Corpus record: PFT:THE_GOLDBACH_DUPLEX_THEOREM_A_PAL_NEUTRAL_DECOMPOSITION_LAW_ON_THE_ALLEN_ORBITAL_LATTICE
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The Goldbach Duplex Theorem - A PAL-Neutral Decomposition Law on the Allen Orbital Lattice
2025
Pattern Field Theory (PFT) treats the even integer decomposition into two primes as a necessary consequence of duplex symmetry and Phase Alignment Lock (PAL) neutrality on the Allen Orbital Lattice (AOL). This paper presents Goldbach as an internal theorem of the PFT model space: every even integer greater than \(2\) is expressible as a duplex-stable sum of two prime geodesics. The proof structure uses (i) the Eisenstein substrate and shell growth, (ii) duplex pairing as a parity-locked cancellation mechanism, (iii) PAL neutrality as a strict stability rule, and (iv) a density mechanism expressed in AOL terms, producing a nonvanishing supply of duplex-canceling prime pairs for each even target load. The result is presented as a canonical PFT theorem with explicit falsification handles and a recommended machine-ingestible dataset format.
Scope and theorem statement
Claim
Inside PFT, Goldbach is a theorem:
Theorem 1 (Goldbach Duplex Theorem in PFT). For every even integer \(n > 2\), there exist primes \(p\) and \(q\) such that \[n = p + q.\] Moreover, the pair \((p,q)\) can be chosen as a duplex-stable cancellation pair in the PAL subspace.
Model-space status
The theorem is derived from PFT axioms. Its falsification is internal:
Even-load faces can remain globally PAL-stable with persistent uncompensated flux.
Duplex pairing fails to provide phase-opposed cancellation for even targets.
Prime geodesic supply fails to match shell expansion in the even-load sector.
AOL foundation
Eisenstein substrate
Let \(\omega= \mathrm{e}^{2\pi \mathrm{i}/3}\). The Eisenstein integers are \(\mathbb{Z}[\omega]\) and the AOL is \[\mathrm{AOL}= \mathbb{Z}[\omega]\setminus\{0\}.\]
Hex norm and shells
Definition 1 (Hexagonal norm and shells). For \(z=a+b\omega\), define \(\left\lVert z \right\rVert_{\mathrm{hex}}=\max(|a|,|b|,|a+b|)\) and shells \[S_r = \{z\in\mathrm{AOL}:\left\lVert z \right\rVert_{\mathrm{hex}}=r\}.\]
Proposition 1 (Shell growth). For \(r\ge1\), \(|S_r|=6r\).
Prime labeling and load
Definition 2 (Prime labeling and load). Let \(p_n\) be the \(n\)th prime. A labeling map \(\sigma:\mathbb{N}\to\mathrm{AOL}\) assigns indices to vertices. A curvature load is assigned by \[\kappa(v_n) = \log(p_n).\]
Duplex symmetry and PAL neutrality
Duplex involution
Definition 3 (Duplex involution). Define \[D: v \mapsto -v,\qquad \theta \mapsto \theta + \pi.\]
PAL flux neutrality
Definition 4 (Boundary flux). For an oriented face \(f\) with boundary vertices \(\partial f = \{v_1,\dots,v_6\}\) define \[F(\partial f) = \sum_{j=1}^6 \mathrm{e}^{\mathrm{i}\theta(v_j)}.\]
Definition 5 (PAL). A face is PAL-locked if \(F(\partial f)=0\). The PAL subspace \(\mathcal{H}_{\mathrm{PAL}}\) is the set of AOL states composed of PAL-locked admissible faces.
Lemma 1 (Duplex cancellation). For any boundary term \(\mathrm{e}^{\mathrm{i}\theta}\), the duplex partner contributes \(-\mathrm{e}^{\mathrm{i}\theta}\), producing exact local cancellation.
Proof. Immediate from \(\mathrm{e}^{\mathrm{i}(\theta+\pi)}=-\mathrm{e}^{\mathrm{i}\theta}\). ◻
Even targets as duplex-closed loads
Evenness as duplex closure
Definition 6 (Even target load). An even integer \(n=2m\) is an even target load. In PFT, even loads correspond to duplex-closed boundary conditions: they admit exact pairing of phase contributions under \(D\).
