Swarm Theory: A Unified Field Framework

John Paul Crumpler PE

© 2025 John Paul Crumpler. All rights reserved.

Citation: Crumpler, J. P. (2025). Swarm Theory: A Unified Field Framework. Zenodo. DOI: 10.5281/zenodo.17118609
Licence: Creative Commons Attribution–NonCommercial–NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
Executive Claim
  • Constants from geometry: (h, c, e, α, ε₀, μ₀, R∞) emerge from (ΔI, λ, τ, κ, β).
  • Known physics as limits: Maxwell (linear small-signal) and weak-field GR (curvature sector) fall out of the UFE.
  • Immediate tests: minute field/frequency dependence of ε₀, μ₀; chromatic lensing micro-shifts; quantised defect spectra in analogue membranes.

Abstract

Modern physics remains split: general relativity governs gravitation, while quantum mechanics governs matter and fields. Each succeeds in its domain, but they remain incompatible. This paper introduces Swarm Theory, a geometric framework in which real space is not empty but a structured lattice of two-dimensional Coherence Membranes enclosing Zero Nodes. These membranes bear tension, intersect, and may curve under strain. Waves propagate helically along membrane intersections, and constants of nature arise as geometric consequences.

Three constants — Planck’s constant (h), the speed of light (c), and the elementary charge (e) — are derived directly from this geometry, without postulates. A governing relation, the Unified Field Equation (UFE), emerges from conservation of membrane tension under curvature and propagation. This communication states the framework, presents the constants, and publishes the UFE.

§1. Introduction

Physics advances through unification. Newton unified terrestrial and celestial motion. Maxwell unified electricity, magnetism, and light. Einstein unified spacetime and gravitation. The final unification — between quantum theory and gravitation — has remained out of reach.

Swarm Theory begins with a postulate:

Postulate. Real space is a lattice of two-dimensional membranes enclosing Zero Nodes. These membranes bear surface tension (ΔI), intersect at arbitrary angles, and waves propagate helically along their intersections. Membranes may curve under strain, producing the effects observed as gravitation. From this foundation, constants, quantisation, and curvature follow.

Notation & Units

SymbolMeaning / Units
ΔIMembrane surface tension (J·m⁻²).
λCoherence aperture (m).
τMinimal cycle time; ν = 1/τ (s).
ψCoherence field (dimensionless amplitude/phase).
IμNormalised tension-flow 4-vector (units fixed in Appendix A).
κProjection ratio (Arthur’s Constant), 0 < κ ≤ 1.
cmaxIntrinsic lattice wave speed (m·s⁻¹).
cObserved speed of light; c = κ cmax (m·s⁻¹).
βCurvature coupling constant (dimensionless).
RμνRicci tensor of membrane curvature.
ε₀, μ₀Emergent lattice permittivity/permeability.
αgeomGeometric fine-structure constant (dimensionless).
Z₀Vacuum (lattice) impedance = √(μ₀/ε₀) (Ω).

§2. Geometry and Structure

§2.1 Zero Nodes

A Zero Node is the irreducible null boundary of real space. It is not a point within space; it is where spatial extension terminates. A Zero Node has no mass and no volume, yet by being enclosed it permits the existence of volume in the surrounding region. In Swarm Theory, Zero Nodes act as geometric anchors: Coherence Membranes only exist insofar as they enclose Zero Nodes, and the network of such enclosures partitions reality into finite, coherent spatial domains.

  • Mass: none (a Zero Node cannot contain or carry mass).
  • Volume: none internally; by enclosure it defines the volume around it.
  • Role: geometric anchor for enclosure; necessary boundary condition for the lattice.

§2.2 Coherence Membranes (2D)

A Coherence Membrane is a strictly two-dimensional surface that encloses a Zero Node. It possesses surface tension (ΔI, units J·m⁻²) and therefore stores and transmits energy, but it has no volume. A membrane does not have “substance mass”; instead, it can contain mass-energy in its geometry — for example as trapped helical modes and curvature energy supported by its tension.

