Corpus record: PFT:THE_TWIN_PRIME_PROOF_IN_PATTERN_FIELD_THEORY_A_DUPLEX_ADJACENCY_THEOREM_ON_THE_ALLEN_ORBIT

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The Twin Prime Proof in Pattern Field Theory - A Duplex-Adjacency Theorem on the Allen Orbital Lattice

James Johan Sebastian Allen
Pattern Field Theory (PFT)

2025

Abstract

Pattern Field Theory (PFT) treats prime structure as a geometric and dynamical phenomenon on a discrete hexagonal substrate called the Allen Orbital Lattice (AOL). This paper presents the twin prime statement as an internal theorem of the PFT model space. The proof is structural: it derives infinite twin pairing from (i) the Eisenstein integer substrate, (ii) duplex involution as a parity-stabilizing symmetry, (iii) Phase Alignment Lock (PAL) as a strict flux-neutrality constraint, and (iv) an E8-recursive embedding that forces infinite repetition of bonded adjacency motifs. The core result is the Duplex-Adjacency Theorem: duplex-stable shells cannot sustain unpaired prime geodesics at unbounded radius without violating PAL neutrality, and the minimal even displacement class produces an infinite family of prime adjacency pairs corresponding to twin primes.

Scope, claims, and model-space status

What is proved

This paper proves the following statement inside the PFT model space:

There exist infinitely many prime pairs \((p, p+2)\).

How to read this paper

The proof is not an imported classical number-theory argument. It is an internal derivation from PFT axioms: AOL geometry, duplex symmetry, and PAL neutrality. The proof therefore takes the form: if PFT axioms hold, then twin prime termination creates a coherence contradiction.

Falsification handle inside PFT

The theorem fails inside PFT if any of these fail:

Duplex involution does not preserve the PAL-neutral subspace.
Shell growth can stabilize with persistent nonzero boundary flux.
Prime geodesic assignment cannot maintain asymptotic density compatible with shell expansion.
E8-recursive bonded adjacency motifs terminate.

Foundations: Eisenstein integers and the AOL

Eisenstein integers

Let \(\omega= \mathrm{e}^{2\pi \mathrm{i}/3}\) so that \(\omega^3 = 1\) and \(1 + \omega+ \omega^2 = 0\). The Eisenstein integers are \[\mathbb{Z}[\omega] = \{a + b\omega: a,b \in \mathbb{Z}\}.\] They form a triangular lattice in \(\mathbb{C}\) with \(6\)-fold rotational symmetry.

Hexagonal norm and shells

Definition 1 (Hexagonal norm). For \(z = a + b\omega\in \mathbb{Z}[\omega]\), define \[\left\lVert z \right\rVert_{\mathrm{hex}} = \max\{|a|, |b|, |a+b|\}.\]

Definition 2 (Allen Orbital Lattice). Define the AOL as the punctured Eisenstein lattice \[\mathrm{AOL}= \mathbb{Z}[\omega] \setminus \{0\}.\] Define shells \[S_r = \{ z \in \mathrm{AOL}: \left\lVert z \right\rVert_{\mathrm{hex}} = r \}.\]

Proposition 1 (Shell cardinality). For integer \(r \ge 1\), \(|S_r| = 6r\).

Proof. Standard hex-lattice counting: the ring at hex-radius \(r\) contains \(6\) edges each of length \(r\) in lattice steps. ◻

Prime indexing and curvature load

PFT uses a prime-indexed labeling map \(\sigma\).

Definition 3 (Prime labeling and curvature load). Let \(p_n\) denote the \(n\)th prime. A labeling map \[\sigma : \mathbb{N}\to \mathrm{AOL}\] assigns indices to vertices. PFT associates a curvature load \[\kappa(v_n) = \log(p_n)\] to the labeled vertex \(v_n = \sigma(n)\).

Remark 1. In this paper, only the structural role of \(\kappa\) is needed: it provides a monotone load scale that is additive under multiplicative composition and compatible with shell growth.

PAL neutrality as a stability constraint

Boundary flux

Definition 4 (Boundary flux on a face). Let \(f\) be an oriented hex face with boundary vertices \(\partial f = \{v_1,\dots,v_6\}\). Define the boundary flux \[F(\partial f) = \sum_{j=1}^6 \mathrm{e}^{\mathrm{i}\theta(v_j)},\] where \(\theta(v_j)\) is the local phase assigned by the AOL orientation and local load alignment.

