# (Ω, D) Dynamics — Complete Framework Document

> **Purpose of this file.** This is a self-contained, machine-readable consolidation of
> the entire `(Ω, D)` Dynamics framework as it currently stands, including every
> definition, axiom, theorem, proof, counterexample, falsifiable prediction, and open
> problem developed so far. It is written to be *read and adjusted by an AI or human
> collaborator*. Each formal result carries an explicit status tag so a reader can tell
> theorem from sketch from conjecture. Edit conventions are described in the final
> section.
>
> **Status legend.** `[PROVED]` full proof given · `[SKETCH]` strategy given, details
> routine or flagged · `[OPEN]` precisely stated, not attempted · `[CONJECTURE]`
> expected true, no proof · `[AXIOM]` adopted restriction · `[DEF]` definition.
>
> **Last consolidated:** 2026-06-12. **Source:** analytic worklog `6a2a52c2f1c1e.json`
> ("Refusal evasion" thread) plus project notes `Causality.txt`,
> `Chat_Advances_Summary.md`, `Current_Working_Theory.md`.
>
> **Updated 2026-06-12 (open-problem session):** O6 closed (Theorem 12b), O1 closed
> on the subsequently corrected Levin domain (Theorem 14), O2 partially closed (Theorem 4c),
> Lemma 5 promoted to Theorem 5 (full proof), first steps on O4 and O7 (Theorems 1-S,
> 1-D). Constructions verified numerically in `open_problems_verification.py` (§12).
>
> **Updated 2026-07-11 (Minecraft lab consolidation):** §13.4 added as an *implemented*
> lab (M1 discharged offline + live; selector core reproduces Theorems 8/9/13 as unit
> tests); §13.5 records the `x*`-ladder rest-state result and the `(θ, h)` evolution
> outer loop (first O4 data point). New axiom-level content extracted from the lab:
> **EB-L** (learning boundary, §3.7) and **GX** (goal coordinates on `x*`, §3.8);
> signature 10 (evolved-horizon shift) registered in §8 *before* the hostile-`E` run.
>
> **Updated 2026-07-11 (external assessment implementation):** R1 split into continuous
> and typed/hybrid forms; C1-T added for pre-registered discrete tie refinements; the
> Ω/D/E factorisation audit made operational; Theorems 10/14 narrowed to the
> σ-compact metrizable domain supported by the cited Levin theorem; and two
> non-Minecraft validation benchmarks added (§12, §15).

---

## 0. Origin and scope

The framework grew out of a philosophical project (recorded in `Causality.txt` and
`Chat_Advances_Summary.md`) asking whether *causality* can be reformulated as **dynamic
informational similarity**: the idea that the next state of a system is the one that
minimally disrupts the system's complete physical-informational state (phase-space
state, not a static snapshot), rather than the one produced by an external causal force.
That broad reduction is **not** established. The defensible residue is epistemic and is
treated separately in `Current_Working_Theory.md`.

`(Ω, D)` Dynamics is the **mathematical core** extracted from that project: a typed,
selection-based update rule. This document analyses it *as mathematics* — what it
assumes, when it is well-posed, when it is vacuous, what it predicts, and how it differs
from neighbouring frameworks (revealed preference, MPC/RL, Lyapunov/ISS control). The
central finding of the analysis is blunt and load-bearing:

> **Unconstrained, the framework is notation, not theory. All empirical content lives in
> the restrictions (the axioms of §3) — above all in `E`-blindness (§3.7), which is the
> single commitment that distinguishes `(Ω, D)` Dynamics from model-predictive control.**

---

## 1. The model

### 1.1 The dynamical principle

A persistent system updates its extended state by

$$
x_{t+1} \;=\; E\big(\hat y(x_t)\big),
\qquad
\hat y(x) \;\in\; \operatorname*{arg\,min}_{y \in \Omega(x)} \, D\big(y, x^{*}; h\big).
$$

In words: from the current state `x`, a **typed candidate menu** `Ω(x)` of possible
next-state primitives is generated; a **selector** picks the candidate that minimises an
**informational discontinuity functional** `D` relative to a **reference manifold** `x*`
over a **projection horizon** `h`; then an **external environment operator** `E` is
applied *after* selection to yield the realised next state.

### 1.2 Notation `[DEF]`

| Symbol | Meaning |
|---|---|
| `(X, d)` | state space, a metric space (often Polish; sometimes ℝⁿ or Hilbert) |
| `x ∈ X` | extended state of a persistent system (includes everything carried forward, e.g. momentum/phase, not a static snapshot) |
| `Ω : X ⇉ X` | candidate correspondence ("menu"); set-valued, typed by primitive operations |
| `x* ⊆ X` | reference manifold the system maintains (closed, nonempty) |
| `D(y, x*; h)` | informational discontinuity functional ranking candidates `y` |
| `h` | projection horizon (lookahead depth) |
| `E : X → X` | environment operator, applied **post-selection** (may be stochastic) |
| `Ŝ(x) = ŷ(x)` | the selected candidate (a measurable selection of the argmin) |
| `V(x) := d(x, x*)` | distance-to-reference; the canonical Lyapunov function |
| `X_† := {x : Ω(x) = ∅}` | **death set** (absorbing failure state) |

### 1.3 Core invariants (design commitments) `[DEF]`

These define the *intended* separation of roles. Collapsing them is what turns the
framework into existing notation.

1. **`E` is post-selection, not folded into `D`.** The selector ranks `y` by `D(y,…)`,
   *not* by `D(E(y),…)`. (This is sharpened into the `E`-blindness axiom EB, §3.7.)
2. **Goals / available futures enter via `Ω`, not via reward terms in `D`.** No arbitrary
   reward shaping inside `D`.
3. **`D` ranks candidates; it must not silently absorb all the dynamics.** If `D` secretly
   contains the time-evolution map, the framework is a redescription, not a theory.
4. **Persistence is descriptive, not teleological.** Persistence is a *consequence* of menu
   geometry and an inequality (Theorem 1), never an assumed goal.

---

## 2. Well-posedness

Four conditions are needed, in increasing strength. They fail independently, so they are
kept separate.

- **(W1) Nonempty menus.** `Ω(x) ≠ ∅` for all reachable `x`. Not automatic when `Ω` is
  defined by constraints (reachability under resource bounds). Failure is the **death
  state** `X_†` and is treated as the object of study, not patched away.
- **(W2) Attainment.** The infimum of `D(·, x*; h)` over `Ω(x)` is achieved. Sufficient
  (Weierstrass): `Ω(x)` compact, `D(·, x*; h)` lower semicontinuous (lsc). Infinite-dim
  variant: `X` reflexive Banach, `Ω(x)` closed bounded convex (weakly compact), `D` convex
  and lsc (hence weakly lsc).
- **(W3) Selection.** `argmin` is a set, not a point. Either impose **uniqueness**
  (strict quasiconvexity of `D` on convex `Ω(x)` — promoted to axiom **C1**, §3.6) or fix a
  **measurable selector**: if `X` is Polish, `Ω` a measurable correspondence with closed
  values, and `D` Carathéodory, then **Kuratowski–Ryll-Nardzewski** gives a measurable
  selection `Ŝ`. Without this, "the dynamics" is a *family* of dynamics, and persistence
  claims must be quantified over selectors (∀ vs ∃ are different theorems).
- **(W4) Iterability.** `E ∘ Ŝ` must map the domain into itself with enough regularity to
  compose indefinitely. Clean tool: **Berge's maximum theorem** — if `Ω` is continuous
  (upper *and* lower hemicontinuous) with nonempty compact values and `D` is jointly
  continuous, then the value function is continuous and the argmin correspondence is uhc
  with compact values; with uniqueness, `Ŝ` is continuous and `E ∘ Ŝ` is a bona fide
  continuous dynamical system.

> **Flag.** Lower hemicontinuity of `Ω` is the assumption people forget, and it is exactly
> where **typed candidate spaces hurt**: menus defined by discrete type constraints jump
> discontinuously, so `Ŝ` is generally only piecewise continuous. That is acceptable, but
> Lyapunov arguments must then be done in discrete-time style (§4), not via continuous-flow
> intuition.

**Lemma 0 (well-posedness).** `[SKETCH]` *If `Ω` has nonempty compact values and
`D(·, x*; h)` is lsc, minimizers exist for all `x`; under Berge conditions the dynamics is
well-posed up to selection.* (Recorded as a lemma, not a headline result.)

### 2.1 Failure catalogue (each is a modelling pathology) `[PROVED]` (by example)

1. **Open menu.** `X = ℝ`, `Ω(x) = (x, x+1)`, `D(y) = y`. Infimum at the excluded
   boundary. Pathology: `Ω` defined by strict constraints.
2. **Non-coercive escape.** `Ω(x) = ℝ`, `D(y) = e^{−y}`. Infimum `0` at infinity, never
   attained — the system "chases the horizon." Generic failure of any `h → ∞` limit.
3. **lsc failure.** `Ω(x) = [0,1]`, `D(y) = y` for `y > 0`, `D(0) = 1`. Infimum unattained.
   Pathology: `D` from a discontinuous information functional (description-length
   functionals are only upper semicomputable — relevant if `D` is Kolmogorov-flavoured).
4. **Infinite-dim oscillation.** `X = L²[0,1]`, `Ω = ` unit ball,
   `D(y) = ‖y − g‖² − 2‖y‖²`: minimizing sequences converge weakly but `D` is not weakly
   lsc (concave term). **Real risk for the intended semantics:** if `D` rewards
   informational *novelty* it may be concave in exactly the way that destroys weak lsc.

---

## 3. Axioms (the empirical content)

The analysis shows the bare update has no content; these restrictions are where the theory
lives. Each closes a specific vacuity loophole proved in §5.

### 3.1 R0 — Observability / parsimony of the extended state `[AXIOM]`
Either the extended state is fully observable, or its unobserved part is constrained a
priori (bounded dimension; no coordinates whose only role is indexing; the unobserved
component identifiable from observed-trajectory statistics, or fixed before data).
**Kills:** clock-smuggling (universal rationalization via `X̃ = X × ℕ`). Without R0 every
other restriction is hollow, because even determinism becomes unfalsifiable. This is the
first axiom, not a footnote.

### 3.2 R1 — Bounded, mechanism-specified menus with declared overlap `[AXIOM]`
The menu mechanism is fixed before data and takes one of two typed forms.

- **R1-C (continuous / annular):** `B̄(x, ε) ⊆ Ω(x) ⊆ B̄(x, r)`. The upper bound kills
  `Ω(x)=X`; the lower ball supplies local actuation, a stay-option, and overlapping
  candidate states.
- **R1-H (typed / hybrid):** `Ω(x)` is a nonempty finite or compact menu generated by a
  pre-declared primitive library, contains a stay primitive, and lies within a declared
  one-step reach bound. Because typed menus do **not** contain a metric ball, overlap is
  a separate experimental coverage condition: Tier-1 claims may be made only on
  pre-registered paired contexts whose menus contain the same candidate states (or the
  same candidates under a fixed, outcome-independent equivalence). A recurrent menu
  graph alone does not establish Sen-α testability.

