Pattern Field Theory

Abstract

Every change that preserves identity continuity follows one sequence on the Allen Orbital Lattice (AOL): Differentiat confirms continuity-preserving options and the Equilibrion Gate permits change; the trajectory-space is prepared (Vector0); all involved observer patterns snap into a shared phase condition (Phase Alignment Lock, PAL); one permitted trajectory executes (cascade) within a window budget W; and the system returns to Equilibrion. This article formalises PAL as the start signal, the continuity constraints, and the selection over the set V with allowed subset V′ and executed path v_k ∈ V′. It clarifies the Split Test prerequisite, the game-theoretic structure of vector loading, and shows how micro- cascades compose smooth movement and cross-scale phenomena (biological, social, engineered).

1. Introduction — Where Change Begins

Every observable action that preserves identity continuity begins the same way. This framework identifies where it begins, how it starts, and why it does not run away. The five phases are operational states that appear in movement, perception, coordination, decision onsets, and social ignition.

Running thesis. Change under continuity is discrete: permission (Gate), preparation (Vector0), phase alignment (PAL), bounded execution (v_k within W), and return (Equilibrion).

Running examples

• Patrol contact: a squad moves silently; a micro-freeze precedes coordinated motion.
• Horse spook: the horse loads early; the rider aligns at ignition.
• Musical drop: crowd breath and timing align just before the chorus hits.
• Everyday movement: standing up and walking, composed of micro-cascades.

Short Definitions (Pattern Field Theory)

Observer Pattern. Identity-stable pattern participating in field interaction and continuity across time.

Substrate. Underlying continuity field within which patterns exist.

Differentiat. Continuity governor that permits or prevents events by assigning a window budget W and prohibiting runaway cascades that threaten substrate continuity.

Equilibrion. Stable identity state of an observer pattern (at any scale).

Equilibrion Gate. Permission moment: Differentiat has confirmed continuity-preserving options and Equilibrion authorises change.

Vector0. Index-zero ready-state where all permitted trajectories are prepared but none has executed.

Phase Alignment Lock (PAL). Instantaneous resonance alignment (no duration in ordinary time) that synchronises all involved observer patterns to the same start phase; this is the start signal.

Continuity-Preserving Path (CPP). A permitted trajectory v_k that preserves the observer pattern’s identity across the event.

Window Budget (W). Upper bound on cascade length or steps assigned by Differentiat to ensure continuity and forbid runaway.

Split Test. Multiple feasible continuations exist and identity continuity is maintainable across them; the Gate opens only after the Split Test is satisfied.

Dominion. Organism-level controller maintaining internal coherence within events permitted by Differentiat and carried within Equilibrion.

Event Cascades on the AOL. Unified process from permission through start, execution, and return to Equilibrion.

Acronym Expansions

• AOL — Allen Orbital Lattice, the structural field substrate.
• PFT — Pattern Field Theory, the unified framework used here.
• PAL — Phase Alignment Lock, instantaneous ignition-phase synchronisation.
• CPP — Continuity-Preserving Path, permitted identity-stable continuation.
• W — Window Budget, maximum cascade length allowed.
• S_i — Strategy set of observer pattern o_i.
• F(v) — Selection functional over V′.
• EMG — Electromyography (muscle activation timing).
• IMU — Inertial Measurement Unit (micro-movement coherence).
• RIP — Respiratory Inductance Plethysmography (breathing-phase alignment).
• Δϕ — Phase offset; PAL reduces Δϕ to zero at ignition.
• o_i — Observer pattern i, identity-bearing pattern participating in the cascade.

2. Split Test and Game-Theoretic Structure of Vector Loading

The Split Test requires that there exist multiple continuity-preserving continuations before the Equilibrion Gate can open. Let V = {v₁, …, v_n} denote all feasible continuations and let V′ ⊆ V be the subset that preserves identity under the assigned window budget W.

Split Test (formal). The Gate opens only if |V′| ≥ 2. If |V′| = 0, change would break identity or exceed W, and the Gate remains closed. If |V′| = 1, no timing coordination is needed and PAL does not occur. PAL only applies when multiple continuity-preserving paths exist.

Game-theoretic interpretation

The set V is a strategy space. Each v_k ∈ V corresponds to a feasible continuation of the observer pattern under identity continuity. The Split Test verifies that at least two strategies remain valid after pruning by continuity constraints.

Vector0 is the index-zero state where all strategies in V′ are prepared, but none have executed:

Vector0: V′ loaded, vk not yet selected.

PAL then acts as ignition: Phase Alignment Lock synchronises all involved observer patterns into a shared phase state and collapses the loaded strategy space into one executed continuation v_k ∈ V′.

Gate → Vector0 → PAL → vk → Return.

