Derivation of Planck Action, Elementary Electric Charge and Gravitation
Year: 2026
Type: preprint
Abstract: Derivation of Planck Action, Elementary Electric Charge and GravitationMatter Manifestation, Native Transport, Duplex Charge and the Universal Lattice Gravitational ArchitectureGravity is Transport. Charge is asymmetry. Matter is what persists. This paper presents a unified derivational account of Planck action, elementary electric charge and gravitation within Pattern Field Theory (PFT), developed by James Johan Sebastian Allen. It establishes the Allen Orbital Lattice (AOL) as the Universal Lattice Gravitational Architecture and develops the physical relationships connecting native Transport, identity, action, charge, matter manifestation and gravitational response. Pattern Field Theory derives its native constructions independently of measured target constants. Observer-side measurements are introduced through Mensura for calibration, correspondence and independent comparison. The Universal Lattice Gravitational Architecture The Allen Orbital Lattice is not a container within which physical processes occur. Its relational architecture constitutes native Transport itself. The Universe contains no primitive empty space. Physical manifestation develops through admissible closure, continuation, relational differentiation and the accumulation of persistent identities. The foundational Base-Pi construction distinguishes sixfold local closure from the continuing relationship carried by the Successful Differentiator. Following closure, continuation establishes either another closure or a new relationship between persistent identities. These outcomes define the architecture of the Action Branch and the Loading Branch. The Action Branch: Planck Action The Action Branch develops the inherited Logical Flow recurrence, Rationic scaling and native action invariance. Committed identities continue through successive admissible events. Rationic scaling establishes reciprocal transformations of energy and duration under the specified preservation conditions. The product of energy and duration remains invariant through the corresponding successful recurrence. A complete cyclic phase recurrence supplies the native structural correspondence with Planck action. Mensura connects the native recurrence with observer measurement units without introducing the measured Planck constant as an originating physical premise. The derivation distinguishes the native action structure from the independent determination of its absolute numerical scale. The Loading Branch: Charge, Matter and Gravitation The Loading Branch concerns the establishment, admission and accumulation of relationships between persistent identities. These relationships carry Manifestation Load and contribute to the configuration through which physical identity continues. Electric charge arises through signed Duplex asymmetry around the Equi balance condition. A balanced Duplex relationship is electrically neutral. A persistent relational asymmetry provides the native structure of charge. Integer-valued deviation accounting establishes discrete charge sectors. Admissible elementary charge requires persistence through the PAL/EQUI continuation architecture. Net charge accumulates through signed elementary relational contributions. Native gravity is Transport itself. Manifested gravitation arises through the accumulated participation and loading of persistent identities within that continuing architecture. Effective gravitational curvature is treated as a downstream rendering of native Transport under Manifestation Load, rather than as the originating physical mechanism. The paper also examines Nikola Tesla's 1932 criticism of assigning physical properties to empty space and develops PFT's constructive response through the Universal Lattice Gravitational Architecture. The Action and Loading branches converge through the proposed native relationship between action, Duplex charge and quadratic transfer cost. Mathematical and Computational Results The paper presents: AOL closure and continuation: Sixfold relational closure, the Successful Differentiator, inherited identity recurrence and admissible inward subdivision. Native action invariance: Rationic scaling, reciprocal energy-duration transformation and complete cyclic action recurrence. Discrete charge: Integer deviation accounting, Duplex symmetry, elementary charge cardinality and accumulated relational charge. Inverse-square allocation: Exact AOL disk and shell counting, conserved participation and the asymptotic inverse-square distribution of accessible participation. Safe Equilibrium: Uniform participation under stated equilibrium conditions, shell allocation and Basel-series normalization. Fine-structure coupling: The QUART chamber construction, its corrected numerical candidate and quantitative comparison with published electromagnetic measurements. Ring-7 chamber calculations: A 169-site