# Phase 3 Boundary Theorem Draft > Phase 3 artifact for > [`COARSE_GRAINING_PROOF_ROADMAP.md`](../COARSE_GRAINING_PROOF_ROADMAP.md). > Status: reviewed, closed positive, 2026-05-16. Phase 2 closed positive in > [`PHASE2_MDP.md`](PHASE2_MDP.md). This proves the negative direction > (deterministic and randomized `F_σ` policies) and maps the theorem onto the > pushable-occluder falsification slate. The reviewer pass is complete (see > *Exit Status*); Phase 4 is unblocked. ## Entry And Gate Phase 3 enters because Phase 2 closed positive. It imports the Phase 2 theorem: finite-MDP sufficiency holds exactly when every `Φ`-fiber over the occupancy support admits a common optimal action. The pre-registered negative for this phase is: > If a flat `𝓕_σ`-controller *does* solve a task whose decisive bit is provably > not `𝓕_σ`-measurable, the boundary is wrong and Postulate 1's failure > prediction is falsified — record, do not rescue. Draft verdict: **negative not triggered.** The boundary is the contrapositive of Phase 2: when a positive-support signature fiber has no common optimal action, no policy measurable with respect to the current signature sigma-algebra can be optimal. ## Local Symbols | Symbol | Definition | | --- | --- | | `b(x)` | A control-relevant bit or latent variable. | | `F_σ` | The sigma-algebra generated by `Φ`; equivalently, the information available to a flat signature policy at the decision point. ASCII rendering of Phase 0's `𝓕_σ` — same object, not a new one. | | `R*` | Occupancy support from Phase 2: states reached with positive probability under an optimal policy from the admitted initial distribution. | | `a_prep` | A preparatory world-changing action, such as pushing an occluder. | | `π*_0` | The optimal first decision at the boundary state/fiber. | ## Policy Class "Flat `F_σ` policy" means a policy measurable with respect to the **current** signature map `Φ` — deterministic `g : Σ → A` or randomized `g : Σ → Δ(A)`. This is the Phase 0 randomized-policy clause carried forward. A finite-memory or belief-augmented controller whose memory is admitted into `X` is, by the Phase 0 decision-information-state convention, a policy over an *enriched* signature: it falls outside the current `F_σ` by construction. Such a controller is therefore not a counterexample to the theorem below — it is exactly the sanctioned "make the missing bit measurable / add memory" escape named in the Pushable-Occluder Mapping, achieved by changing `Φ`/`X`, not by beating a fixed `Φ`. ## Theorem — Nonmeasurable Decisive Bit Let `Φ : X → Σ` be the current signature map. Suppose there exists a signature fiber `C = {x ∈ R* : Φ(x)=σ}` with positive evaluation/occupancy mass on which the optimal first action is not fiber-constant: ```text ∃x0, x1 ∈ C : Φ(x0) = Φ(x1) = σ and A*(x0) ∩ A*(x1) = ∅. ``` Then no flat `F_σ` policy is optimal on that fiber. Write `b(x)` for any bit that distinguishes such `x0, x1`; it is the *witness* of the conflict and is, by construction, not `F_σ`-measurable on `C` — non-measurability of `b` is a consequence of the two conditions above, not an independent hypothesis. Equivalently, if the decisive bit only becomes observable after a preparatory world-changing action, but the current decision already requires choosing among different preparatory actions or between prep/no-prep based on that bit, then a flat policy over the current signature cannot be Bayes-optimal. ## Proof **Deterministic case.** Any deterministic `F_σ` policy is constant on the fiber `C`: for some `g`, it selects `g(σ)` for every `x ∈ C`. Optimality at `x0` requires `g(σ) ∈ A*(x0)`; optimality at `x1` requires `g(σ) ∈ A*(x1)`. Since `A*(x0) ∩ A*(x1) = ∅`, no single action satisfies both, so no deterministic `F_σ` policy is optimal on `C`. **Randomized case.** A randomized `F_σ` policy is likewise fiber-constant: it applies the same kernel `g(σ) ∈ Δ(A)` to every `x ∈ C`. In a finite MDP every action in the support of an optimal stationary policy at a state lies in that state's `A*` (any off-`A*` action has strictly lower `Q*` and would reduce expected return). Optimality at `x0` therefore needs `supp g(σ) ⊆ A*(x0)`, and at `x1` needs `supp g(σ) ⊆ A*(x1)`; with `A*(x0) ∩ A*(x1) = ∅` the support must be empty, which is impossible. So no flat `F_σ` policy — deterministic or randomized — is optimal on `C`. This is exactly the contrapositive of Phase 2's fiber theorem. A signature is control-sufficient only when the collapsed states agree on at least one optimal action. If the collapsed states disagree on the action, the signature has discarded a control-relevant bit. ## Scope Guard The theorem