of moving; a trajectory is a history following that way. Different sections can reach, miss, or only approach the same end.
The cylinder and three policies are this essay’s own illustrative construction, not examples attributed to Myers. Exact formulas drive the animation; there is no numerical integration error hidden by the picture. No correspondence with the finite brick model has been proved.
Generate a route, then build it
Every step is selected from the computed transition graph. A and B are separate cylindrical installations, each with its own course support. Only the hoist pool is shared.
□ Not placed■ In placement■ Course curing■ Placed / ready
AN IMAGINED MANORWATER OUTPOST
A hilltop. Two towers. One shared hoist.
Both rings must rise. But the next lift needs a free hoist—and a cured course beneath it.
▱ Pale outline: still to build▰ Gold: lifting◌ Purple: curing▰ Brick: placed
Tiny brick-built peel-tower forms in a fictional Borders setting. This is the same generated route and model state as “Build & paths”; the loop starts partway through so the site is visible immediately. Each model sector represents a chunk of brickwork. The landscape, brick texture and hoist travel add atmosphere, not extra work or timing rules. The foundation outlines are outside the modelled work. Hoist travel illustrates the move towards the next tick; “Inspect current tick” shows the current tick boundary.
A picture is not automatically a planning state
Choose what the planner may distinguish. The test checks all reachable states, not only the displayed path. States in one class must agree on completion, admissible commands and the next class for each command.
What was discovered—and what was supplied?
The algorithm discovers the coarsest stable partition refining the chosen summary within the supplied finite model. It does not infer physics from a drawing, find unknown real-world variables or invent a sensor. A class is calculated from the full model state. If its necessary distinctions cannot be observed or tracked, this planner cannot run on the coarse readout alone.
Ready/not-ready can preserve today's choices yet lose tomorrow's: one and two curing ticks remaining both look “not ready”. Refinement carries future distinctions backwards until every labelled continuation is preserved.
Does the abstraction survive different wiring?
A summary derived after composition may exploit the current bottleneck. Add a hoist, and a difference previously hidden by compulsory waiting may matter again.
Can the stated way deliver the end?
Yes for this fully specified finite toy—but the blueprint alone is insufficient, and trajectories are not themselves sections of the action bundle.
Original ambition
Constructed result
Boundary
Blueprint becomes project state
Construct reachable dynamics from component rules; refine their summary to preserve admissible futures.
Consistent geometry does not supply timers, active work or laws of change.
Dynamics and possible trajectories
Full reachable graph, stable quotient, completion policy, exact shortest-path count and concrete replay.
Finite deterministic rules and unit ticks; not calibrated engineering.
Lenses and open-system wiring
Local updates exchange start, release and reservation information with one pool and one clock.
A specific composition, not a general wiring-language implementation. Rewiring requires a fresh abstraction test.
Paths as tangent-bundle sections
A finite action bundle; a section chooses a policy; iteration yields a path.
A path is not a vector field. Finite action fibres are not tangent vector spaces.
Working visuals and analytics
All views are computed from the component rules.
No manually planted outcomes; no real-world success claim.
Ends, ways and means
End: a state representation from which valid full-completion paths can be generated. Way: specify local dynamics and interfaces, wire them together, then derive sufficient state and a completing policy. Means: finite enumeration, refinement, counterexamples and replay. A useful picture is a motivator; delivery of the full goal is the test.
What had to be added
Placement and support rules; placement and curing durations; one authoritative pool; shared tick semantics; the full goal; and the observation-preservation criterion. The blueprint and lens notation do not supply these ingredients. With them supplied, the finite construction reaches the end rather than merely depicting it.
What the negative cases establish
Without capacity constraints, local work can over-reserve the hoist pool.
Blueprint or ready-flags alone collapse distinct futures.
Close access to B: the solver reports no completion route.
Add a hoist: an abstraction exploiting the old bottleneck may cease to preserve behaviour.
The companion essays include separate continuous slot-observation models. This essay does not complete that optional continuous extension.
The executable construction
State
S is the reachable subset of SA × SB × H. Each front stores brick masks, curing ticks and an active placement. H stores the authoritative reservations.
New bricks extend either end of a contiguous cyclic run. Direction may change between placements; this is not a fixed clockwise or anticlockwise course. Four bricks give 16 allowed orders per course (rotations count separately), whereas fixing one direction gives 8. With two or three bricks, adjacency adds no exclusion. A new course waits for the previous course to be complete and cured. The goal includes final curing.
Interface and update
r(s) exposes completion status, local admissible requests, releases and occupancy. I(r(s)) contains feasible joint requests. u(s,i) advances both fronts and the pool on one tick.
Myers convention: (u,r): S/S → I/O in the dependent deterministic theory. Inputs lie over outputs. The blueprint q(s) is a separate proposed summary, not silently the full interface.
Policy and trajectory
E = {(s,a): a admissible at s}; p(s,a)=s; t(s,a)=u(s,a). A policy σ chooses from each nonterminal state's fibre. Then sn+1=t(σ(sn)).
The completion policy is defined on the goal-reaching region. A completed trajectory stops at the terminal goal.
The implemented wiring
Workfront A readiness · finish signalWorkfront B readiness · finish signalHoist pool H current reservations
↓ admissible local requests + shared capacity ↓
Joint input relation → choose a feasible command
↓ starts to fronts and pool · releases to pool ↓
Local update ALocal update BPool update H
One shared tick → next global state → read again
Capacity filters possible joint commands; it does not greedily choose which front wins. The solver selects a policy only after the complete composed dynamics exist. This is one explicit finite wiring, not a universal diagram interpreter.
Refinement and preservation
Start: s ~₀ t iff q(s) = q(t).
Split classes when members disagree on:
goal status, enabled labels, or successor classes.
Stop when no class splits.
Then q*(u(s,a)) = ū(q*(s),a).
Solve shortest distance to a goal on the quotient.
Choose a transition reducing distance by one.
Lift and replay it in the detailed system.
Refinement only splits a finite set, so terminates. Any stable refinement must separate every pair separated at each round, by induction. The result is the coarsest stable refinement of the starting summary. It preserves action labels, goal and unit costs—not every possible reward or output, and not arbitrary replacement contexts.
Tick semantics
Reserve at tick start. Advance active/new placements and existing curing timers once. Release finishing jobs at tick end; newly completed courses start their full cure period then. A hoist released at the end cannot be reused at that tick's beginning. Waiting through curing is not an identity morphism.
Research discipline
State the end before choosing a mechanism. Record assumptions and distinguish an implemented toy from a validated project model. Reject a proposed summary when two reachable states sharing it permit different futures; reject a completion claim when a generated path fails support, capacity, curing or the full goal. The unresolved question is which necessary distinctions can be observed in practice—not whether an attractive diagram can be drawn.
Source mechanisms
Spivak (2019), Lenses: applications and generalizations: readout/update and product-plus-wiring. Author-hosted slides.
Myers (2021), Double Categories of Open Dynamical Systems, §§1–3, 5–6: dependent interfaces, variable substitution and trajectories as behaviours. Paper.
Myers (2023), Categorical Systems Theory, §3.5.3 and §4.2: dependent deterministic systems and preservation under wiring. Refinement is an additional construction here.
Coecke (2009), process composition; Coecke–Fritz–Spekkens (2014), resource theory: parallel composition does not duplicate a physical hoist.
Lynch (2022), Relational Composition of Physical Systems: explicit shared-variable/interface semantics.
Dynamics, proof obligations and limitations: method note. This is not a new theorem of category theory.