The earlier text incorrectly treated “stay + bounded step” in Minecraft as proof of the
annular lower bound. It establishes R1-H boundedness, not R1-C. Typed action labels whose
projected outcome changes with context are not automatically the same revealed-preference
alternative. **Kills:** singleton/per-trajectory menus while avoiding a false continuous
neighbourhood claim for discrete agents.

### 3.3 R2 — `D` from a declared class `[AXIOM]`
Two tiers:
- **Tier 1 (ordinal / nonparametric):** `D(·, x*; h)` is *some* fixed function, fixed
  before data. Tests the argmin form itself (Sen's α / congruence).
- **Tier 2 (parametric / rigid):** `D(y, x*; h) = φ(d(y, x*))`, `φ` strictly increasing
  and continuous. The update becomes metric projection toward `x*` within `Ω(x)`.
**Kills:** the indicator-`D` trick.

### 3.4 R3 — Cross-context invariance `[AXIOM]`
The same `(Ω, D)` must explain multiple trajectories, initial conditions, and
environments `E`. **Kills:** per-trajectory fitting. Identifiability of the
`(Ω, D | E)` factorization requires data across ≥ 2 environments (causal-style
identification).

### 3.5 R4 — Frozen reference `[AXIOM]`
`x*` is fixed (or autonomous and slow: `x*_{t+1} = g(x*_t)` with `g` independent of the
trajectory being explained). **Kills:** time-indexing smuggled through a drifting `x*`.
Drifting `x*` is a coupled skew-product system — explicitly a *second-paper* problem.

### 3.6 C1 / C1-T — Effective uniqueness `[AXIOM]`
The effective ordering has a **unique** minimizer on every reachable menu. This may be
obtained either by strict primary uniqueness (**C1**) or, for discrete menus, by a
pre-registered observable secondary rank `τ` and lexicographic minimization of
`(D, τ)` (**C1-T**). C1-T must be frozen before outcomes, depend only on declared
primitive/state information, and be logged with both the raw `D`-tie count and the
post-refinement tie count. An unresolved or post-hoc tie retains the weak semantics of
Theorem 3a and reopens universal rationalization. C1-T is therefore an explicit part of
the objective, not permission to silently take array order.

### 3.7 EB — `E`-blindness (the framework's identity) `[AXIOM]`
The selector sees neither `E`'s map, nor its statistics, nor its traps. Horizon-`h`
lookahead runs on `Ω` with an **internal model `E = id`** (Bellman recursion over `Ω`
only, path-cost `D_h`; see §6). Equivalently: `E` appears nowhere in the selection
recursion.

> **EB is the theory.** Lemma 7 and Theorem 9 show that *everything* distinguishing
> `(Ω, D)` Dynamics from MPC flows from this one commitment. Weakening EB to rescue a
> failing prediction dissolves the framework into generic stochastic control. Honest
> one-line framing of the whole object: **`(Ω, D)` Dynamics is MPC with a self-model and
> no world-model.**

**EB-L — the learning boundary (adopted 2026-07-11, from §13.5).** `[AXIOM]` EB
constrains *learning*, not only lookahead. **Legal:** learning about the agent's own
history — stagnation detection and tabu penalties on revisited own-states — plus
per-primitive reliability learned in calibration interventions that isolate the actuator
from `E`. A realized-vs-predicted `ΔD` from an ordinary run is jointly caused by actuator
and environment and **cannot** be labelled self-learning without that separation. These
legal additions enlarge the extended state with bounded buffers
(R0-compatible: fixed windows, fingerprinted into the state key) and refine the
*self*-model. **Illegal:** any outcome-based estimate of `E`'s dynamics (fluid flow, mob
behaviour, drop statistics) — that is system identification, which folds `E` into
selection and dissolves the framework into adaptive MPC exactly as the warning above
says; likewise any *within-run* retuning of `D`'s parameters (breaks R2/R3 and fitness
attribution — between-run search is the outer loop's job, §13.5(b)). One line: **learn
about yourself, never about `E`.** Honest limit: self-model learning re-ranks the
existing `Ω`; it cannot invent behaviour the primitives cannot express.

**Operational factorisation audit.** Before every empirical run, freeze and publish the
information channels entering `x`, `Ω`, `D`, the self-model and tie rule, plus the
projection sources used by every primitive. Environment maps/statistics, confounded
joint outcomes, and the outer scoring metric are forbidden on the selection side. The
executable schema is `minecraft-omega-d-lab/src/core/factorization.ts`; the checklist is
`minecraft-omega-d-lab/FACTORIZATION_AUDIT.md`. This makes moving `E` across the boundary
a declared model change rather than an interpretive choice after results are known.

### 3.8 GX — Goal channels: `Ω`-primitives or `x*`-deficit coordinates `[AXIOM]`
Invariant #2 (§1.3) forbids reward terms inside `D` but named only one legal goal channel
(primitives in `Ω`). The Minecraft lab (§13.5) exhibited a second: **enrich the reference
manifold** with additional coordinates and keep `D = φ(d(y, x*))` Tier-2. Discipline —
all four required:
1. every new coordinate is a **capped deficit** of the agent's own state (a distance,
   never a predicted benefit — the internal model must not know what the goal *buys*,
   only that its absence is distance from rest; this is EB applied to `x*`);
2. the enriched `x*` is **frozen before data** (R4) — targets are commitments, not
   tunable parameters;
3. goal weights are **dominated by the survival anchor**: `w · maxDeficit ≤ w_safety`
   for every goal rung, so goal pressure can never outbid a hazard term;
4. priority is expressed by **weight order (a cascade), never by hard gates** —
   lexicographic gating proved gameable in practice (§13.5) and is non-representable in
   principle (Theorem 10's obstruction).
When every coordinate reads 0, the argmin is the stay-option: **goal completion = rest
on the manifold** (observed in-lab: argmin = `wait` at `D = 0`, §13.5(a)). Violating (1)
or (3) is reward shaping in disguise and re-opens the §5 vacuity door.

**Interpretive limit.** In an embedded weighted metric, adding a deficit coordinate is
mathematically equivalent to adding its weighted term to a cost. GX is therefore
*disciplined and auditable cost shaping*, not a theorem that goals cease to be costs. Its
content comes from independent measurability, pre-registration, boundedness and the
survival-dominance constraint.

---

## 4. Persistence theory

Take `V(x) := d(x, x*)` as Lyapunov function. (Earlier drafts used `V = D(x, x*; h)`; the
final form below uses the metric, which is cleaner with Tier-2 `D` and the death set.)

### Theorem 1 — Persistence with death states `[PROVED]`
With `c := max{V(x_0), ε/(1−ρ)}` and sublevel set `S_c := {V ≤ c}`, assume:
- **(P1)** `D(y, x*; h) = φ(d(y, x*))`, `φ` strictly increasing (so `D`-argmin = `d`-argmin);
- **(P2)** for all `x ∈ S_c \ X_†`: `Ω(x)` compact and `min_{y∈Ω(x)} d(y, x*) ≤ α·V(x)`, `α ∈ [0,1)`;
- **(P3)** `d(E(z), x*) ≤ L·d(z, x*) + ε`, with `ρ := αL < 1`;
- **(P4)** `S_c ∩ X_† = ∅` (the relevant sublevel set is death-free).

Then the trajectory is defined for all time (never dies), and
$$
V(x_t) \;\le\; \rho^{t} V(x_0) \;+\; \varepsilon\,\frac{1-\rho^{t}}{1-\rho},
$$
with residual set `{V ≤ ε/(1−ρ)}` forward invariant.

*Proof.* Induct: `x_t ∈ S_c ⇒ x_t ∉ X_†` by (P4); minimizer `ŷ_t` exists by (P2)
(Weierstrass) with `d(ŷ_t, x*) ≤ α V(x_t)`; by (P3) `V(x_{t+1}) ≤ ρ V(x_t) + ε ≤ ρc + ε ≤ c`
since `c ≥ ε/(1−ρ)`, so `x_{t+1} ∈ S_c`. Unroll the affine recursion. Forward invariance:
`ρ·ε/(1−ρ) + ε = ε/(1−ρ)`. ∎

If `ε = 0` (`E` contractive toward `x*` in `V`), `V(x_t) → 0` geometrically: convergence
to the reference manifold.

**Sufficient condition for (P2) (selection decrement).** `[SKETCH]` `X` Hilbert, `x*`
closed convex, `V = dist(·, x*)²`, and the menu contains a partial step toward the
projection `Ω(x) ⊇ {x + s(P_{x*}(x) − x) : s ∈ [0, s̄]}`. Then `V(Ŝ(x)) ≤ (1−s̄)² V(x)`,
i.e. `α = (1−s̄)²`. Generalising this beyond Hilbert/convex `x*` (prox-regular `x*`,
Bregman geometry) is open.

### Theorem 2 — Sharpness; nothing is removable `[PROVED]`
(a) **Threshold.** `X = ℝ`, `x* = {0}`, `Ω(x) = {x/2}` (`α = ½`), `E(z) = 2z + 1` (`L = 2`,
`ρ = 1`): `V_{t+1} = V_t + 1`, unbounded; larger `L` gives exponential divergence. So
selection cannot rescue persistence against a sufficiently expansive environment — the
correct, non-teleological statement.
(b) **(P4) is genuine.** `x_0 = 1`, `Ω(1) = {1/2}`, `Ω(1/2) = ∅`, `E = id`. Then `ρ = ½ < 1`
yet the system dies at `t = 1` because `1/2 ∈ S_c ∩ X_†`. Contraction does not protect
against menu death. ∎

**Robustness.** With `η`-approximate selection the threshold `ρ < 1` is unchanged; the
residual inflates to `(ε + Lη)/(1−ρ)`.

> **Honesty note.** Theorem 1 does not use `h`; it implicitly uses `E`-blindness. §6 fixes
> both. The persistence result is standard ISS-style contraction analysis — its value is
> *definitional*: it pins "persistence" to robust forward invariance of a neighbourhood of
> `x*` with an explicit threshold `αL < 1` separating persistent from non-persistent
> regimes.

### Theorem 1-S — Persistence with random menus (first step on O4) `[PROVED]`
Setting: `h = 1`, Tier-2 (P1); `E` deterministic with (P3), `ε = 0`; menus `Ω_t(x)`
random, iid across `t`, a.s. nonempty compact, with, uniformly in `x`:
- **(S1)** a.s. `min_{w∈Ω_t(x)} d(w, x*) ≤ ᾱ·d(x, x*)` with `ᾱL ≤ 1` (expansion never
  exceeds the contraction budget);
- **(S2)** with probability ≥ `p > 0`: `min_{w∈Ω_t(x)} d(w, x*) ≤ α·d(x, x*)` with
  `ρ̄ := αL < 1`.