does **not** say every task with a preparatory action is impossible for a signature policy. If the common optimal first action over the whole fiber is `a_prep`, then `a_prep` is measurable and a signature policy can choose it. After the preparatory action, the process may enter a new information state with a richer signature, and later control may become solvable. The boundary is sharper: the current signature is insufficient when the hidden bit changes which preparatory action is optimal, or whether preparation is optimal at all, while the signature remains the same. ## Pushable-Occluder Mapping The pushable-occluder design extends the photometric alignment task with a block whose geometry is not in the flat controller's observation channel. The flat controller sees detector intensities, mirror joint state, and push-effector proprioception; the oracle sees block geometry. For the Phase 3 theorem to apply, the diagnostic-positive seed slate must instantiate a fiber conflict: ```text Φ(x0) = Φ(x1) but oracle-required preparatory action differs: push direction, push/no-push decision, or clearance sequence. ``` Under that condition, the expected flat-controller failure is a theorem-predicted boundary. It is not a surprise after the data; it is the negative branch of the sufficiency predicate: - the oracle can solve because it observes the missing bit; - a flat `F_σ` policy cannot be optimal because it must choose the same preparatory action across states that require different optimal preparation; - a hierarchical controller or redesigned signature can succeed only by making the missing bit measurable, adding memory/belief state, or changing the agent class. If the flat controller succeeds on a non-trivial fraction of diagnostic-positive seeds, one of two things happened: 1. the design did not actually create a nonmeasurable fiber conflict; some photometric/proprioceptive feature leaked the decisive bit into `Φ`; or 2. the theorem boundary is wrong for the admitted task. The pre-registered disposition is to record that result, not rescue the boundary by renaming the signature after the fact. ## Cross-Filed Statement Safe statement for the design and analogy docs: > The pushable-occluder slate is Phase 3's theorem-predicted boundary: when > matched states share the flat photometric/proprioceptive signature but require > different preparatory block actions, no current-signature policy can be > optimal. A success by the flat controller would mean the decisive bit leaked > into the signature, or the boundary theorem is wrong. ## Exit Status Phase 3 proof: **positive.** Pre-registered negative: **not triggered.** The impossibility result is the Phase 2 contrapositive, extended to randomized `F_σ` policies; the pushable-occluder mapping is conditional on the design instantiating a genuine nonmeasurable fiber conflict. Artifacts required by the roadmap are present: 1. negative-direction proof (deterministic and randomized); 2. explicit scope guard for preparatory-action tasks; 3. pushable-occluder theorem-predicted boundary statement, cross-filed in [`PHASE2_BLOCKS_DESIGN.md`](../PHASE2_BLOCKS_DESIGN.md) and the river-and-dam entry of [`analogies.md`](../../internal/theory/analogies.md); 4. disposition branch for flat-controller success. Reviewer pass: **complete, 2026-05-16.** The contrapositive proof, the scope guard, the gate quote (verbatim against roadmap), and both cross-filings were checked. The randomized-policy completeness gap and the policy-class scoping were flagged and are now applied (new *Policy Class* section + randomized case in the proof); the `b(x)` hypothesis is reframed as the conflict witness. Resolved review-item dispositions: 1. **Cross-filed statement is adequately hedged.** It is conditional on a genuine fiber conflict and pre-registers both branches without asserting the empirical setup has one — no overclaim before the seed audit. Confirmed as written; both cross-filings match. 2. **"Flat `F_σ` policy" = measurable wrt the current `Φ`** (deterministic or randomized), resolved in the new *Policy Class* section. A finite-memory / belief-augmented controller whose memory enters `X` is a policy over an *enriched* signature, outside the current `F_σ` by the Phase 0 convention — the sanctioned escape, not a counterexample. 3. **Phase 4 reuses this vocabulary — pre-committed.** Phase 4's measurable/off-measurable cell split uses the same fiber-conflict definition (fiber has / lacks a common optimal action), so the empirical gate and the theorem share one language (spec self-consistency). Phase 3 closes **positive**; Phase 4 entry (Phase 3 exit) is satisfied.