RAPT Theorems

Influence between elements is restricted to bounded neighbor relations. Establishes discrete neighborhoods and finite locality of effect — the precondition under which “near” and “far” are meaningful. Not space, distance, or metric.

Closed traversals may carry invariant structural effect. Establishes path-dependence and loop memory — returning to the same place is not equivalent to never having left. Not curvature, field, or gauge structure.

Unresolved configurations are not equivalent to resolved ones with respect to permitted transitions. Establishes directionality without time. Not energy, force, entropy, or gradient.

A structure may bear relations to the effects of its own prior states. Establishes feedback eligibility and recursive closure as possibilities. Makes recursion possible; ignition makes it occur.

Differences may endure. Establishes inside-vs-outside as a meaningful relation, interface stability, and protection from total diffusion. Not a membrane, wall, or act of formation.

The continued existence of any structure is conditional on ambient inflow. Establishes non-self-sufficiency and the impossibility of closed-system persistence. Not energy, supply, or resource.

Recursion is admissible if and only if all six Tier-0Ṛ primitives are simultaneously satisfied within a bounded logic field. An admissibility statement, not an occurrence statement.

Where distinctions endure, unresolved ≠ resolved, and existence is conditional on inflow, persistence carries a maintenance condition that cannot be self-supplied.

The identity of a self-referential structure is sustained only while its boundary remains intact; distinction loss terminates recursive identity.

Continued operation of a self-referential bounded structure incurs a non-zero maintenance condition; free amplification is not admissible. Whatever runs, runs at cost.

Admissibility decomposes into six non-equivalent forms; conclusions under one form do not transfer to another.

Structural admissibility and temporal admissibility are independent (neither implies the other). Stable attractor behavior over time requires both classes jointly.

Trajectory admissibility does not require pointwise state admissibility at every instant. Bounded transient deficit, survived with identity intact, is consistent with sustained persistence. Persistence is a window-based condition.

Stable attractor behavior over time holds iff both the structural and temporal admissibility classes are satisfied. Attractor formation and attractor persistence are orthogonal requirements; neither alone suffices.

A recursive attractor exists as an operational dynamical structure iff IST holds, the trajectory has entered the basin, and the recursive loop has closed. Three regimes exhaust its status: pre-attractor (mathematical possibility only), active (extant, self-maintaining), and post-loss (ceased; only α-trace remains). Existence is regime-discrete, not gradient.

An attractor's organizing invariants are observable primarily along basin-resident trajectories. Trajectories outside the basin sample only transient states and cannot reliably infer the attractor's internal organizing logic.

Time, history, and trace are not primitives. Under co-presence of T0-χ∧T0-P∧T0-B∧T0-M in persistent recursion, they are generated as downstream consequences. Nothing flows upward: the substrate does not contain time; recursion generates it.

A loss event is β-classified when a boundary defining recursive identity is violated, the prior configuration cannot be restored under permitted transitions, and re-establishment would require re-ignition rather than continuation. A classification outcome, not a force or process. When β applies, ADM-A₅ trajectory admissibility has terminated.

For autocatalytic systems the six ADM forms are not equally weighted: ADM-A₅ (trajectory) is primary, ADM-A₄ (transition) is the per-update identity gate, ADM-A₁ (structural) is the precondition. A prioritization construct, not a new decomposition.

After β-loss, recursion may not resume freely: any subsequent autocatalytic recursion in the same substrate must re-satisfy IST, re-pay initiation cost, and re-establish recursive identity. Identity is not inherited across β — the new instance is genuinely new. History matters in autocatalysis.

A seven-rule classification pack governing degradation, residue, and restoration downstream of β and NFR:

An ordered sequence A₀, A₁, … where each layer is admissible under a primitive-preserving extension of the one below. Ordered, DAG-consistent, additive, monotonic, non-destructive to lower layers.

Higher recursion layers cannot access lower attractors' internal transaction histories, formation pathways, or accumulated descriptor content. A cross-recursion-depth observability constraint.

The formal characterization of how an attractor's accumulated history is compressed into its present structure.

Candidate Tier-2 construct (status pending adversarial testing).

The named SI-1–SI-5 stack-integrity invariants, with a validity condition and the L2-RSI / L2-HCM audit operators.

Three classes: ATX-BASE, ATX-COORD, ATX-SOV. The classification of attractors by bootstrapping-signature class.

Eighteen emergent recursive bootstrapping modes forming a partially-ordered path from ATX-BASE to ATX-SOV. Nine spine modes, eight branch (substrate) modes, with AC₁₁ (Boundary-Stabilizing Autocatalysis) as the sovereignty pivot and AC₁₇ (Hierarchical Autocatalytic Stack) as the apex.

Attractor evolution proceeds through a formation phase (mutation-class operators) and a stacking phase (crossover-class operators); an operator admissible in one may be destructive in the other.

Self-amplification is non-diagnostic of admissibility; the four-way classification of self-reinforcing phenomena.

A binary test: Recursion Lock, Internal Recurcline Persistence, Boundary Retention, Maintenance-Bearing Continuation. A configuration is sovereign or it is not.

Sovereignty realized collectively across constituent attractors.

The diagnostic procedure applying the four SOV conditions.

A non-recursive configuration that mimics attractor behavior through external coupling. The contrast category to sovereign attractors — not a kind of attractor.

A five-component descriptive profile (Pₖₗ, Tₖₗ, Rₖₗ, Bₖₗ, Sₖₗ) of a sovereign attractor's continuation characteristics.

The derived accounting quantity for stored recursive compression within an active sovereign attractor. Scalar magnitude plus gradient structure. Defined only on the bootstrapping interval Iₖ.

The durable structural record left by recursive selection. Live (active filter) or inert (static residue). Records what survived selection; does not itself perform selection.

Resistance of a sovereign structure to admissible reconfiguration under recurcline pressure. Not physical mass.

Distance between an attractor's current state and the nearest admissibility boundary, in transaction-cost space. Not thermodynamic entropy.

Transaction burden required to sustain recursive persistence per unit continuation. The operational form of MBC.

Minimum condition for a specific recursive behavior to remain admissible (Θᵐ maintenance, Θʳ reconfiguration, Θᵣ rupture). Classificatory, not causal.

How densely historical compression is encoded within an attractor's α-trace.

The degree to which a sovereign attractor internally produces its own structuring rather than drawing it pre-formed from ambient inflow.

Accounting categories — not energies, forces, or causal drivers. The set is closed; no T₇ may be added without a MAJOR governance action.

Mechanical/structural rupture as a β/T₅ transition between admissible kinematic sets, with descriptive fields C/B/J and holonomic memory (KR-10) bound to T0-χ.

A reflexive-coupling instantiation with descriptive fields E (Exploitation Load), K (Remaining Capacity), and ρ (Reflexive Coupling).