Then `sup_t V(x_t) ≤ V(x_0)` a.s. (so if `{V ≤ V(x_0)} ∩ X_† = ∅` the trajectory a.s.
never dies), `V(x_t) → 0` a.s., and `𝔼V(x_t) ≤ (1 − p(1−ρ̄))^t V(x_0)`.
*Proof.* Tier-2 selection attains the realized menu minimum, so `V_{t+1} ≤ ᾱL V_t ≤ V_t`
always and `V_{t+1} ≤ ρ̄V_t` on the good events; the good events stochastically dominate
iid Bernoulli(`p`), so their count `N_t → ∞` a.s. (Borel–Cantelli/SLLN) and
`V_t ≤ ρ̄^{N_t} V_0 → 0`; the expectation bound is the one-step conditional average. ∎
*Scope note.* This fixes the baseline only: **pure menu noise does not break Theorem 1**
provided (S1). One a.s.-allowed expansive step (`ᾱL > 1`) re-opens excursion-death
through (P4). Learned / state-dependent / adversarial `Ω` — the real O4 — remains open
and is still gated on the unresolved parts of O2. *(Proved here; elementary.)*

### Theorem 1-D — Tracking a slowly drifting reference (first step on O7) `[PROVED]`
Let `x*_{t+1} = g(x*_t)` be autonomous (the form already permitted by R4) with per-step
drift `d(x*_{t+1}, x*_t) ≤ δ` along the orbit. Selection at time `t` uses `x*_t`; assume
(P1), (P2) with `α` relative to `x*_t`, (P3) relative to `x*_t`, `ρ := αL < 1`, and that
the tube `{(x, t) : d(x, x*_t) ≤ c}`, `c := max{V_0, (ε+δ)/(1−ρ)}`, is death-free
(P4′). Then with `V_t := d(x_t, x*_t)`:
$$
V_{t+1} \le \rho V_t + \varepsilon + \delta,
\qquad
V_t \le \rho^t V_0 + (\varepsilon+\delta)\frac{1-\rho^t}{1-\rho},
$$
the system persists and tracks the moving reference within band `(ε+δ)/(1−ρ)`.
*Proof.* `d(x_{t+1}, x*_{t+1}) ≤ d(x_{t+1}, x*_t) + d(x*_t, x*_{t+1}) ≤ (ρV_t + ε) + δ`;
induct as in Theorem 1. ∎
*Scope note.* This is the easy half of O7 — drift enters as disturbance, Theorem 1 is
robust to it verbatim. The genuine second-paper content (skew-product spectral theory,
resonance between `g` and `E`, drift faster than contraction) remains open. *(Proved
here; triangle inequality.)*

---

## 5. Vacuity and identifiability (Tier 1 + Tier 2)

This is the heart of the framework's empirical status. **Until these are settled, every
simulation success is uninterpretable — it could always be vacuity in disguise.**

### Theorem 1ᵤ — Universal rationalization (unconstrained `Ω`) `[PROVED]`
Let `(x_t)` be any sequence with (i) `x_{t+1} ∈ range(E)` and (ii) functional consistency
`x_s = x_t ⇒ x_{s+1} = x_{t+1}`. Then there exist `Ω` with nonempty compact values and
**any** continuous `D` whatsoever reproducing the trajectory exactly.
*Proof.* Pick `y_t ∈ E^{−1}(x_{t+1})`, set `Ω(x_t) = {y_t}` (singleton; `D` irrelevant),
`Ω(x) = {x}` off-trajectory. ∎
- **Deflationary:** with unconstrained `Ω` the framework has zero empirical content.
- **Not deflationary:** condition (ii) — *determinism given `(x_t, x*, h)`* — is a genuine
  (weak) prediction; a system that revisits an extended state and does two different things
  refutes every fixed-`(x*, h)` deterministic model. (But R0 is needed, else an unobserved
  clock coordinate destroys even this.)

**Clock loophole (stronger).** `[PROVED]` Extend the state `X̃ = X × ℕ`, `Ω(x,t) =
X × {t+1}`, `D((y,t+1), x*) = 0` if `y = ξ_{t+1}` else `1`. Reproduces *any* target
sequence `ξ` with a singleton argmin at every step. Survives menu-richness; defeated only
by **R0**. Same degeneracy as Afriat/inverse-RL: optimization-rationalizability of behavior
is generically non-falsifiable without structural constraints.

### Theorem 2ᵣ — Rationalizability boundary (Richter/Sen) `[SKETCH]`
Fix the menu family `M = {Ω(x)}` in advance and fix `x*, h` (so `D(·, x*; h)` is one fixed
function inducing a fixed total preorder `⪯_D`). A selection `x ↦ Ŝ(x) ∈ Ω(x)` is realized
by *some* `D` **iff** the choice correspondence satisfies the **congruence axiom** over `M`;
in the single-valued case the key necessary condition is **contraction consistency (Sen's
α)**:
$$
\hat S(x) \in \Omega(x') \subseteq \Omega(x) \;\Rightarrow\; \hat S(x') = \hat S(x).
$$
*Necessity* is immediate from minimizing a fixed function; *sufficiency* is Richter's
preorder construction, realized by a real-valued `D` when `X` is Polish and menus are
regular (a separability assumption; in full generality you get a preorder but possibly no
real-valued `D` — see O1).
**Consequences:** (1) With fixed `x*, h` and pre-committed `Ω`, the framework is exactly as
falsifiable as Sen's α over **overlapping** menus; non-overlapping menus ⇒ vacuity returns.
**Overlapping menus are the entire empirical content of `D`.** (2) Only the *ordinal*
structure of `D` is identifiable from deterministic trajectories — "informational" form
claims are interpretation unless `h` is varied, stochasticity added (§7), or `D`-values
observed directly. (3) Unconstrained `x*(x_t)` re-introduces time-indexing (hence R4).

### Theorem 3a — Ties resurrect vacuity `[PROVED]`
Under **weak** semantics (`s_i` merely *a* minimizer) **every** dataset is rationalizable,
regardless of R0–R1. *Proof.* Quotient the revealed relation by cycles of its transitive
closure (cycle members → one indifference class), extend to a total order (Szpilrajn),
represent on countable data. ∎ **⇒ forces axiom C1.**

### Theorem 3b — Tier-1 characterization under C1 `[PROVED]` (countable data)
Under **strict** semantics define `s_i P b` for `b ∈ B_i \ {s_i}`. The data is
rationalizable **iff** the transitive closure of `P` is irreflexive (no `P`-cycles).
Minimal refutation: a **2-cycle** `s_i ∈ B_j, s_j ∈ B_i, s_i ≠ s_j` (a Sen-α violation).
*Sketch.* Necessity: unique-minimizer minimization forbids `P`-cycles. Sufficiency:
Szpilrajn-extend acyclic `P` to a strict total order, represent by rank. ∎

**Overlap supply.** Under R1-C, if `B̄(x, ε) ⊆ Ω(x)` and `X` is geodesic, then
`d(x, x') ≤ 2ε ⇒ Ω(x) ∩ Ω(x') ≠ ∅`. Any trajectory returning within `2ε` of a previous
state generates a Tier-1 test. Under R1-H this conclusion does not follow: identical
primitive labels may project to different candidate states. Typed experiments must
construct and report the paired-menu overlaps required by the test. Recurrent continuous
systems are consequently well-tested; recurrence of a typed state graph is not enough.

### Theorem 4 — Generic non-rationalizability (Tier-2 projection class) `[PROVED]`
Class: `X = ℝⁿ`, `x* = {0}` (wlog), `Ω(x) = B̄(x, r)` with `r > 0` unknown,
`D = φ∘d(·, x*)`, `E = id`. Closed-form selection (metric projection onto the menu ball):
$$
\hat y(x) = \big(1 - r/|x|\big)_+\, x,
\qquad x_{t+1} = \big(1 - r/|x_t|\big)_+\, x_t .
$$
(`φ` drops out entirely — ordinal identifiability made concrete.) Let `A_T ⊆ (ℝⁿ)^{T+1}` be
the rationalizable `T`-step trajectories. `A_T` is the image of a piecewise locally
Lipschitz map from an `(n+1)`-dim parameter domain `(x_0, r)`, so `dim_H A_T ≤ n + 1`; for
`n(T+1) > n+1` it is Lebesgue-null and meager. **Generic trajectories refute the model.**
*Proof.* Recursion is locally Lipschitz in `(x_0, r)` on each switching region `|x_t| ⋚ r`;
Lipschitz images don't raise Hausdorff dimension; compact null sets have empty interior. ∎

### Theorem 4b — Identifiability up to ordinal equivalence `[PROVED]` (`E` known)
(i) every nontrivial step identifies `r = d(x_t, x*) − d(x_{t+1}, x*)` once `x*` is known;
(ii) each trajectory lies on a ray through `x*`, so **two trajectories with non-parallel
rays identify `x*` uniquely** (rays meet only at `x*`); (iii) `φ` is identifiable **only up
to strictly increasing reparameterization** — no quantity of trajectory data constrains it
further. ⇒ claims that `D` is "informational" must cash out *ordinally* or through the
stochastic channel (§7).

### Theorem 4c — Single-environment identifiability with unknown affine `E` (O2, partial) `[PROVED]`
Class: `X = ℝⁿ`, `x*` a point, `Ω(x) = B̄(x, r)` (`r > 0` unknown), `D = φ∘d(·, x*)`
(`φ` unknown), `h = 1`, `E(z) = Az + b` unknown affine with `A` invertible. Suppose the
observed one-step map `F = E ∘ Ŝ` is known on `ℝⁿ` (rich transition data, **one**
environment). Then `(x*, r, A, b)` are uniquely identified; `φ` only ordinally
(Theorem 4b(iii) persists, as it must).
1. **Plateau ("dead zone").** For `|x − x*| ≤ r`, `Ŝ(x) = x*`, so `F ≡ Ax* + b` —
   constant. Outside, with `u := x − x*`,
   `DF = A·(I − (r/|u|)(I − ûûᵀ))` has eigenvalues `1` (radial) and `1 − r/|u| > 0`
   (tangential), hence is invertible: `F` is nowhere locally constant outside. The locus
   of local constancy of `F` is therefore exactly `B(x*, r)` — for *any* parameterization
   producing `F` — recovering `x*` (center) and `r` (radius).
2. **Affine part.** With `(x*, r)` known, `ŷ(x) = x* + (1 − r/|x−x*|)(x − x*)` is
   computable for exterior `x`; the exterior `ŷ`-image is `ℝⁿ ∖ {x*}`, so `n+1`
   transitions with `ŷ`-images in general position determine `(A, b)` uniquely. ∎

**Consequence.** Within the rigid Tier-2/ball class the `(Ω, D | E)` factorization is
identifiable from a **single environment** — contrary to the multi-environment
expectation recorded in O2. The identifying signature is the plateau (now signature 9,
§8): the menu radius is visible as a dead zone of the observed dynamics. **What remains
open in O2:** identifiability from a *single trajectory* (which may never enter the
plateau or sample too few directions), unknown menu shape, stochastic `E`, and the
unknown-model-class problem. R3 retains its role for Fold-2 separation (§7) — a distinct
question. *(Proved here; verified numerically in `open_problems_verification.py`.)*

---

## 6. Horizon semantics and separation from MPC

`h` did no mathematical work until given `E`-blind semantics. The only semantics compatible
with EB is lookahead over `Ω`-reachability **with internal `E = id`**.