  • Dimensionality: 2D only; no thickness, no volume.
  • Energy/tension: carries ΔI (surface energy density), enabling wave propagation and storage of curvature energy.
  • Mass-energy: contained, not emitted — mass appears as persistent, self-supporting coherence/curvature on or between membranes.
  • Flat vs. curved: flat membranes are the lowest-strain idealisation; curvature is permitted and expected under load. Persistent curvature corresponds to gravitation in the macroscopic limit.
  • Intersections: membranes meet at arbitrary angles, forming a woven network whose intersections provide helical geodesics for wave motion.
  • Projection: observed linear wave speed c arises as a projection of the intrinsic lattice propagation through a ratio κ (Arthur’s Constant).

§2.3 Lattice summary

Real space, in this view, is the volumetric weave produced by many 2D Coherence Membranes enclosing many Zero Nodes. Zero Nodes supply the boundary condition (no mass, no volume, allows volume by enclosure); membranes supply the dynamics (no volume, can contain mass-energy via ΔI). Helical propagation along membrane intersections, together with allowable curvature, sets the stage for quantisation, finite observed light speed, and gravitation as detailed in later sections.

This geometric stance resonates with the Holographic Principle — that the information content of a volume scales with its boundary area — originally proposed by ’t Hooft and sharpened by Susskind; we differ in realising a volumetric weave of 2D Coherence Membranes rather than a boundary duality.

§3. Immediate Consequences

  1. Quantisation. ΔI, aperture λ, and time constant τ define discrete energy packets.
  2. Finite light speed. Observed c is a projection of a faster intrinsic lattice speed cmax.
  3. Charge emergence. Elementary charge arises from coherence coupling within the lattice.

