The loop, stated precisely.

Def. 1.1 State space

Let \(S\) be the state space: the set of all configurations a governed system may occupy at a given time. A state \(x \in S\) carries whatever is needed to describe the system completely at that instant — no external memory is assumed.

Def. 1.2 Evidence store

Let \(E\) be the evidence store: the accumulated record of what has been observed, measured, or logged about the system's history. \(E\) is a set (or append-only sequence) of evidence items; it is distinct from \(S\) because a system can change state without forgetting what happened.

Def. 1.3 Action

An action is a map \(a: S \to S\). Applying \(a\) to a state produces a new state. Actions are the only way \(S\) changes; nothing outside an action is permitted to mutate state.

Def. 1.4 Snapshot

A snapshot \(\sigma\) is a recorded copy of a state \(x \in S\) taken before an action is applied, together with enough context to attempt reconstruction of \(x\) from \(\sigma\) alone. Snapshots are evidence items: taking one adds to \(E\), it never modifies \(S\).

Def. 1.5 The AVPT cycle operator

The AVPT cycle operator is the composite map

\[ T \;=\; \mathrm{Track} \circ \mathrm{Preserve} \circ \mathrm{Validate} \circ \mathrm{Attempt} \]

Read right to left, as function composition: a cycle first Attempts an action, then Validates its result, then Preserves the evidence produced, then Tracks the outcome into the record. \(T\) is the single object every later section builds on.

Def. 3.1 The labeling map

Let a causal claim be decomposed into terms \(\{c_1, c_2, \dots, c_n\}\). RFPA defines a labeling map

\[ \ell : \{c_1, \dots, c_n\} \to \{\text{measured}, \text{inferred}, \text{assumed}\} \]

that classifies every term of the claim into exactly one of three classes: measured (directly observed), inferred (derived from measured terms by a stated method), or assumed (asserted without direct support). Recover, Field, and Principal name the roles terms play in the claim; Assumption names the class a term falls into when it cannot be recovered from the field or from a principal source.

Inv. 3.2 Domination condition

Partition the labeled terms into \(M = \ell^{-1}(\text{measured})\), \(I = \ell^{-1}(\text{inferred})\), and \(A = \ell^{-1}(\text{assumed})\). A change built on the claim is admissible only when

\[ A \preceq M \]

meaning the assumed set is dominated by the measured set under the claim's stated ordering — informally, what is asserted without support may never outweigh what has actually been observed. A claim that fails this condition is not blocked from being stated; it is blocked from being acted on.

Def. 4.1 Type 1 — invertible

An action \(a\) is Type 1 if an inverse \(a^{-1}\) exists such that \(a^{-1}(a(x)) = x\) for all reachable \(x\), and applying \(a^{-1}\) is cheap — bounded cost, no external dependency.

Def. 4.2 Type 2 — invertible through snapshots

An action \(a\) is Type 2 if no direct \(a^{-1}\) exists, but recovery is possible through a snapshot taken before \(a\) was applied:

\[ \mathrm{recover} = \sigma^{-1} \]

Recovery here carries real cost — reconstruction time, partial fidelity — and must be demonstrated, not assumed: a Type 2 classification requires a shown recovery path, not a claim that one probably exists.

Def. 4.3 Type 3 — no inverse

Type 3 actions have no inverse and no snapshot-based recovery path. They require explicit human authorization before \(\mathrm{Attempt}\) may run at all — the cycle does not gate these after the fact, it gates them before.

Doc. 4.4 Gates as predicates

A change proceeds if and only if every gate predicate holds:

\[ \bigwedge_{i} g_i(x) = \top \]

Each \(g_i\) is a named, checkable predicate over the state (reversibility class, validation status, authorization present, evidence domination from §3). The conjunction is total: one failing gate blocks the change regardless of how many others pass.

Def. 5.1 Evidence-preserving loop

The Infinity Engine is the unbounded iteration of the AVPT cycle operator,

\[ x_0 \xrightarrow{T} x_1 \xrightarrow{T} x_2 \xrightarrow{T} \cdots \]

framed as an \(\infty\)-iteration under supersession discipline: a prior state \(x_t\) is never erased when \(x_{t+1}\) is produced. It is superseded — kept in the evidence store and marked as no longer current, but still present.

Inv. 5.2 Supersession, not deletion

For all \(t\), the transition \(x_t \to x_{t+1}\) satisfies the preservation invariant of §2 (\(E_t \subseteq E_{t+1}\)) and additionally records \(x_t\) as superseded rather than removing any trace of it from \(E_{t+1}\). This is what makes the loop's history auditable at any point, not only at its current state.

Def. 5.3 Fixed points

A state \(x^{*}\) is a fixed point of the loop when \(T(x^{*}) = x^{*}\) — equivalently, when a cycle at \(x^{*}\) runs \(\mathrm{Validate}\) and finds nothing new to add to \(E\). Reaching a fixed point is not termination of the loop; it is the loop confirming, cycle after cycle, that the current evidence already accounts for the state.

Def. 7.1 Posterior propagation

Motion forecasting is treated as propagation of a posterior density matrix \(\rho(t)\) forward in time under a stated Hamiltonian \(H\) and stated environment assumptions, together with measurement backaction and covariance constraints on the estimate. The output is a predictive distribution over future states — an envelope — not a single predicted position.