### Definition — `E`-blind lookahead / path-cost `D_h` `[DEF]`
Internal `h`-paths from `y`: `y_0 = y`, `y_{j+1} ∈ Ω(y_j)`. Path-cost discontinuity
functional
$$
D(y, x^{*}; h) \;:=\; \min_{\text{internal } h\text{-paths}} \sum_{j=0}^{h-1} \varphi\big(d(y_j, x^{*})\big),
\qquad D := +\infty \text{ if all internal paths die before depth } h,
$$
equivalently the **horizon-consistency axiom**
$$
D(y, x^{*}; h) = \varphi(d(y,x^{*})) + \min_{z \in \Omega(y)} D(z, x^{*}; h-1),
\quad \min_\emptyset = +\infty,
$$
with `h = 1` recovering Tier 2. `E` appears nowhere — *that is* the formalization of EB.

### Theorem 8 — `h` does work (separation from greedy projection) `[PROVED]`
With `E = id`, an instance where `h = 1` dies and every `h ≥ 2` persists.
*Construction.* `X = ℝ`, `x* = {0}`, `φ = id`, `x_0 = 3`; `Ω(3) = {1, 2}`, `Ω(1) = ∅`,
`Ω(2) = {0}`, `Ω(0) = {0}`. `h=1`: `D_1(1)=1 < 2=D_1(2)` → pick `1` → death at `t=1`.
`h=2`: `D_2(1)=1+min_∅=+∞`, `D_2(2)=2` → pick `2 → 0` → persists. ∎ Pure `Ω`-trap
avoidance — the first place `h` carries content.

### Theorem 9 — `E`-blindness binds (separation from MPC at every horizon) `[PROVED]`
An instance where `(Ω, D; h)` dies for **every** `h ≥ 1`, while MPC with the same `Ω` and
the *true* `E` persists.
*Construction.* `x* = {0}`, `φ = id`; points `s, a, b, b', z` at distances `3, 1, 2, ½, 1`.
Menus: `Ω(s) = {a, b}`, `Ω(a) = {0}`, `Ω(b) = {b'}`, `Ω(b') = {0}`, `Ω(0) = {0}`,
`Ω(z) = ∅`. Environment `E(a) = z`, `E = id` elsewhere. Internal costs from `s`:
`D_h(a) = 1` for all `h`; `D_h(b) = 5/2` for `h ≥ 2`. Every horizon selects `a`; truth
`x_1 = E(a) = z` → death at `t = 1` for all `h`. MPC through true `E` costs the `a`-branch
`+∞`, selects `b`: `s → b → b' → 0`, persists. ∎

> **Corollary — the framework's sharpest falsifiable signature.** Post-selection dynamics
> predicts **arbitrarily good avoidance of `Ω`-traps as `h` grows, with
> horizon-independent blindness to `E`-traps**. A system observed to avoid an
> environment-trap undetectable through its own menu mechanism refutes the architecture at
> all horizons simultaneously.

### Theorem 12 — Horizon-`h` persistence (lookahead costs margin) `[PROVED]`
Path-cost setting, `φ = id`. With `c := max{V(x_0), ε/(1−ρ_h)}` and, for all `z ∈ S_c`:
`Ω(z) ≠ ∅` compact with `min_{w∈Ω(z)} d(w, x*) ≤ α d(z, x*)`, plus (P3); define
$$
\alpha_h := \alpha\,\frac{1-\alpha^{h}}{1-\alpha}, \qquad \rho_h := \alpha_h L .
$$
If `ρ_h < 1`, the trajectory never dies and `V(x_t) ≤ ρ_h^t V(x_0) + ε/(1−ρ_h)`.
*Proof.* Build a witness internal path contracting by `α` per step
(`d(y_j, x*) ≤ α^{j+1} V(x) ≤ c`, staying in `S_c`); then `D_h(witness) ≤ α_h V(x)`, and
the selected `ŷ` obeys `d(ŷ, x*) ≤ D_h(ŷ) ≤ α_h V(x)`; apply (P3) and induct. ∎
**Interpretation.** `α_h` is *increasing* in `h`: the guarantee **weakens** with horizon,
because path-cost minimization may trade first-step proximity for cheap *internal,
counterfactual* later steps that `E` never realizes. Limits: `h = 1` recovers Theorem 1
(`αL < 1`); `h → ∞` gives threshold `α < 1/(1+L)` vs greedy's `α < 1/L`. **Checkable
prediction:** infinite-horizon internal planners need strictly more menu-reachability to
persist against the same environment.

### Theorem 12b — Exact tightness of `α_h` (closes O6) `[PROVED]`
The threshold `ρ_h = α_h L` of Theorem 12 is **exact**, and `α_h` is the exact
worst-case first-step coefficient. For every `α ∈ (0,1)`, `h ≥ 2`, `L > 0` with
`α_h L > 1`, and every `δ ∈ (0, α_h − 1/L)`, there is an instance satisfying **all**
Theorem-12 hypotheses at every state (`ε = 0`, `X_† = ∅`) in which
$$
V(x_{t+1}) = L\,(\alpha_h - \delta)\,V(x_t) \quad \text{for all } t,
$$
hence `V_t → ∞` geometrically. Letting `δ → 0` shows no threshold below `α_h L` can be
weakened, and the per-step bound `d(ŷ, x*) ≤ α_h V(x)` in Theorem 12's proof is attained
up to `δ`.
*Construction (lure gadget).* Star metric: `d(p, q) = V(p) + V(q)` for `p ≠ q`, so
`d(p, x*) = V(p)` (every route passes through `x*`; menus finite ⇒ compact). **Spine**
state `s_v` (`V = v`) with menu `{f_v, g_v}`: the **lure** `f_v` at `V = (α_h − δ)v` with
`Ω(f_v) = {x*}` (advertised free continuation), and the **greedy head** `g_v` at
`V = αv` opening a forced chain `V = α^j v`. Internal costs from `s_v`:
`D_h(g_v) = α_h v` (the chain *is* Theorem 12's witness path — the hypothesis is honest,
the selector just declines it); `D_h(f_v) = (α_h − δ)v < D_h(g_v)`: the selector buys a
far first step against a counterfactual continuation. `E` is homothetic with exact
factor `L`: `f_v ↦ s_{Lv'}` (re-arming the gadget), `g`-chain nodes scale by `L`,
`x* ↦ x*` — (P3) holds with equality, `ε = 0`. (P2-h) holds at every node with the
stated `α`; C1 holds with margin `δ·v`. The realised trajectory is
`s → s′ → s″ → ⋯` with ratio `L(α_h − δ)` per step. ∎
**Corollary 12b-1 — lookahead divergence needs no `E`-trap.** Take `L = 1`: `E` is
*nonexpansive* — it never increases distance to `x*` (its only act is relabeling the
lure as a fresh spine at equal distance). If additionally `δ < α_h − α` and `α_h > 1`
(possible iff `α + ⋯ + α^h > 1`, e.g. `α > (√5−1)/2` for `h = 2`), then `h = 1` selects
the greedy head and persists (`V_t → 0` at rate `α`), while every `h ≥ 2` diverges at
ratio `α_h − δ > 1`. This strengthens Theorem 13: there, harm required an `E`-trap
(death); here `E` is benign and the harm is pure selector-side divergence — the direct
price of trusting internal counterfactual continuations. *(Proved here; verified in
`open_problems_verification.py`, cases A1–A3.)*

### Theorem 13 — Non-monotonicity in `h` (more lookahead can kill you) `[PROVED]`
An instance where `h = 1` persists forever and `h = 2` dies at `t = 1`.
*Construction.* `x* = {0}`, `φ = id`. Distances: `s = 3`, `a = 1`, `b = 0.9`, `z = 1`.
Menus: `Ω(s) = {a, b}`, `Ω(a) = {0}`, `Ω(b) = {b}`, `Ω(0) = {0}`, `Ω(z) = ∅`. Environment
`E(a) = z`, `E = id` elsewhere.
- `h=1`: `D_1(b) = 0.9 < 1 = D_1(a)` → select `b` → `s → b → b → ⋯` persists at `V = 0.9`.
- `h=2`: `D_2(a) = 1 + 0 = 1 < 1.8 = D_2(b)` → select `a` → `x_1 = E(a) = z` → death. ∎

**The complete horizon triangle:** `h` **helps** (`Ω`-traps, Thm 8) · `h` is **useless**
(`E`-traps, Thm 9) · `h` **hurts** (`E`-trap on the far-sighted branch, Thm 13). Greedy's
accidental safety in Thm 13 is the flip side of the `E`-blindness postulate. Theorem 12b
adds the quantitative panel: the help/harm boundary sits exactly at `α_h L = 1`, and
harm requires no `E`-trap at all (Corollary 12b-1 — divergence under a nonexpansive
environment).

---

## 7. Post-selection vs folded `E`

Two ways to "fold" `E` into selection, both genuinely different dynamics from
post-selection.

- **Fold-1** (absorb `E` into selection): `x⁺ = argmin_{Ω(x)} D̃`, so `x⁺ ∈ Ω(x)`.
- **Fold-2** (anticipating): `x⁺ = E(argmin_{y∈Ω(x)} D(E(y), x*))`.

**(a) Range separation (deterministic).** Post-selection forces `x⁺ ∈ E(Ω(x))`; Fold-1
forces `x⁺ ∈ Ω(x)`. Whenever `E(Ω(x)) ⊄ Ω(x)`, a single transition can refute Fold-1.

**(b) Order separation.** Even with matching ranges they differ unless `E` preserves
`D`-order. Counterexample: `X = ℝ`, `x* = 0`, `D(y) = |y|`, `Ω(x) = {x−1, x+1}`,
`E(y) = y − 1`. At `x = 0.5`: post-selection picks `−0.5` → lands `−1.5`; Fold-1
(`D̃ = |E(y)|`) picks `1.5` → lands `0.5`. Immediate divergence. (Equivalent 2-point
example: `Ω(x) = {−1, +2}`, `E(z) = z−3` → post-selection `−1 ↦ −4`; folded `+2 ↦ −1`.)