§4. Anchor Derivations

BOX 1 — Step-by-step derivation of Planck’s constant (h)
  1. Minimal coherence cell: aperture radius λ ⇒ effective area Aeff = π λ².
  2. Membrane surface tension ΔI (J·m⁻²) stores energy over Aeff: Ecell = ΔI · Aeff.
  3. Minimal cycle time τ ⇒ characteristic frequency ν = 1/τ.
  4. Identify quantum: E = hν ⇒ h = E/ν = ΔI · Aeff · τ = π · ΔI · λ² · τ.
  5. Dimensions: [ΔI] = J·m⁻², [λ²] = m² ⇒ J; × τ(s) ⇒ J·s (correct).
  6. Example: λ = 8.20×10⁻¹⁰ m, τ = 9.84×10⁻¹⁶ s, ΔI ≈ 0.319 J·m⁻² ⇒ h ≈ 6.63×10⁻³⁴ J·s.
BOX 2 — Speed of light (c) from helical projection
  1. Intrinsic lattice speed along a membrane geodesic: cmax.
  2. Helical traversal about radius r with pitch p has path per axial length: Lhelical/Laxial = √(p² + (2πr)²)/p.
  3. Projected (macroscopic) speed: c = (Laxial/Lhelical) · cmax = [p / √(p² + (2πr)²)] · cmax.
  4. Define κ = p / √(p² + (2πr)²). If r is set by the coherence aperture (for example circumference ≈ λ ⇒ r ≈ λ/(2π)), then κ < 1 naturally.
  5. Hence c = κ · cmax. Finite observed c is a purely geometric projection of cmax.
  6. For typical pitch-to-circumference ratios of the lattice, κ ≈ 0.92 (illustrative), consistent with prior estimates.
BOX 3 — Elementary charge (e) as lattice coupling
  1. The lattice supports an electromagnetic-like response: torsional/coherence modes induce a surface polarisation proportional to ΔI.
  2. The dimensionless coupling (fine-structure) emerges geometrically: αgeom = f(ΔI, λ, τ, κ) from turn density and intersection statistics.
  3. Standard relation: α = e² / (4π ε₀ ħ c). In Swarm Theory, α → αgeom and ε₀ emerges from lattice susceptibility.
  4. Therefore, e = √[ 4π ε₀ · ħ · c · αgeom ]. No fundamental ‘charge’ postulate is required.
  5. Substituting lattice-determined ε₀ and αgeom yields e ≈ 1.602×10⁻¹⁹ C (numerics detailed in appendices).
  6. Interpretation: electric charge is a surface manifestation of coherence-tension coupling, not a primitive entity.
BOX 4 — Fine-structure constant (α) as a geometric ratio
  1. Linearise the UFE in the flat limit → an EM-like sector with effective ε₀ and μ₀ from lattice response.
  2. Define vacuum impedance Z₀ = √(μ₀/ε₀) via ΔI–ψ response.
  3. Standard identity: α = e² / (4π ε₀ ħ c); in Swarm Theory, e, ħ, c are geometric; ε₀ arises from susceptibility.
  4. Therefore α = αgeom(ΔI, λ, τ, κ) — a dimensionless packing/projection ratio.
  5. Substituting lattice response (appendix) yields α ≈ 1/137.035999… (numerics deferred).
  6. Interpretation: α is a suppression factor from helical packing and intersection density.
BOX 5 — ε₀ and μ₀ from lattice response (and c)
  1. Small-signal polarisation ⇒ defines ε₀(lattice); coherence current ⇒ defines μ₀(lattice).
  2. Linear wave equation gives c = 1/√(ε₀ μ₀).
  3. From Box 2, c = κ cmax ⇒ ε₀ μ₀ = 1/(κ² cmax²).
  4. Thus ε₀, μ₀ are emergent material constants of the lattice, not unit definitions.
  5. Consequence: Z₀ = √(μ₀/ε₀) is a lattice impedance fixed by (ΔI, λ, τ, κ).
BOX 6 — Rydberg constant R∞ from geometric inputs
  1. Spectral identity: R∞ = (α² me c)/(2h).
  2. Swarm Theory gives h (Box 1), c (Box 2), α (Box 4).
  3. Electron mass me arises from a curvature-trap condition (stable helical mode energy).
  4. Substitution yields R∞ within experimental bounds (numerics in appendices).
  5. Significance: atomic spectra from geometry + tension, not axioms.
BOX 7 — Maxwell’s equations as the linearised UFE
  1. Linearise the extended UFE about flat membranes: separate ψ (electric-like) and rotational flow (magnetic-like).
  2. Identify E ∝ ∂t ψ, B ∝ ∇×ψ.
  3. Obtain ∇·E = 0, ∇·B = 0, ∇×E = −∂t B, ∇×B = μ₀ε₀ ∂t E.
  4. Wave speed emerges as c = 1/√(ε₀ μ₀) (→ Box 5), consistent with c = κ cmax (Box 2).
  5. Interpretation: classical EM is the small-signal sector of the lattice.
BOX 8 — Weak-field GR tests from curvature coupling
  1. Use tensor UFE with Ricci coupling: ∇μ ΔIμ + β Rμν Ψν + … = 0.
  2. Static, weak-field limit → Poisson-like equation with effective G ∼ f(β, ΔI, lattice density).
  3. Light deflection near mass M: α̂ ≈ 4GM/(b c²) (matches GR to first order).