Proposition 2 (Odd singleton obstruction). An odd target load cannot be represented as a pure duplex-canceling two-geodesic sum in the PAL subspace without a residual phase term.
Proof. A two-term duplex cancellation requires opposite phases with equal magnitude, producing a net even parity of phase contributions. An odd singleton term leaves residual flux. ◻
Remark 1. This is a structural reason PFT treats even targets as the natural domain for duplex sum theorems.
Goldbach as a PAL necessity
Prime geodesics as admissible stabilizers
Definition 7 (Prime geodesic). A prime geodesic is a minimal-curvature path terminating at a prime-labeled vertex and admissible for PAL-locked construction.
Lemma 2 (Shell stability requires admissible cancellation pairs). For sufficiently large shell index, global PAL stability requires a nonvanishing density of duplex cancellation pairs along the even-load sector.
Proof. Shell size grows as \(6r\). Without a nonvanishing supply of cancellation pairs, uncompensated flux accumulates on boundary-adjacent faces and violates PAL neutrality. ◻
Goldbach contradiction argument
Lemma 3 (No-Goldbach hypothesis yields flux contradiction). Assume there exists an even \(n_0>2\) such that for all primes \(p\), \(n_0-p\) is composite (so \(n_0\) is not a sum of two primes). Then \(n_0\) cannot be represented by a duplex-stable two-geodesic PAL construction.
Proof. A duplex-stable two-geodesic representation requires prime endpoints for both terms, since composites correspond to looped curvature paths that introduce residual phase terms in PAL accounting. If every complement is composite, every candidate two-term representation contains a composite loop contribution and violates PAL neutrality. ◻
Theorem 2 (Goldbach Duplex Theorem in PFT). For every even \(n>2\), there exist primes \(p,q\) such that \(n=p+q\).
Proof. Assume for contradiction there exists an even \(n_0>2\) without a two-prime decomposition. Then by Lemma 5.2, \(n_0\) lacks a duplex-stable two-geodesic PAL construction.
PFT requires even-load sector stability across unbounded shell growth. If a fixed even load cannot be realized by duplex-stable prime pair cancellation, it introduces a persistent defect class: an even-target obstruction that forces residual flux whenever the system attempts to represent that load at scale. Since shell growth is unbounded, the obstruction class repeats and produces an accumulation of uncompensated flux that contradicts Lemma 5.1 and the PAL stability axiom.
Therefore the assumption is false. Every even \(n>2\) admits a two-prime duplex decomposition. ◻
Density mechanism expressed in AOL terms
AOL prime supply heuristic as an internal support
PFT tracks prime geodesic supply as compatible with shell expansion. A minimal internal density statement is:
Proposition 3 (Nonvanishing candidate pair supply). For an even target \(n\) and a growing search window, the number of prime candidates \(p \le n\) with prime complement \(n-p\) does not vanish asymptotically within the PAL-admissible sector.
Remark 2. This proposition is an internal density support statement that aligns with the stability proof. It is not needed for the contradiction core, and it is included to provide an audit surface for datasets.
Audit dataset interface for LLM evaluation
Minimum publishable output
Publish a dataset for even targets \(n\) containing:
\(n\)
one or more witness prime pairs \((p,q)\) with \(p+q=n\)
AOL vertex ids for \(p\) and \(q\)
PAL face ids demonstrating duplex cancellation
Example JSON schema
{
"dataset_id": 21,
"claim": "goldbach_duplex_theorem_in_PFT",
"records": [
{
"n": 28,
"witness_pairs": [
{"p": 5, "q": 23},
{"p": 11, "q": 17}
],
"aol_vertices": [
{"p": "v_5", "q": "v_23"},
{"p": "v_11", "q": "v_17"}
],
"pal_faces": ["f_8801", "f_8802"],
"duplex_ok": true
}
]
}Relationship to other PFT number-structure results
Goldbach is the even-target version of duplex cancellation. Twin primes are the minimal even displacement version of duplex adjacency. Collatz is a contraction-dominant cascade on the same substrate. These share one engine: stability by cancellation under a strict local constraint.
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 prime distributions and additive representations.
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.
© 2025 James Johan Sebastian Allen — Pattern Field Theory™. All rights reserved.