AOL chamber, 462 lateral edges, Duplex-odd network response and calculated Duplex transfer resistances. Charge and flux: A PAL-compatible unit-charge graph configuration, minimum-dissipation transfer and a discrete Gauss-law representation. QUART recurrence: The distinction between 32-step modular compatibility and 96-step rotational recurrence, including the exact interlayer-offset period condition. Gravitational response: Native participation accounting, gravitational normalization and the relationship between Transport distribution and weighted-network response. Moore–Penrose inverse: The weighted Transport Laplacian, its constant equilibrium mode, budget-conserving transfer sector and effective resistance construction. Action-charge correspondence: The relationship between committed transfer expenditure, Duplex response, recurrence action and electromagnetic Mensural representation. Predictive Independence Measured constants must not be used to construct their own predictions. A measured quantity introduced as an upstream input, calibration or normalization cannot subsequently be presented as an independent numerical prediction of that same quantity. Pattern Field Theory distinguishes native derivation, mathematical identity, numerical computation, Mensural calibration and empirical comparison. The paper identifies the mathematical constructions established within its stated premises and the additional physical specifications required for their complete quantitative realization. These include the native commitment expenditure, persistent charge-bearing continuation, complete physical recurrence, electromagnetic coupling and independent absolute Mensural normalization. Reproducibility The manuscript presents mathematical definitions, derivational dependencies, numerical results, physical assumptions and references to the supporting Pattern Field Theory research corpus. The accompanying computational work provides reproducible chamber calculations, including Duplex resistances, Phase Alignment Locking (PAL)-compatible transfer configurations, discrete flux accounting and rotational recurrence tests. The research is developed within an Engineering Physics methodology emphasising mathematical consistency, operational construction, reproducibility, systems integration and cross-domain physical correspondence. This work contributes to Pattern Field Theory's continuing programme of deriving physical phenomena through a common architecture of Identity, Relationship, Transport, admissibility, continuation and accumulated manifestation. Pattern Field TheoryPatternFieldTheory.comFiles------------------------------------------------- """pft_chamber_calculations.py Reproduces every computed value in the section "Chamber and Tier Calculations" ofJames Johan Sebastian Allen, "Derivation of Planck Action, Elementary Electric Chargeand Gravitation", Pattern Field Theory, 2026. Computes: - element orders of the chamber structural group D6 x Z2 x Z6 - Duplex symmetry of the ring-7 AOL chamber Laplacian and its odd sector - Duplex resistances R_Duplex by ring, and the boundary-pair value by chamber radius - the PAL closed-traversal test for the unit-charge transfer - the discrete Gauss-law flux - Duplex preservation under tiered handover and the interlayer period theorem Requirements: Python 3.8+, numpy. Run: python3 pft_chamber_calculations.py"""import numpy as np, itertools, math# ---- Chamber group: element orders in D6 x Z2 x Z6def d6_orders(): out=[] for k in range(6): out.append(6//math.gcd(k,6)) # rotations out += [2]*6 # reflections return outorders=set()for a in d6_orders(): for b in (1,2): for c in [6//math.gcd(k,6) for k in range(6)]: orders.add(math.lcm(a,b,c))print("Chamber group element orders in D6xZ2xZ6:",sorted(orders),"max",max(orders))print("Period 96 available inside chamber group:",96 in orders,"; order-3 elements exist:",3 in orders) # ---- AOL ring-7 chamber (radius-7 disk), lateral edges only (new IIC weight 0 at SE)R=7S=[(q,r) for q in range(-R,R+1) for r in range(-R,R+1) if max(abs(q),abs(r),abs(q+r))<=R]ix={s:i for i,s in enumerate(S)}; N=len(S)E=[]for (q,r),i in ix.items(): for d in [(1,0),(0,1),(-1,1)]: j=ix.get((q+d[0],r+d[1])) if j is not None: E.append((i,j))B=np.zeros((N,len(E)))for e,(i,j) in enumerate(E): B[i,e]=1; B[j,e]=-1g=np.ones(len(E)); [email protected](g)@B.TD=np.zeros((N,N))for (q,r),i in ix.items(): D[ix[(-q,-r)],i]=1 # Duplex = antipodal map v -> -vprint("N sites",N,"edges",len(E),"[D,L]=0:",np.allclose(D@L,L@D))w,V=np.linalg.eigh((np.eye(N)-D)/2); Podd=V[:,w>0.5]Lm=Podd.T@L@Poddprint("odd-sector dim",Podd.shape[1],"min eigen",round(min(np.linalg.eigvalsh(Lm)),6))def Rdup(a): s=np.zeros(N); s[ix[a]]=1; s[ix[(-a[0],-a[1])]]-=1 so=Podd.T@s; return [email protected](Lm,so), sprint("Duplex resistance R_Duplex(a), a on ring m along an axis:")for m in range(1,R+1): r,_
Author: Allen, James Johan Sebastian
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224854148