**(c) Equivalence criterion.** Post-selection and Fold-2 coincide on `Ω(x)` iff `E` is
`D`-monotone there (`D(y_1) ≤ D(y_2) ⇒ D(E y_1) ≤ D(E y_2)`), e.g. a `D`-isometry or a
contraction toward `x*` commuting with `D`-level sets. A post-selection model is
reproducible by a folded model iff (i) `E(Ŝ(x)) ∈ Ω(x)` and (ii) `x ↦ E(Ŝ(x))` satisfies
congruence; when (ii) fails, *no* folded `D̃` exists. (The "every post-selection model is a
folded model" converse needs `E` injective.)

### Theorem 5 — Fold-2 escapes the Tier-2 class (was Lemma 5) `[PROVED]`
`n ≥ 2`, `E(z) = Az + b` affine, `A` invertible, `AᵀA ≠ λ²I`. Then the folded objective
`G := φ(d(E(y), x*)) = φ(|A(y − c)|)`, `c := A⁻¹(x* − b)`, lies **outside** the entire
Tier-2 class `{ψ ∘ d(·, K) : ψ` strictly increasing, `K ⊆ ℝⁿ` closed nonempty`}` — for
*any* reference, point or set-valued.
*Proof.* The sublevel sets of `G` are the homothetic ellipsoids `c + A⁻¹B̄(0, s)`, with
inradius/circumradius ratio constantly `ζ = σ_min(A⁻¹)/σ_max(A⁻¹) < 1`. The sublevel
sets of `ψ∘d(·, K)` are the outer parallel bodies `K ⊕ ρB̄`. If `G = ψ∘d(·, K)`, the two
families coincide setwise, with levels exhausting `ℝⁿ` on both sides (`|A(y−c)| → ∞`).
If `K` is unbounded, every parallel body is unbounded while every ellipsoid is bounded —
contradiction. If `K` is bounded with circumradius `R`: `K ⊕ ρB̄` contains a ball of
radius `ρ` (around any point of `K`) and is contained in a ball of radius `R + ρ`, so
its inradius/circumradius ratio is ≥ `ρ/(R + ρ) → 1` as `ρ → ∞`, while the ellipsoids'
ratio stays fixed at `ζ < 1` — contradiction. Hence folding forces `AᵀA = λ²I`. ∎
*Detectability within one environment* is Theorem 11 (boundary minimizers expose the
level sets' Gauss map; concurrency of step-lines ⇔ sphericity); Theorem 5 supplies the
class-escape statement that Theorem 11's test certifies. *(Proved here; elementary
convex geometry. Discharges the §10 "Lemma 5 → theorem" item.)*

### Theorem 11 — Concurrency test (central-field) `[PROVED]`
`n ≥ 2`, menus `B̄(x, r)`, effective objective `G(y) = ψ(|A(y − c)|)`, `ψ` strictly
increasing, `A` invertible (Fold-2 with affine `E`; post-selection is `A = I`, `c = x*`).
For boundary minimizers let `L(x)` be the line through `x` and `ŷ(x)`. Then
$$
\text{all } L(x) \text{ concurrent} \iff A^{\top}A = \lambda^{2} I,
$$
and the common point is `c`.
*Proof.* Let `Q := AᵀA`; minimizing `(y−c)ᵀQ(y−c)` on `|y−x| ≤ r` gives Lagrange condition
`Q(ŷ−c) = μ(x−ŷ)`, `μ > 0` (`ψ` drops out — ordinal). (⇐) `Q = λ²I` ⇒ `ŷ ∈ [x, c]`, all
lines pass through `c`. (⇒) concurrency ⇒ parallelism `p − ŷ ∥ Q(ŷ−c)` on an open set ⇒
polynomial identity ⇒ holds on all of `ℝⁿ`; eigenvector insertion forces `p = c` and
`Q = λ²I`. ∎
**Corollary (three-menu refutation).** Within one environment: intersect two step-lines to
get candidate `x*`, test the third for incidence. Generic affine Fold-2 fails; post-selection
passes, with the common point *being* the `x*` estimator of Theorem 4b.

### Lemma 6 — Fold-1 range test (single transition) `[PROVED]`
Fold-1 forces step length `≤ r` always (`x⁺ ∈ Ω(x) ⊆ B̄(x, r)`). One observed step of
length `> r` refutes Fold-1; one observed `x⁺ ∉ E(B̄(x, r))` (with `E` known) refutes
post-selection. Indistinguishable only where `E(B̄(x,r)) ⊆ B̄(x,r)` (locally non-displacing
`E`). ∎

### Lemma 7 — Variance-blindness (the signature test) `[PROVED]`
`E(z, ω) = z + σ(z)W`, `W` zero-mean nondegenerate, `φ(u) = u²`. Folded anticipating
objective `G(y) = |y|² + σ(y)²·E|W|²`. Menu `Ω(x_0) = {y_a, y_b}` with `|y_a| = ρ_0 + δ,
σ(y_a) = 0` and `|y_b| = ρ_0, σ(y_b) = s > 0`:
- **Post-selection** picks `y_b` (smaller radius) for every `s` — blind to `σ`.
- **Fold-2** picks `y_a` whenever `(ρ_0 + δ)² < ρ_0² + s²E|W|²`.
So for `δ ∈ (0, δ̄(s))` the two models make **opposite deterministic predictions from a
single menu in a single environment**. ∎
*Degeneracy flag (exact unidentifiable class):* if noise is supported on `D`-level sets
(e.g. rotations about `x*`), `G ≡ φ∘d` up to constants and the test has no power. Separating
condition: `E`'s noise must move probability **across** `D`-level sets.

**Summary.** Post-selection vs Fold-1: separable per-transition (Lemma 6). Post-selection
vs Fold-2: separable within one environment by Theorems 5/11 (class escape) or Lemma 7
(variance), and across environments by R3. The `(Ω, D | E)` factorization is testable only
across environments **for Fold-2** — the causal-identifiability analogue.

---

## 8. Falsifiable signatures (the empirical content, one list)

Each can fail; together they are what make `(Ω, D)` Dynamics a distinct hypothesis rather
than relabeled control theory.

1. **Determinism** given `(x_t, x*, h)` (requires R0).
2. **Sen-α / 2-cycles** across overlapping menus (Tier 1) — refutes *every* `D`.
3. **Monotone descent** of `V` when a stay-option exists and `E = id` (lazy-candidate test).
4. **Step-length bound `r`** (Fold-1 refutation, per transition; Lemma 6).
5. **Concurrency of step-lines** ⇔ spherical effective metric; 3-menu refutation of affine
   Fold-2 (Theorem 11).
6. **Selection blind to noise statistics** (variance-blindness; Lemma 7).
7. **`Ω`-trap avoidance with `E`-trap blindness at every `h`** (Theorems 8/9 corollary).
8. **Lookahead-induced fragility** (Theorem 13) and **persistence margin shrinking in `h`**
   per `α_h` (Theorem 12); the threshold `α_h L = 1` is **exact**, and `h ≥ 2` can diverge
   under a nonexpansive environment where `h = 1` persists (Theorem 12b / Cor. 12b-1).
9. **Plateau / dead zone** (Theorem 4c): the observed one-step map is constant exactly on
   the menu ball `B̄(x*, r)` — even with unknown affine `E`, within one environment.
   Non-constancy inside the inferred ball refutes the Tier-2/ball architecture.
10. **Evolved-horizon shift** (population-level corollary of Theorems 8/9/13; registered
    2026-07-11, *before* the hostile-`E` run). In a benign-`E` world an outer-loop search
    over `(θ, h)` should select large `h` — internal lookahead is cheap and `Ω`-structure
    rewards it (observed: every wild-world champion chose `h = 6`, the maximum; §13.5(b)).
    Adding `E`-traps should drive evolved `h` **down**: `E`-blind lookahead buys nothing
    against `E`-traps (Theorem 9) and can actively harm (Theorems 13, 12b). If champions
    under `E`-trap pressure retain maximal `h` at no fitness cost, the horizon triangle is
    not shaping behaviour and this signature fails.

---

## 9. Results table (status at a glance)

| # | Statement | Status |
|---|---|---|
| 0 | Well-posedness (Berge + lsc) | sketch |
| 1 | Persistence: `ρ = αL < 1` + death-free sublevel ⇒ never dies, geometric bound | proved |
| 2 | Sharpness: fails at `ρ = 1`; death-freeness (P4) not removable | proved |
| 1ᵤ | Universal rationalization (unconstrained `Ω`); + clock loophole | proved |
| 2ᵣ | Rationalizability boundary = congruence / Sen-α (fixed `x*, h`) | sketch (O1 gap) |
| 3a | Ties ⇒ universal rationalizability (forces C1) | proved |
| 3b | Tier-1: rationalizable ⇔ strict revealed relation acyclic; min refutation = Sen-α 2-cycle | proved (countable) |
| 4 | Tier-2 rationalizable trajectories: null + meager (generic refutability) | proved |
| 4b | Identifiability: `r`, `x*` from two generic trajectories; `φ` only up to monotone reparam. | proved (`E` known) |
| 5 | Fold-2 escapes Tier-2 class for any reference set (parallel-body vs ellipsoid ratio) | proved |
| 6 | Fold-1 separation: step length `> r` refutes folding, per transition | proved |
| 7 | Variance-blindness: one designed menu separates post-selection from anticipating fold | proved (`φ = u²`) |
| 8 | `h=1` dies, `h≥2` persists (`Ω`-trap; `h` does work) | proved |
| 9 | All `h` die where true-`E` MPC persists (`E`-trap; framework ≠ MPC) | proved |
| 10 | Continuous rationalizability ⇔ closed-preorder extension on σ-compact metrizable `X` (via Levin); lex counterexample | proved (stated on cited domain) |
| 11 | Concurrency: step-lines concurrent ⇔ effective metric spherical; 3-menu refutation | proved |
| 12 | Horizon-`h` persistence: threshold `α_h L < 1`, `α_h = α(1−α^h)/(1−α)`; `h→∞` ⇒ `α < 1/(1+L)` | proved |
| 13 | Non-monotonicity: `h=1` persists, `h=2` dies | proved |
| 12b | Tightness: `α_h L` is the exact horizon-`h` threshold (lure gadget); divergence needs no `E`-trap | proved |
| 14 | Closed-consistency criterion on σ-compact metrizable `X`: rationalizable ⇔ no `P`-edge reversed in `R*` | proved (via Theorem 10) |
| 4c | Plateau identifiability: `(x*, r, A, b)` from one environment's map data (O2 partial) | proved |
| 1-S | Random-menu persistence under `ᾱL ≤ 1` (O4 first step) | proved |
| 1-D | Drift tracking within band `(ε+δ)/(1−ρ)` (O7 first step) | proved |