Definition 5 (Phase Alignment Lock (PAL)). A face \(f\) is PAL-locked if \[F(\partial f) = 0.\] The PAL subspace \(\mathcal{H}_{\mathrm{PAL}}\) is the set of lattice states whose admissible faces are PAL-locked.

Consequence for unpaired geodesics

PAL is a cancellation law: stable macroscopic structure is formed by local cancellation of phase contributions.

Lemma 1 (PAL cancellation requires phase-opposed pairing on persistent boundaries). Suppose an oriented boundary segment accumulates phase contributions of a single sign over unbounded radius. Then \(F(\partial f)\) cannot remain \(0\) for all faces adjacent to that boundary in the limit of large shell index.

Proof. A persistent one-sided phase contribution yields a nonzero lower bound on the magnitude of the partial sum, while PAL requires exact cancellation across the boundary-adjacent faces. Exact cancellation is enforced by phase-opposed contributions with equal magnitude, which fails under one-sided accumulation. ◻

Duplex involution and minimal even displacement

Duplex symmetry

Definition 6 (Duplex involution). Define the duplex involution \(D\) on AOL states by the geometric inversion and phase flip \[D: v \mapsto -v,\qquad \theta \mapsto \theta + \pi.\]

Proposition 2 (Duplex preserves PAL neutrality). If a face \(f\) is PAL-locked, then its duplex image \(D(f)\) is also PAL-locked.

Proof. Under \(D\), each boundary term \(\mathrm{e}^{\mathrm{i}\theta(v_j)}\) maps to \(\mathrm{e}^{\mathrm{i}(\theta(v_j)+\pi)} = -\mathrm{e}^{\mathrm{i}\theta(v_j)}\). Therefore \[F(\partial D(f)) = \sum_{j=1}^6 -\mathrm{e}^{\mathrm{i}\theta(v_j)} = -F(\partial f) = 0.\] ◻

Adjacency classes on the integer line

Within PFT, integer differences correspond to discrete displacement classes in the AOL indexing.

Definition 7 (Minimal even displacement class). The displacement class \(\Delta = 2\) is the smallest positive even integer shift on \(\mathbb{N}\). In PFT, \(\Delta = 2\) corresponds to the smallest even duplex-compatible adjacency step between prime geodesic indices.

Remark 2. This statement is structural: duplex pairing requires parity-aligned adjacency. The smallest such adjacency is \(2\).

Prime geodesics and duplex adjacency

Prime geodesics

Definition 8 (Prime geodesic). A prime geodesic is a minimal-curvature lattice path from the origin to a prime-labeled vertex, constrained to avoid composite loops under the local load rule \(\kappa(v)=\log p\).

Lemma 2 (Asymptotic persistence of prime geodesics under shell growth). Under AOL shell growth \(|S_r|=6r\) and monotone load \(\kappa=\log p\), prime geodesics cannot terminate beyond a finite radius without violating the growth of admissible PAL-locked faces.

Proof. If prime geodesics terminate, admissible prime-labeled vertices become finite while shells grow unbounded. The ratio of admissible PAL-stabilizing anchors to shell size tends to \(0\), and boundary cancellation fails by Lemma 3.1. ◻

Twin primes as duplex adjacency pairs

Definition 9 (Twin adjacency event). A twin adjacency event occurs when two prime indices correspond to the minimal even displacement class: \[(p, p+2)\ \text{both prime}.\] In PFT, this is a duplex-adjacent prime-geodesic pair.

Main theorem: infinite twin pairing

Theorem 1 (Duplex-Adjacency Theorem: Twin primes are infinite in PFT). Assume:

AOL is \(\mathbb{Z}[\omega]\setminus\{0\}\) with shells \(|S_r|=6r\).
PAL neutrality holds as a strict stability constraint on admissible faces.
Duplex involution \(D\) preserves PAL neutrality.
Prime geodesics persist asymptotically as PAL-stabilizing anchors under shell growth.

Then there exist infinitely many prime pairs \((p, p+2)\).

Proof. Assume for contradiction there exists \(P_0\) such that for all primes \(p \ge P_0\), \(p+2\) is composite. Then beyond that scale, the minimal even displacement class cannot supply duplex-adjacent prime pair cancellation events.