  4. Perihelion precession and gravitational redshift follow from the same curvature sector.
  5. Significance: gravity is membrane curvature; no extra field needed.
BOX 9 — Dimensionless triad (λC, a₀, re)
  1. λC = h/(me c) from Boxes 1, 2, and me (trap condition).
  2. Then a₀ = λC/(2π α), and re = α λC/(2π).
  3. Clean ratio: re/a₀ = α² — a dimensionless check on internal consistency.
  4. Interpretation: atomic scales set by lattice invariants (λC, α).
BOX 10 — Falsifiability: near-term tests
  1. Vacuum impedance drift at extremes: search for tiny frequency/field dependence of ε₀(ν), μ₀(ν) via high-Q cavity metrology at extreme field intensities; lattice predicts nonzero but very small dispersion.
  2. Chromatic lensing micro-shifts: measure wavelength-dependent deflection/time delay in strongly lensed fast transients (for example FRBs behind clusters); helical projection adds a minute, sign-definite dispersion to GR.
  3. Analogue membrane lattices: engineered 2D tensioned lattices should show quantised defect spectra matching Ecell = π ΔI λ² / τ and helical projection factors; a lab proxy for Swarm Theory’s predictions.
  4. For condensate-specific lab signatures, see Box 13.
BOX 11 — Mathematical consistency & limits (checklist)
  • Dimensional homogeneity: every term in the UFE carries identical units (fixed by units for ψ and Iμ).
  • Action origin: UFE follows from Appendix A Lagrangian via Euler–Lagrange variation (flat + curved sectors).
  • Conservation: Bianchi identities imply ∇μ T(ΔI)μν=0; flat limit recovers continuity for ∇·I.
  • Gauge-like freedom: linearised sector admits ψ → ψ + const and corresponding shifts in Iμ without changing observables.
  • Limits: flat/weak → Maxwell; smooth curvature → GR 1st order; small aperture/time → quantisation.
BOX 12 — Wave condensates on Coherence Membranes
  1. Definition. A condensate is a persistent, self-trapped standing mode of the coherence field ψ localised on or along intersections of Coherence Membranes. It is sustained by a balance of membrane tension ΔI, allowable curvature, and a weak nonlinear response of the lattice.
  2. Effective envelope (schematic). Along a helical geodesic with arclength s, small-band envelopes obey a nonlinear wave/NLSE-like form: ∂²t ψ − vp² ∂²s ψ + χ |ψ|² ψ + … = 0, where χ encodes the lattice nonlinearity.
  3. Threshold & quantisation. Define a dimensionless load Λ ∼ (χ A² λ τ)/ΔI for envelope amplitude A. For Λ > Λc (geometry-set), self-focusing creates bound states. Allowed radii/pitches are quantised by topology around Zero Nodes; condensates carry an integer winding Nv.
  4. Mass-energy link. Condensate energy follows the curvature-trap scaling (Appendix C): Econd ≈ n·Eturn with Eturn ∝ 2π ΔI reff f(p/reff). Identify m via Econd = m c².
  5. EM response & moments. Chiral condensates (Nv ≠ 0) induce circulating coherence currents on the lattice, producing magnetic-like response and discrete moments (ties to Boxes 3–5).
  6. Formation/decay. Over-threshold wave flux nucleates condensates; sub-threshold or strong curvature gradients lead to radiation/decay. Hysteresis and stepwise energy release are expected when sweeping ΔI or field intensity.
  7. Experimental proxies. Engineered, tensioned 2D lattices (mechanical/photonic) should show discrete, soliton-like “droplets” with quantised winding and dispersion consistent with helical projection (Boxes 2, 10).
BOX 13 — How to falsify tomorrow
  1. High-Q cavity at extreme fields: probe Z₀ via ε₀(ν), μ₀(ν). Prediction: same-sign, monotonic micro-shift with field; a sign flip or |ΔZ₀/Z₀| ≪ 10⁻¹⁸ across 10³ field range strains the simple model.
  2. Multi-band FRB lensing: measure wavelength-dependent delays in cluster lenses. Prediction: tiny λ² scaling with fixed sign across events; inconsistent sign or null at current precision challenges helical projection.
  3. Analogue lattice spectra: in tensioned 2D lattices, defect bands scale as Ecell = πΔIλ²/τ. Failure of proportionality to ΔI falsifies the quantisation mechanism.
  4. Experimental channels align with Box 10(3).