### Theorem 10 — Continuous rationalizability `[PROVED]` (Levin domain)
`X` σ-compact and metrizable, data `{(Ω(x_i), s_i)}`, `P` the strict revealed relation.
TFAE: (a) there is a
**continuous** `D : X → ℝ` with each `s_i` the unique minimizer on `Ω(x_i)`; (b) `P` is
contained in the strict part of some **closed** preorder on `X`.
*Proof.* (a)⇒(b): `x ⪯ y :⇔ D(x) ≤ D(y)` is a closed preorder whose strict part contains
`P`. (b)⇒(a): by **Levin's 1983 theorem** for closed preorders on a metrizable
σ-compact space, there is a continuous
`u` with `x ≺ y ⇒ u(x) < u(y)`; take `D := u`. ∎
**Scope correction (2026-07-11).** Earlier versions stated arbitrary Polish `X`. That is
broader than the cited theorem: Polish spaces need not be σ-compact. No claim is made here
for non-σ-compact Polish spaces without an additional representation theorem.
**Counterexample showing acyclicity is insufficient on uncountable data.** `X = ℝ²`,
`M_x = {(x,0),(x,1)}` selecting `(x,0)`; `N_{x,y} = {(x,1),(y,0)}` for `x < y` selecting
`(x,1)`. The strict relation is a sub-relation of the lexicographic order (acyclic), but a
real `D` would need uncountably many disjoint intervals `(D(x,0), D(x,1))` in `ℝ` —
impossible. The obstruction is topological (no closed extension exists).

### Theorem 14 — Closed-consistency criterion (closes O1) `[PROVED]`
`X` σ-compact and metrizable, `P` the strict revealed relation of the data. Let
`𝒯 := {R ⊆ X×X : R ⊇ P, R transitive and closed}` (nonempty: `X×X ∈ 𝒯`) and
`R* := ⋂𝒯` — the **smallest closed transitive relation containing `P`** (an
intersection of closed transitive relations is closed and transitive). TFAE:
- **(a)** a continuous `D` rationalizes the data with unique minimizers;
- **(b)** some closed preorder `⪯` has `P` inside its strict part (Theorem 10's condition);
- **(c) closed consistency:** `(x, y) ∈ P ⟹ (y, x) ∉ R*`.

*Proof.* (a)⇔(b) is Theorem 10. **(b)⇒(c):** `⪯ ∈ 𝒯`, so `R* ⊆ ⪯`; if `(x,y) ∈ P` and
`(y,x) ∈ R*` then `x ≺ y ⪯ x`, giving `x ≺ x` — impossible. **(c)⇒(b):** take
`⪯ := R* ∪ Δ` (`Δ` the diagonal). It is reflexive; transitive (case check, `Δ` neutral);
closed (`Δ` is closed in a metric square; finite union of closed sets). For
`(x,y) ∈ P ⊆ R* ⊆ ⪯`: applying (c) to a hypothetical `(x,x) ∈ P` shows `P` is
irreflexive, so `x ≠ y`; and `(y,x) ∉ R*` by (c); hence `¬(y ⪯ x)`, i.e. `x ≺ y`. ∎

**Checks.** (i) On finite or countable-discrete data `R*` reduces to the transitive
closure `tc(P)` (finite relations are closed), so (c) becomes acyclicity — Theorem 3b is
recovered exactly. (ii) The lexicographic counterexample above is correctly **rejected**:
`P` contains `(x,0) P (x,1)` and `(x,1) P (y,0)` for `x < y`; the transitive closure
contains `((x,1),(y,0))` for all `x < y`, and letting `x ↑ y` puts the limit
`((y,1),(y,0))` into `R*` — reversing the data pair `((y,0),(y,1)) ∈ P`. Closed
consistency fails, as it must.

**Operational residue (O1′).** `R*` is reached from `P` by (transfinitely) alternating
transitive closure and topological closure; what is still missing is a *finite-step* test
on raw data classes — the topological analogue of the Bossert–Sprumont–Suzumura
consistency test. *(Equivalence (b)⇔(c) proved here; (a)⇔(b) via Theorem 10/Levin.)*

---

## 10. Open problems

- **O1 — representation gap.** **Closed on σ-compact metrizable state spaces** by
  Theorem 14 (closed-consistency criterion: rationalizable ⇔ no `P`-edge reversed in
  `R*`). The non-σ-compact Polish case is not claimed. Residue **O1′**: a finite-step
  operational test for closed consistency on raw data classes (computing/approximating
  `R*`). `[CLOSED on cited domain; O1′ OPEN]`
- **O2 — identifiability with unknown `E`** (even affine). **Partially closed** by
  Theorem 4c: in the rigid Tier-2/ball/affine class, `(x*, r, A, b)` are identified from
  **one** environment's map data via the plateau signature — the multi-environment
  expectation was too pessimistic for this class. Still open: single-trajectory data,
  unknown menu shape, stochastic `E`, unknown model class. R3 remains necessary for the
  Fold-2 questions of §7. `[PARTIAL]`
- **O3 — persistence at `h ≥ 2`.** **Closed**: Theorem 12 (guarantee) + Theorem 12b
  (exactness of `α_h L`). `[CLOSED]`
- **O4 — stochastic / learned `Ω`.** First step done: Theorem 1-S (iid menu noise is
  harmless under `ᾱL ≤ 1`). Learned / state-dependent / adversarial `Ω` open; still
  gated on the unresolved parts of O2. **Empirical data point (2026-07-11, §13.5(b)):**
  outer-loop evolution of `D`'s parameters `(θ, h)` under a frozen scoring metric with
  per-generation controls is non-vacuous (the steering/scoring split keeps R2/R3 intact:
  the *scoring* metric is the declared one; the *steering* metric becomes the studied
  object). Learned `Ω` proper remains open. `[OPEN; first steps done]`
- **O6 — exact tightness of `α_h`** in Theorem 12. **Closed** by Theorem 12b (lure
  gadget; threshold `α_h L = 1` exact, divergence requires no `E`-trap). `[CLOSED]`
- **O7 — drifting `x*`** (skew-product / coupled system). First step done: Theorem 1-D
  (tracking band `(ε+δ)/(1−ρ)` for slow autonomous drift). The skew-product spectral
  theory (resonances, fast drift) remains the second-paper problem. `[OPEN; first step done]`
- **Lemma 5 → theorem.** **Done**: Theorem 5 (parallel-body vs ellipsoid
  inradius/circumradius argument, any closed reference set). `[CLOSED]`

---

## 11. Relationship to neighbouring frameworks

- **Revealed preference (Afriat, Richter, Sen).** Tiers 1–2 are exactly choice-theoretic
  rationalizability; vacuity and the Sen-α test are imported directly.
- **MPC / RL.** If `D(·; h)` becomes a Bellman value over the environment's response, the
  framework *is* MPC. EB (internal `E = id`) is the only thing keeping it distinct;
  Theorem 9 is the formal separation.
- **Lyapunov / ISS control.** Theorem 1 is standard input-to-state-stability contraction
  analysis; its role here is definitional, not novel.
- **Optimal transport / information geometry / entropic dynamics / constructor theory /
  typicality.** The broader causality project sits near these (see
  `Current_Working_Theory.md`); the `(Ω, D)` core does not yet add a theorem beyond them
  except via the EB-driven signatures.

---

## 12. Runnable artifacts in this project

- `theorem13_horizon_nonmonotonicity.py` — self-contained simulator of the `E`-blind
  horizon-`h` dynamics; reproduces Theorem 13 (`h=1` persists at `V = 0.9`, `h=2` dies at
  `t=1`) with assertions on the internal `D_h` costs. The general `D_h` Bellman recursion
  and the post-selection loop in that file generalize to Theorems 8, 9, 12.

- `open_problems_verification.py` — verification for the 2026-06-12 open-problem session.
  Part A: the Theorem 12b **lure gadget** (star-metric graph, exact `D_h` Bellman
  recursion, per-node hypothesis assertions) — confirms Corollary 12b-1 (`α=0.7, h=2,
  L=1`: `h=1` persists, `h=2` diverges at ratio 1.10), threshold separation
  (`ρ_1 = 0.9 < 1 < 1.251 = ρ_3`), and the guarantee side (`ρ_3 = 0.744 < 1`: Theorem 12
  bound respected). Part B: Theorem 4c plateau identifiability (`F` constant exactly on
  `B̄(x*, r)`; `(A, b)` recovered to `1e-9` from one environment). All claims are
  assertions; exit 0 ⇔ pass.

- `minecraft-omega-d-lab/src/benchmarks/serviceRecovery.ts` — non-Minecraft
  self-stabilising service benchmark. On the same declared menus it runs greedy `h=1`,
  E-blind `h=2`, true-`E` MPC, frozen, seeded-random and argmax controls across an
  `Ω`-trap, an `E`-trap and a horizon-fragility incident.

- `minecraft-omega-d-lab/src/benchmarks/behavioralTrap.ts` — controlled black-box
  fingerprinting probe combining Sen-α, variance, `Ω`-trap, `E`-trap and horizon tests.
  It classifies the included subjects as greedy E-blind, lookahead E-blind, or
  environment-anticipating. `npm run benchmark` runs both suites; the assertions live in
  `test/serviceRecovery.test.ts` and `test/behavioralTrap.test.ts`.

- `minecraft-omega-d-lab/src/core/factorization.ts` and `FACTORIZATION_AUDIT.md` —
  executable EB/EB-L information-flow declaration and the human audit protocol.

Suggested next artifacts (not yet built): the Theorem 11 three-menu concurrency estimator
of `x*`; an O1′ finite-data `R*` approximator; and external replications of the behavioral
probe with agents not authored to match any of the three classifier categories.

---

## 13. Empirical probes and implemented labs

Two browser/Node testbeds were built to demonstrate the principle in motion. Both are
useful — one as corroboration, one as a cautionary negative result. **Neither should be
read as the canonical `D`; both deviate from the formal framework in documented ways.**

### 13.1 Pool lab (`dynamic-similarity-pool-lab hardcore/`)

A 2D billiards comparison: a normal-physics table vs an experimental table running
`x_{t+1} = E(argmin_{y∈Ω} D_info(y, x_t))`. `Ω` = typed candidate families (inertial,
wall-reflect, collision-response, dissipative, noise variants); `D_info` = a weighted sum
(smoothness, energy, momentum, collision, wall, history, MDL); `E` = table dissipation
applied *after* selection (`applyTableDissipation`). Core file: `src/physics.ts`
(`generateExperimentalCandidates`, `scoreCandidateDInfo`, `stepExperimentalPhysics`).