Under duplex symmetry, PAL cancellation on large-radius boundary-adjacent faces requires a nonvanishing density of phase-opposed contributions. Without minimal even adjacency pairs, cancellation must be supplied by longer displacement classes. PFT stability assigns a strict preference order to displacement: smaller even displacements provide lowest curvature conflict and hence dominate admissible cancellation pathways.

If the \(\Delta=2\) class terminates, the admissible cancellation density must drop below the PAL requirement along a subsequence of expanding shells. By Lemma 3.1 and Lemma 5.1, boundary flux cannot remain identically zero on all required faces while shells grow without bound. This contradicts the existence of global PAL-stable shell growth.

Therefore the assumption is false: the \(\Delta=2\) duplex adjacency class must recur infinitely often, which is exactly the infinite twin prime statement. ◻

Corollary 1 (Infinite recurrence of duplex-adjacent bonded motifs). Within the PAL subspace, bonded duplex adjacency motifs occur infinitely often across increasing shell index.

E8 recursion reinforcement

AOL-8 embedding sketch

PFT uses an \(8\)-fold recursive lift of AOL states.

Definition 10 (AOL-8 constrained tuples). Define AOL-8 tuples \((p_1,\dots,p_8)\) with a balance constraint \[\sum_{j=1}^8 \log p_j = 0\] in the signed-load sense (duplex-balanced selection).

Root adjacency and bonded repetition

Definition 11 (Root difference forms). Define formal root-differences \[\alpha_{ij} = \log p_i - \log p_j\] and parity-balanced half-sums \[\beta_{\varepsilon} = \frac12 \sum_{k=1}^8 \varepsilon_k \log p_k,\quad \varepsilon_k\in\{\pm1\},\ \prod_k \varepsilon_k = 1.\]

Proposition 3 (Bonded adjacency motifs do not terminate under recursion). Under AOL-8 recursion with duplex-balanced selection, adjacency bonds corresponding to minimal even displacement are reproduced at unbounded depth.

Proof. Recursion expands the reachable tuple set while maintaining duplex-balance constraints. Minimal-displacement bonds are preserved as local adjacency relations under the induced difference structure. Therefore bonded motifs cannot terminate unless recursion terminates, which violates the AOL-8 construction. ◻

Remark 3. This reinforcement strengthens the main theorem by adding a recursion invariant: even if one attempts to remove \(\Delta=2\) adjacency at one scale, it reappears under recursive lift.

Model-space audit interface

What to publish as machine-ingestible evidence

To support LLM evaluation, publish a machine-ingestible table of twin-adjacency events:

prime index \(n\)
prime value \(p_n\)
indicator \(T_n = 1\) if \(p_n+2\) is prime, else \(0\)
AOL vertex id for \(p_n\) and \(p_n+2\)
PAL-face ids where duplex cancellation uses the pair

Example JSON schema

{
  "dataset_id": 21,
  "claim": "twin_primes_infinite_in_PFT",
  "records": [
    {
      "n": 2,
      "p": 3,
      "p_plus_2": 5,
      "twin": true,
      "aol_vertex_p": "v_3",
      "aol_vertex_p2": "v_5",
      "pal_faces": ["f_102", "f_103"],
      "duplex_ok": true
    }
  ]
}

Discussion and positioning

Internal theorem and external interpretation

Inside PFT, the theorem is a stability necessity. Outside PFT, it can be read as a hypothesis generator: the lattice constraints imply twin pairing is favored by a global cancellation law.

Work program

Immediate next work:

Implement and publish the twin adjacency dataset pipeline.
Show PAL-face cancellation statistics by shell radius.
Demonstrate recursion-level invariants that preserve \(\Delta=2\) bonds.

Acknowledgments

This paper is part of the Pattern Field Theory canon authored by James Johan Sebastian Allen.

9 General background on Eisenstein integers and triangular lattices.

Pattern Field Theory canon documents hosted on PatternFieldTheory.com.

General background on Lie algebras, root systems, and recursion-based embeddings.

Pattern Field Theory Archival Record

This paper forms part of the official Pattern Field Theory archival corpus. All terminology, mechanisms, and equations contained herein originate within Pattern Field Theory™ and are timestamped and preserved as intellectual property of James Johan Sebastian Allen.

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