Equilibrion and the Elias Model (λΦ Duplex Resonance)

Pattern Field Theory™ (PFT™) models reality as recursive pattern dynamics in the Metacontinuum™. Using the Allen Orbital Lattice™ with λΦ duplex resonance, Equilibrion selects stable minimal-curvature realizations. π projected to AOL rings exhibits an equilibrium signature — the Pi–Lattice Equilibrion Correspondence.

Allen Orbital Lattice, RAGC, and the Pi–Lattice Equilibrion Correspondence

Independent derivation of the Allen Orbital Lattice (AOL) and the Elias Model (λΦ Duplex Alternation), with Equilibrion and the Riemann Active Generative Constraint (RAGC), and an empirical Pi–Lattice Equilibrion Correspondence.

Summary

Claim. The Allen Orbital Lattice (AOL) and the Elias Model (λΦ Duplex Alternation Mechanism) were derived independently, prior to any engagement with Eisenstein or modular–form literature. After establishing AOL and the duplex rule, the fastest modern π methods (Chudnovsky–Ramanujan; Heegner 163) were reviewed and found to rely on a hexagonal complex-multiplication lattice—confirming the AOL prediction rather than informing its construction. Mapping π’s digits to concentric rings on AOL yields a stable equilibrium signature (uniform per-ring frequencies, high and steady entropy, no prime-ring effect): the Pi–Lattice Equilibrion Correspondence.

Background

Motivation. PFT models structure as a pattern field resolved into form by a stability-selecting operator (Equilibrion). AOL is the discrete operator domain observed when growth must remain coherent under curvature pressure. The central question here: does π, when expressed on that domain, exhibit the same equilibrium signature predicted by the theory?

Derivation Record (independent origin)

1. Construct AOL. Concentric hexagonal rings of size 6n form the minimal-curvature discrete domain for outward growth with correction.
2. Derive Elias Duplex Rule. Growth and restoration must alternate to prevent drift; parity is fixed by construction at the center.
3. Prediction. Any correct computation of π must match the AOL curvature equilibrium when projected to rings.
4. Validation step. Only after the above did we examine modern π computation; the Heegner-163/hexagonal link corroborates the prediction.

Elias Model — Minimal Derivation

• Requirements: (i) allow outward increase of curvature (formation), (ii) prevent runaway by periodic restoration.
• Assign phases on ring index n: constructive \( \lambda=+1 \) on even rings, compensatory \( \Phi=-1 \) on odd rings; equivalently \( \sigma(n)=(-1)^n \) with \( \sigma(0)=+1 \).
• Consequence: duplex parity keeps curvature centered while permitting growth (biological morphogenesis, spectral centering, lattice stability).

Operators

Equilibrion. A resolving operator that selects the minimal-curvature, stability-preserving realization from a relational pattern to spatial form. Informally, \( E:\; \text{logical layer} \rightarrow \text{spatial layer} \) with stability constraint.

Riemann Active Generative Constraint (RAGC). A generative constraint on coherent outward growth that enforces duplex parity and spectral centering, preventing curvature drift under expansion.

Procedure: Pi–Lattice Equilibrion Correspondence

1. Digits. Use the fractional digits of π at target depths (10k–100k).
2. Ring mapping. Place digits one-by-one onto AOL rings in a fixed axial sweep; capacity after radius R: \(1+3R(R+1)\).
3. Per-ring stats. For each ring compute digit frequencies (0–9), mean digit, standard deviation, and Shannon entropy.
4. Uniformity & stability. Check per-ring uniformity; track entropy across rings; compare prime vs non-prime ring aggregates.

Results (concise)

• Uniformity. Per-ring digit frequencies align with uniform expectation; no persistent banding.
• Entropy. Shannon entropy rises quickly and remains stable with radius at large depths.
• Prime effect. No significant differences between prime and non-prime rings in means or entropies.

Implications

1. Validation of AOL. The equilibrium pattern observed on AOL rings is consistent with the domain being the correct operator geometry for curvature-conserving growth.
2. Consistency with computation. The modern fast π computation’s hexagonal basis is consistent with, not causal of, the AOL prediction.
3. Elias Duplex Necessity. Alternating \( \lambda/\Phi \) parity is the minimal mechanism that permits growth without drift; removing it breaks equilibrium.

Definitions (expanded on first use)

Allen Orbital Lattice (AOL). Discrete hexagonal operator domain with rings of size 6n; domain for the operator H.

Explicit-Formula Trace Correspondence (ALEF). Alignment between trace peaks of \( e^{itH} \) on AOL and the \( k\log p \) terms in the explicit formula.

AOL Critical Line Theorem (ACLT). Duplex symmetry centers unfolded spectra as if on \( \Re(s)=\tfrac12 \) and yields Gaussian Unitary Ensemble (GUE) spacings.

Elias Model (λΦ Duplex Resonance Model). Alternation \( \lambda=+1 \) (constructive curvature) on even rings, \( \Phi=-1 \) (compensatory curvature) on odd rings.

Riemann Active Generative Constraint (RAGC). Constraint that coherent outward growth with curvature minimization adopts duplex parity and prevents drift.

Equilibrion. Resolving operator selecting minimal-curvature, stable spatial realization from logical structure.

Reproducibility

• Traversal. Fixed axial sweep order per ring; origin holds the first fractional digit.
• Capacity. After radius R, capacity \(1+3R(R+1)\); ring size for n≥1 is \(6n\).
• Artifacts. Provide CSVs for ring mapping and ring stats, plus the four figures shown above.

Original work of James Johan Sebastian Allen. PatternFieldTheory.com

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