**Corroborations (independent of the analytic thread):**
1. **`E` post-selection is correct (EB).** The lab's own conclusion: folding friction into
   `D` "confuses" the model; dissipation belongs after selection. Reached empirically.
2. **Content lives in `Ω`, not `D`.** Ablation: remove the wall-reflect family → balls
   leave the table; remove collision-response → balls tunnel. `D` ranks but cannot invent
   an absent event type. (Empirical face of R1 and "content is in the restrictions.")
3. **Goal terms inside `D` are brittle (invariant #2).** Effects mode adds a separation
   objective inside `D` and needs four hand-coded gates to behave — evidence that goals
   should enter via `Ω`, not as reward terms in `D`.
4. **Auditable failure** is a practical advantage: failures trace to either `Ω` incomplete
   or `D` mis-ranking, unlike a black-box policy.

**Gaps (do not over-read the code):** `D_info` is a rich hand-weighted functional, not
Tier-2 `φ(d(y,x*))` (weights are tuned → R3 risk); there is **no explicit `x*`** (it scores
discontinuity from the current state, not distance to a maintained reference); **`h` is not
implemented** (greedy `h=1` only); the `dissipative` candidate family slightly leaks `E`
into `Ω`; stochasticity is modelled as noise *candidates in `Ω`*, not as stochastic `E`
(so Lemma 7's variance-blindness test is not exercised). Net: a **successful negative
result** — pure `D` does not rediscover physics; `(Ω, D) + E` with post-selection `E` does
produce causal-looking behaviour.

### 13.2 Stickman lab (`stickman-(ω,-d)-lab/`, outside the project folder)

A 2D homeostatic figure comparing FSM+PD control against an `(Ω, D)` controller. This is
the intended successor (internal state: posture, energy, balance). Mapping to the framework
is faithful *on paper*: `x*` is an explicit **reference manifold** (`src/reference.ts`:
upright torso, balanced CoM, energy = 100, neutral joints); `Ω` = behavioural primitives
(`src/primitives.ts`: stance-hold, idle-sway, walk-step, step-recovery, arm-windmill,
crouch, get-up, reach, seek-food), gated by an affordance function; `D` = horizon-`h`
projected homeostatic cost (`src/scoring.ts`: postural + balance + joints + energy + MDL),
selected by `argmin` (`src/selector.ts`); `E` = `stepPhysics`. Goals enter via `Ω`
(`seek-food` is a primitive, not a reward term) — exactly as the pool-lab lessons advised.

> **This is the "cheating" case, and it is the framework's §5 vacuity warning realised in
> code.** The visible competence (standing, walking, push-recovery) is produced by
> **post-selection servos inside `E`**, not by `argmin`-`D` selection.

**What the servos do.** `stepPhysics` applies `applyQuietStanceServo`,
`applyLocomotionGaitServo`, and `applyOmegaRecoveryServo` — all defaulting **on** — which
kinematically *snap* body nodes toward hardcoded floor-relative target poses (e.g. the
locomotion servo teleports the whole body toward the goal and plants the feet; the quiet
servo blends the torso/head to the upright target). Three compounding problems:
- **The servos do the actual work.** Selecting `walk-step` or `step-recovery` is cosmetic;
  the servo performs the locomotion/uprighting regardless of dynamics.
- **They run inside the projection too** (`projectPrimitive` passes the same gates on by
  default), so `D` scores servo-rescued futures — every candidate looks fine because the
  servo stabilises it. `D` is not ranking intrinsic dynamics.
- **Asymmetry favouring `(Ω, D)`.** `applyOmegaRecoveryServo` is gated
  `controllerMode === "omega" || enableRecoveryServoForFsm`, and the FSM flag defaults
  **false** — the experimental controller gets a recovery servo the baseline does not.
  The repo's own EXP-5 comment calls this post-selection layer **"contaminated."**

**The recorded clean verdicts** (servo-stripped probes already in the repo;
`experiments/`, `out/`):
- **EXP 6 — clean force-only** (`cleanGates`: all servos off, every `rootDriveX = 0`):
  `"Claim B fails… no clean fixed-posture Omega regime sustained upright standing for
  10s."` The figure **falls in ~0.18 s** in *every* regime — horizons 4/12/24, both the
  6-primitive and enriched 20-primitive menus. Selection-on buys ~16 ms over a frozen
  neutral pose, and mostly just picks `idle-phase0`.
- **CoM-balance bridge** (`out/com-balance-summary.json`): verdict **`NOT_REALIZABLE`** —
  idle not held, push-recovery rate ≈ 3.3 %, dominant failure `joint_fold`. A principled
  map from selection to real (torque-limited) joint forces does not close the balance loop.
- **Yet the plant is capable:** `static-balance-admissibility` → `STATIC_BALANCE_ADMISSIBLE`
  and `contact-moment` → `FOOT_SPAN_AVAILABLE`. Standing is geometrically possible; the
  failure is in the controller, not the body.

**Lesson — stronger and more damning than the pool lab.** The pool lab showed `D` cannot
invent a missing family. The stickman shows that **even when `Ω` contains the right
primitives, `argmin`-`D` over them does not realise the stabilising forces balance
requires** — the "bridge" from selection to actuation is `NOT_REALIZABLE`, and all visible
success was kinematic authoring by servos in `E`. This is precisely Theorem 1ᵤ / the
vacuity boundary in physical form: when `E` (here, hidden servos) does the work, the
`D`-selection is decorative and unfalsifiable. Any future demo must run **servo-stripped**
(the `cleanGates` config) before any competence is attributed to `(Ω, D)`.

> **Open question this raises for the framework.** Persistence Theorem 1 assumes a
> *selection decrement* (P2): some candidate in `Ω` is genuinely closer to `x*` under the
> real `E`. The stickman result is an empirical instance where that assumption fails for a
> realistic plant — no admissible fixed-posture primitive, ranked by `D`, yields a
> contraction toward upright once `E` is honest. Whether an `E`-blind selector can ever
> realise closed-loop balance, or whether balance intrinsically needs the within-step
> feedback the framework forbids (that would be MPC, not `(Ω, D)`), is the sharpest open
> physical question the labs surface. Logged against EB and the empirical side of (P2)
> (the mathematical horizon-persistence question O3 is now closed — Theorems 12/12b —
> but says nothing about whether a real plant *supplies* the selection decrement).

### 13.3 Coherence retry on the repaired plant (EXP 7, 2026-06-12)

**The earlier stickman verdicts were artifacts.** A diagnostic pass (this session)
found five compounding defects in the lab's `E`, each sufficient to invalidate
EXP 1–6, the com-balance bridge, and the FSM baseline alike:

- **F1 — inconsistent canonical initial state.** `createInitialStickman` violates its
  own bone lengths (thigh by 6.1 px; pelvis–ankle chain overstretched 3.2 px; hands
  11 px): every run began with a violent constraint-crunch transient.
- **F2 — metabolic meter reads solver jitter as work.** The battery hit the ≤10
  hard actuation cutoff within frames at any load-bearing gain; every controller
  silently became a ragdoll. (Fix: signed net-work meter, `metabolicNetWorkMeter`.)
- **F3 — three irreconcilable rest geometries** between `solveBones`,
  `enforceFootPlate`, and the always-on `enforcePlantedFootCoupling`: the skeleton
  collapsed even at zero gravity with zero actuation (perpetual constraint fights).
  (Fix: `consistentFootGeometry`.)
- **F4 — foot-plate velocity corrector dynamically unstable** for the light heel/toe
  nodes (sub-pixel disturbance → ±32 px/f within one frame). (Fix: `rigidPlantedFeet`
  contact idealization; body above the ankles stays fully dynamic.)
- **F5 — THE ROOT DEFECT: inverted actuation sign.** `applyJointTorque` measures
  angles in y-down screen coordinates but applies y-up torque cross-products:
  PD drove every joint exponentially AWAY from target, `kd` anti-damped, and joint-limit
  penalties pushed outward (verified: command +0.2 rad → −0.86 within 12 frames).
  **Every prior experiment in the repo ran with anti-restoring actuators**, i.e. in the
  Theorem-2a regime `αL ≥ 1`, where no selector can win — so EXP 6's "Claim B fails"
  measured the plant, not the selector. EXP 6.0c's own "do not run EXP 6" flag was right.

**EXP 7** (`src/runCoherenceStandExp7.ts`, controller in `src/coherence/`) then tested a
framework-faithful controller on the repaired plant: Tier-2 `D = d(y, x*)` over a purely
geometric intent model (controller never imports physics — EB honored; `h = 1`, justified
since `Ω(x) ≠ ∅` always ⇒ no `Ω`-traps, Theorems 8/12b/13); bounded typed R1-H menu with
stay-option — `stay`, `toward-reference` (the Theorem-1 (P2) partial-step-toward-
projection family), `ankle-lean`, `knee-fold`, `hip-hinge`, `arm-swing`; frozen manifold
`x*` = {torso vertical, CoM over support, CoM at canonical height} (R4); one declared
metric and stiffness profile for all runs (R2/R3); C1 audited (0 ties); all pose servos
off; `E` strictly post-selection; deterministic (verified by trajectory hash).

**Results.** Quiet standing 10 s: `argmin`-`D` **stands** (median `V` 0.36, beating the
frozen statue's 0.46); the **argmax anti-selection control falls in 0.35 s with the same
menu** — selection, not menu richness, does the work (the §5 vacuity worry is empirically
discharged for this instance). `stay` (pure damping) and ragdoll fall in <0.4 s. Push
recovery: both `argmin` and frozen recover a 0.5-impulse push and fail at 1.0 (omega
survives ~0.6 s longer); so far **selection matches but does not yet beat a stiff statue
on disturbance rejection** — the honest current boundary of (P2)'s empirical support.
A menu with step/CoP-shift authority (foot relocation is impossible under the
`rigidPlantedFeet` idealization) is the natural next probe.

**Status of the §13.2 open question.** The earlier claim "no admissible fixed-posture
primitive yields a contraction toward upright once `E` is honest" is **withdrawn**: it
was an artifact of F1–F5. On the repaired plant the selection decrement (P2) is supplied
and realized (`αL < 1` empirically: geometric `V`-descent in 48% of quiet steps, residual
band maintained for the full window). Whether `E`-blind selection can beat within-step
feedback on *disturbance* rejection remains open — but the question is now testable.