3. Event Cascade Sequence (Chapter Map)

1. Equilibrion Gate. Differentiat validates that multiple continuity-preserving outcomes exist (Split Test) and assigns W; Equilibrion authorises change under these constraints.
2. Vector0. The allowed trajectory-space is prepared.
   Let V = {v₁, …, v_n} and V′ ⊆ V denote the subset allowed under the Identity Continuity Constraint. No v_k has executed.
3. Phase Alignment Lock (PAL). A phase-alignment condition synchronises all involved observer patterns to a shared start phase. PAL has no measurable duration; it is the ignition condition.
4. Cascade Execution (v_k). Exactly one v_k ∈ V′ unfolds within W; Equilibrion protects identity during execution.
5. Return to Equilibrion. Identity continuity is restored; intention-space is released.

The Gate opens only when the Split Test is true (|V′| ≥ 2); PAL then serves as the coordination signal for selecting a single v_k ∈ V′.

What locks where?

• Gate: permission only; no execution, no timing alignment.
• Vector0: trajectory-space V′ exists; no v_k has fired.
• PAL: timing only; all involved observer patterns share one resonance phase; divergence follows immediately.
• Execution: exactly one v_k unfolds within W; identity is protected in-flight.
• Return: Equilibrion restores intention-space and ordinary behaviour.

4. Foundational Clarifications

Gate vs Vector0 vs PAL. Gate = permission (after Split Test and W). Vector0 = strategy-space prepared (no execution). PAL = phase-alignment start tick. Order: Gate → Vector0 → PAL → execution.

Continuity-first. Differentiat never opens the Gate unless at least one CPP exists and W bounds the cascade. Equilibrion and Dominion maintain identity during execution; runaway is forbidden.

Split Test as prerequisite. Feasible multi-path continuation must already be established before the Gate opens. PAL is not prediction; it is continuity validation followed by start.

5. Mathematical Model (Concise)

Sets and constraints

V  = {v₁, …, vₙ}           (all continuations)
V′ ⊆ V                    (allowed by Differentiat under identity continuity)

Gate      : permission once Split Test confirms V′
Vector0   : strategy-space prepared; no vₖ executed
PAL       : phase alignment condition (start signal)
Cascade   : vₖ ∈ V′ executes within W
Return    : Equilibrion restored
                            

Vector loading exposes a strategy space V under constraints (identity continuity, W). Game theory gives the language for the multi-path decision; PAL is the ignition that begins the realised path v_k.

Compact sequence:

Equilibrion Gate → Vector0 —PAL→ vk ∈ V′ —W→ Return to Equilibrion.

6. Split Test (Formal)

Given the strategy space V = {v₁, …, v_n} and constraints C = {identity continuity, window budget W, domain invariants}, the Split Test is true iff there exist at least two distinct continuations v_i, v_j ∈ V, i ≠ j, such that both satisfy C. Equivalently, the allowed subset satisfies |V′| ≥ 2.

If |V′| = 0, no continuity-preserving path exists under W and the Gate remains closed. If |V′| = 1, the event is degenerate and does not require timing coordination, so PAL is unnecessary. PAL is only meaningful when a start signal must coordinate among multiple safe continuations.

For interacting observer patterns o₁, …, o_m, the group Split Test requires a non-empty set of joint CPPs that respect all identities and the assigned W: V′_joint ≠ ∅ and |V′_joint| ≥ 2.

7. Pre-PAL Load State

Once the Split Test is satisfied, the system enters a loaded state in which the allowed trajectory set V′ is real and identity continuity is preserved across all trajectories, but no start signal has yet occurred. This is not execution and not timing alignment; it is the state of being ready-to-diverge.

Gate open ⇒ Vector0 prepared ⇒ Pre-PAL Load State.

Properties of the Pre-PAL Load State:

• Multiple continuity-preserving continuations exist.
• No trajectory has begun.
• No timing synchronisation has occurred.
• The system is sensitive to the structure of V′.

This state is present whenever |V′| ≥ 2. It is the condition that makes PAL meaningful: a start signal is only relevant when there are multiple possible allowed continuations.

8. Game Theory View of Vector Loading

Vector loading exposes a structured decision space before execution; this admits a game-theoretic description constrained by identity continuity and W.

• Players: interacting observer patterns o₁, …, o_m.
• Strategies: each player has a feasible set S_i of micro-continuations that respect its own identity constraints. A joint continuation v_k corresponds to a strategy profile s = (s₁, …, s_m) ∈ S₁ × … × S_m.
• Feasibility: the identity and window constraints prune the joint space:
  V′ = { vk(s) | s ∈ S₁ × … × Sm, vk preserves all identities and fits W }

Utilities (continuity-weighted):

uᵢ(vₖ) = cont(vₖ | oᵢ) + goal(vₖ | oᵢ) − risk(vₖ | W, oᵢ)
                            

Continuity violations set uᵢ = −∞ and eliminate the path. Once the Gate is open and Vector0 has prepared V′, PAL functions as a coordination signal that locks timing across players so that exactly one v_k ∈ V′ is realised. PAL is not a payoff computation; it is the phase alignment that enables selection without conflict.