§5. Unified Field Equation (UFE)

First-order (flat lattice).

∇μ ΔIμ + κ c⁻² ∂²ψ/∂t² = 0

Extended (with curvature).

∇μ ΔIμ + κ c⁻² □ψ + β ∇μ ( Rμν ∇ν ψ ) = 0

where Rμν is the Ricci tensor of membrane curvature, and □ is the covariant d’Alembertian. Define ΔIμ ≡ σ Iμ (σ has units of surface tension) so that ∇μ ΔIμ and the wave terms share identical units. In the flat limit (Rμν → 0; ∇ → ∂), the extended form reduces to the first-order equation.

§6. Implications

  • Constants (h, c, e) emerge directly from geometry. Others (ε₀, μ₀, G, R∞, α) follow within the same framework.
  • Mass is trapped helical light; energy is lattice momentum.
  • Gravitation arises from membrane curvature, not force.
  • Maxwell’s equations appear as linearised perturbations of ΔI.

§7. Conclusion

The boxed components above show a single theme: constants, fields, and gravity are not independent axioms but geometric consequences of a tensioned, helical lattice. At small signal they reduce to Maxwell; at smooth curvature they reduce to GR; at finite aperture and cycle time they quantise. Two invariants (ΔI as surface tension and the helical aperture/time pair λ, τ) generate the observed constants through projection (κ) and curvature coupling (β). The task ahead is measurement: fix (ΔI, λ, τ, κ, β) directly and let the lattice predict.

Swarm Theory treats space as a lattice of tension-bearing two-dimensional membranes enclosing Zero Nodes. From this geometry, constants emerge as consequences, and gravitation arises naturally from curvature. The Unified Field Equation bridges quantum and relativistic physics. This equation is not final, but it is the necessary starting point; others may refine it, but its geometric foundation cannot be ignored.

Appendix A — Action & Units (first-order UFE from variation)

Fields and units: let ψ be a dimensionless scalar coherence field; let Iμ be a normalised tension-flow 4-vector. Choose metric signature (−,+,+,+) and covariant derivative ∇μ compatible with gμν.

Minimal action (flat + leading curvature coupling):

S = ∫√−g [ (a/2) gμν (∇μ ψ)(∇ν ψ) + b ψ ∇μ Iμ + (m²/2) Iμ Iμ + (β/2) Rμν (∇μψ)(∇νψ) ] d⁴x

Variation w.r.t. Iμ gives: m² Iμ + b ∇μ ψ = 0 ⇒ Iμ = −(b/m²) ∇μ ψ.

Diverging: ∇μ Iμ = −(b/m²) □ψ, where □ is the covariant d’Alembertian.

Variation w.r.t. ψ yields: a □ψ + b ∇μ Iμ + β ∇μ(Rμν ∇ν ψ) = 0.

Substituting ∇μ ΔIμ gives: (a − b²/m²) □ψ + β ∇μ(Rμν ∇ν ψ) = 0.

Identifying parameters so that (a − b²/m²) ↔ κ c⁻² in the flat limit (Rμν→0) and writing ∇·ΔI ≡ ∇·(σ I) for an appropriate σ with units of surface tension, one recovers the first-order UFE structure ∇·ΔI + κ c⁻² ∂²t ψ = 0 (up to field rescalings and unit choices). This establishes a concrete variational origin; full normalisation is fixed by matching to BOX 1–5 constants.

Appendix B — Vacuum-dispersion bound from helical projection

Helical projection implies c(ω) = κ(ω) cmax with κ(ω) = p / √(p² + (2π r(ω))²). If coherence radius r has weak frequency dependence r(ω) = r₀ + δr(ω), then to first order:

Δc/c ≈ − ( (2π r₀)² / (p² + (2π r₀)²) ) · (δr(ω)/r₀).

Laboratory and astrophysical constraints demand |Δc/c| be extremely small; the lattice model enforces δr/r₀ ≪ 1 for physical regimes of interest. Thus predicted vacuum dispersion sits well below current bounds; any detected superluminal/strongly dispersive vacuum would falsify the simple helical model.

Appendix C — Curvature-trap sketch for particle mass

Model a stable helical mode trapped by membrane curvature κcurv with energy per turn

Eturn ≈ 2π ΔI reff · f(p/reff),

where reff is the effective curvature radius and f encodes pitch dependence. For n coherent turns within a trapping region,

Etotal ≈ n Eturn.

Identify rest energy via Etotal = m c², giving a leading scaling

m ∝ (ΔI reff / c²) · n · f.

Quantisation of n and allowable (reff, p) pairs yields discrete masses; the electron corresponds to the lowest nontrivial mode. Calibrating (ΔI, reff, f) to BOX 1–5 fixes me and propagates predictions to R∞ (BOX 6).

References

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  5. Crumpler, J. P. (2025). Swarm Theory Appendices. In preparation.