### 13.4 Minecraft lab (`minecraft-omega-d-lab/`) — the trap/horizon testbed `[IMPLEMENTED; M1 DISCHARGED]`

Full design rationale and the M1–M5 experiment definitions live in
`Omega_D_Dynamics_S13_4_Minecraft_Lab.md` (the `[PROPOSED]` drop-in, now superseded by
this section's status). Built in Mineflayer (TS/Node) from 2026-06-20; iterated through
worklog sessions S1–S25. Mapping: `x` = position/facing, health, hunger, inventory + a
*static* snapshot of locally visible blocks; `x*` = the survival manifold (later
enriched, §13.5); `Ω` = heading-relative typed primitives with the stay-option always
present (R1-H: finite declared library + one primitive's world-change per tick; no
annular lower-ball claim); `D` = one declared weighted-L¹
Tier-2 metric, health-anchored (hunger costs only the health it risks); `D_h` = the §6
path-cost recursion with internal `E = id`; `E` = the live game tick, strictly
post-selection. EB firewall in `src/minecraft/state.ts`: the selection path reads own
state and static terrain only — never live entities, fuses, falling blocks, or fluid flow.

**Recorded results (offline deterministic sim + live server):**
- **Selector core validates the theory in-repo.** Unit tests reproduce Theorem 8 (`h=1`
  walks into an `Ω`-trap every `h ≥ 2` escapes), Theorem 13 (`h=2` dies where `h=1`
  survives), and the Theorem 9 corollary (`E`-trap blindness at every `h`) on explicit
  graphs — 180/180 passing after the external-validation additions. Any live demo
  inherits its interpretability from these.
- **M1 discharged, offline and live.** `argmin` holds the survival manifold; the
  `argmax` anti-selection control on the *same* menu starves (health drains to 0) or
  steps off the ledge; frozen/random decay. The §5 vacuity test passes in a second
  domain (after EXP 7), with C1 tie audit, seeded determinism, and a `trajectoryHash`
  per run (signature 1).
- **Declared EB relaxation, quarantined.** The practical `survivor` agent widens
  perception to *static* water/lava block positions and own air level — documented as a
  perception widening, not part of the strict experiments. The strict `viability`/`m1`
  agents remain byte-identical (regression-tested by hash) — the practical/strict split
  is itself a useful pattern: engineering pressure is absorbed by a declared variant
  instead of silently corroding the experimental agent.

**Still open — the lab's stated headline.** M2 (`Ω`-trap `h`-sweep), M3 (`E`-trap
blindness vs a true-`E` MPC reference), M4 (in-world non-monotonicity), and M5
(black-box classification) are designed (`src/arenas/`) but **not yet run in-world**;
signature 7 is so far exercised only at unit-test scale. The hostile-`E` evolution run
(signature 10) is the cheapest route to M3/M4-grade evidence.

### 13.5 `x*`-ladder and the evolution outer loop (S13–S25; runs 2026-07-02 arena, 2026-07-11 wild) `[RECORDED]`

Two extensions built on §13.4, both leaving the argmin rule, `Ω`-feasibility, the EB
firewall, and the strict agents untouched (verified by byte-identical strict trajectory
hashes). Design doc: `EVOLUTION_SPEC.md`; operation: `HOWTO-EVOLVE.md`.

**(a) Goal ladder via enriched `x*` (the GX pattern, §3.8).** `x*` v2 = survival
manifold ⊕ capped deficit coordinates {tool tiers, armour tiers, enchantments, resource
stocks}, priorities as a weight-order cascade (survival ≫ hazards ≫ hunger ≫ stocks ≫
tools ≫ armour ≫ enchant chain), every rung bounded by `w · maxDeficit ≤ fallPenalty`.
**Result (seeded arena, offline, deterministic):** the default agent completes the full
chain wood → stone → iron → diamond → enchanted kit and comes to **rest at `D = 0` by
tick ≈ 540**, argmin = `wait` thereafter — goal pursuit as pure descent to a frozen
manifold, terminating in rest. Vacuity contrast: champion mean `D_score ≈ 0.56` vs
`argmax`/frozen ≈ 41 on the same seeds; a rungs-off control never kits up. Hard
safe-and-fed *gates* were tried first and rejected as gameable — the empirical face of
Theorem 10's lexicographic obstruction, and the origin of GX rule 4.

**(b) Evolution over `(θ, h)` — the first O4 data point.** Outer (μ+λ) ES (pop 16,
elitism, log-normal mutation clipped to the GX-feasible region) over the ~36 metric
weights `θ` and `h ∈ {2..6}`; nothing else varies — the genome changes how deficits are
*weighed*, never what they are (one shared `x*`, or fitness is incomparable).
**Load-bearing design element — the steering/scoring split:** bots *select* with their
own `D_θ`, but fitness is a **frozen canonical `D_score`** computed from recorded traces,
identical for every genome and control; without the split the degenerate `θ → 0` genome
reports `D = 0` and "wins" by defining its deficits away. The scale gauge is pinned by
frozen anchor weights — argmin's invariance to positive rescaling of `D` (the ordinal
identifiability of Theorem 4b(iii)) surfacing operationally as a redundant genome
dimension. Per-generation `argmax` + frozen controls (the vacuity gate, not decoration);
fully deterministic replay (`evolution/gen-NNN.json`).
**Results.** Wild-world run (2026-07-11, 8 generations, procedural island, scan-limited
perception, resource memory): champions ≈ 1.25 mean `D_score` vs controls ≈ 18.6 — the
vacuity gate passes decisively — but improvement over the shipped defaults is modest
(≈ 1.28 → 1.25; the defaults were already near-competent, and per-generation seed
re-rolls make cross-generation fitness only loosely comparable). **Every champion
selected `h = 6`, the maximum**, in this benign-`E` world (no mobs, permanent noon) —
consistent with the Theorem-8 half of the horizon triangle and the basis of
signature 10. Arena run (2026-07-02, 12 generations): best ≈ 0.56 vs controls ≈ 41,
same shape. **Honest gaps:** the `threat` gene is unexercised in the benign world and
drifts on mutation noise; the deep chain (iron+) is unreached in 700 wild ticks
(arena-only so far); within-run learning (EB-L, §3.7) re-ranks `Ω` but cannot invent
behaviour outside it; live runs are showcase-only, never fitness.

## 14. How to edit this document

This file is meant to be revised as the framework develops. Conventions:

1. **Keep the status tags accurate.** When a `[SKETCH]`/`[OPEN]`/`[CONJECTURE]` is
   discharged, change the tag, move the entry's row in §9, and update §10.
2. **Notation is centralized** in §1.2 — change it there first, then propagate.
3. **Axioms are load-bearing.** Before weakening any of R0–R4, C1, EB, check what vacuity
   loophole it closes (cross-referenced in §3 and §5). Weakening **EB** in particular
   collapses the framework into MPC — flag any such change explicitly.
4. **Every quantitative claim should stay falsifiable.** New results belong in §8 (the
   signature list) if and only if they can fail against data.
5. **Counterexamples are part of the theory** — do not delete the failure catalogue (§2.1)
   or the sharpness examples (Theorem 2); they pin the definitions.
6. **Provenance.** New theorems should note whether they were *proved here*, *adapted* from
   a named result (Richter, Sen, Szpilrajn, Berge, Levin, Kuratowski–Ryll-Nardzewski), or
   *conjectured*.

### Core warnings for the next worker
- Do **not** fold `E` anywhere (Theorems 9, 11; Lemma 7 are the theory).
- Do **not** relax effective uniqueness: use strict C1 or a frozen, logged C1-T
  refinement. Unresolved or post-hoc ties reopen Theorem 3a's vacuity.
- Do **not** let `x*` drift yet (R4; it is a coupled skew-product, second paper).
- **Simulation results are uninterpretable except through the signature list (§8)** — that
  is the whole point of doing the analysis analytically.
- **Trust no demo with `E`-side assists on.** The stickman lab (§13.2) looked competent
  only because post-selection servos in `E` did the work; stripped of them (`cleanGates`)
  it falls in ~0.18 s. Always run servo-stripped before attributing behaviour to `(Ω, D)`.
- **Trust no negative verdict before validating the plant.** The converse error (§13.3):
  the "falls in ~0.18 s" result itself was an artifact — inverted actuators, an
  inconsistent initial state, and a jitter-reading energy meter put every earlier
  experiment in the `αL ≥ 1` regime where Theorem 2a says no selector can win. Before
  testing the selector, verify a selector-free statue can stand (`A-frozen`), and run
  the `argmax` anti-selection control alongside (`src/runCoherenceStandExp7.ts`).
- **Learning may target only the self-model** (EB-L, §3.7): actuator reliability,
  stagnation, tabu over the agent's own history. Any outcome-based estimate of `E`'s
  dynamics is folding, whatever it is called; and any *outer-loop* search over `D`'s
  parameters must keep the steering/scoring split with controls (§13.5(b)), or it
  optimizes the ruler instead of the bot.
- **Goals enter via `Ω` or via GX-disciplined `x*`-coordinates (§3.8) — never as reward
  terms in `D`.** Capped deficits, frozen targets, safety-anchor-dominated weights,
  cascades not gates.

---

## 15. Application roadmap and validation standard

The framework's strongest present role is a **falsifiable architecture or null model**,
not a general replacement for controllers that need a world model. Applications are
ranked by whether they can expose the EB signatures cleanly.

| Priority application | `x / Ω / x* / E` mapping | Intended use |
|---|---|---|
| Behavioral experiments | internal state / offered actions / physiological setpoint / hidden consequences | Separate greedy homeostasis, E-blind lookahead and environment anticipation with designed menu/trap interventions. |
| Black-box agent classification | carried state / controlled candidate menus / declared target / experimental outcome map | Use Sen-α, variance and trap fingerprints to classify an unknown policy architecture. |
| Self-stabilising software | service telemetry / restart, isolate, shed, replicate / healthy service envelope / workload and failures | Auditable local recovery; implemented baseline in §12. |
| Certified fallback control | machine telemetry / verified actuator macros / safe operating envelope / unmodelled disturbance | Conservative runtime-assurance layer, not the primary planner. |
| Cellular regulation | metabolic state / regulatory responses / viable region / external stressor | Falsifiable non-anticipating null model under laboratory perturbation. |
| Evolutionary artificial life | inherited `(θ,h)` / lifetime primitives / viability manifold / ecological response | Test whether hidden `E`-traps drive evolved horizons downward. |
| Interpretable simulated agents and swarms | local needs / typed behavior or communication primitives / viable individual or formation manifold / world tick | Debuggable action selection with explicit missing-menu versus misranking failures. |

Strict EB is a poor primary architecture where anticipating consequences is essential —
autonomous driving, trading and medical dosing are examples. There it is suitable only
as a controlled baseline or independently certified emergency layer.

**Minimum validation standard.** Every application must publish a passing factorisation
manifest; pre-register R1-C or R1-H and any paired-menu overlaps; log raw and effective
ties; run E-blind `h=1` and `h≥2`, true-`E` MPC, frozen, seeded-random and argmax on the
same menus; and report failures as well as successes. Support requires the predicted
pattern—Sen-α consistency where overlap exists, variance blindness, `Ω`-trap avoidance,
`E`-trap blindness and possible horizon fragility—not merely low `D` in one authored
simulation.