Selection principle. Execution chooses a single CPP:

vₖ ∈ arg max_{v ∈ V′} F(v)
                            

where F is a domain-appropriate selection functional (risk-bounded success for patrol, stability for balance, entrainment strength for music, etc.). Classical Nash analysis applies to the pruned game on V′. PAL acts like a global coordination signal, analogous to a correlation device enabling immediate selection of a permissible joint action; timing divergence resumes as the cascade unfolds.

Consequences.

• No PAL without multiplicity: the Split Test guards against unnecessary alignment.
• No runaway: W caps path length, independent of payoffs.
• Fast smoothness: micro-cascades realise repeated PAL-driven selections at high rate.

9. Micro-Cascade Continuity (Movement)

Movement is a sequence of micro event cascades. Each micro-cascade follows the five-step sequence and then restores Equilibrion before the next begins.

Why movement feels smooth. Movement is not frame-by-frame, not pixel- by-pixel, and not literally continuous fluid motion. It is:

cascade → Equilibrion → cascade → Equilibrion → …

This repeats so fast and so stably that it feels continuous. The same mechanism underlies:

• eyes tracking motion without motion blur,
• gait maintaining balance,
• speech progressing syllable to syllable,
• music transitioning from tension to release.

All are sequences of micro event cascades, each protected by Equilibrion and initiated by PAL.

10. Case Studies Across Scales (PAL in Practice)

Patrol contact

Gate: multiple safe continuations (hold, peel, break) satisfy W (Split Test true).

Vector0: sectors covered; hand signals prepared.

PAL: the silent micro-freeze — shared phase alignment without speech.

Execution: peel, bound, or hold; exactly one v_k; W limits exposure.

Return: posture normalises; communications resume.

Horse spook

Gate: continuity-preserving options validated for horse and rider.

Vector0: posture, reins, stance prepared.

PAL: the snap moment both feel; ignition without command.

Execution: shying or halt (v_k) within W.

Return: calm breathing; identity continuity intact.

Musical drop

Gate: audience and performers have multiple continuity-preserving entries.

Vector0: timing pre-aligns.

PAL: breathless instant before the chorus.

Execution: the drop (v_k); roles diverge.

Return: tension resolves; baseline timing re-establishes.

Everyday movement

Gate: standing is permitted; W bounds sub-movements.

Vector0: muscle synergies prepared.

PAL: the onset tick.

Execution: rise step as v_k; next steps repeat.

Return: brief stabilisations between micro-cascades.

11. Detection and Instrumentation

PAL is a phase-state. Group PAL can be inferred from coherence markers, including:

• timing alignment across sensors (IMU micro-jitter, sway),
• respiratory phase locking (RIP belts, microphone arrays),
• micro-EMG onset synchrony (surface EMG clusters).

A detector estimates p(PAL) from multi-sensor coherence for training, safety, and coordination systems.

12. Existing Research That Aligns Partial Aspects

• Readiness potential — aligns with Vector0.
• Forward models — simulate V; PAL and W complete the mechanism.
• Coordination dynamics — capture phase-locking; PAL is ignition under Gate.
• Swarm and herd models — alignment / avoidance / cohesion; PAL adds ignition and identity constraints.
• Music entrainment — timing alignment; PAL is the explicit start signal.
• Predictive processing — continuity maintenance maps to Equilibrion; cascades and W added here.

13. Terminology Finalisation

Earlier drafts used the term Cascade-Lock. The final mechanistic term is Phase Alignment Lock (PAL).

14. Implications

1. Universality. The same start signal and constraints apply across domains.
2. Continuity-first. The Gate opens only when CPPs exist and W is assigned; runaway cascades are blocked.
3. Identity protection. Equilibrion and Dominion preserve identity through each v_k.
4. Composition. Smooth behaviour is micro-cascade composition.

15. Conclusion

Event Cascades on the Allen Orbital Lattice proceed through five phases. PAL is the start signal that synchronises observer patterns and permits divergence into a single allowed trajectory. Differentiat guarantees continuity via W; Equilibrion and Dominion protect identity; and micro-cascade composition explains the apparent smoothness of action across scales. The same mechanism applies across movement, perception, social coordination, and engineered systems.

External Reference

A copy of this paper is also available on Academia.edu:

Event Cascades on the Allen Orbital Lattice — Ghost Field, PAL Bridge, Boundary Conduction, and Mechanistic Framework