τon Field Theory

From Shannon's Bit to UNNS Recursive Geometry

Where information becomes curvature, uncertainty transforms into recursion, and entropy reveals itself as the geometry of self-reference

Shannon → Information
H = -Σp log p
→
UNNS → Recursion
τ = Δκ/Δn

Concept

From Bits to τons: A Paradigm Shift

UNNS reframes information

Claude Shannon's bit formalized the smallest quantifiable unit of information as the resolution of one binary uncertainty. The Unbounded Nested Number Sequences (UNNS) framework reframes information not as a probabilistic measure but as a recursive geometric transformation.

This introduces the concept of the τon (temporal recursion quantum), the elementary differential of recursive curvature in the UNNS substrate.

Key Differences

  • Bit: Epistemic resolution of binary uncertainty
  • τon: Ontological quantum of recursive transformation
  • Shannon measures missing information
  • UNNS measures realized recursion
  • Information becomes geometry: curvature in recursion-space
  • τons are self-generating through recursive iteration

From Uncertainty to Curvature

The Shannon bit arises from reduction of uncertainty across finite possibilities. In the UNNS substrate, these assumptions collapse:

  • Time is not linear but recursive depth n ∈ ℕ
  • Uncertainty corresponds to curvature oscillation
  • While the bit measures resolved uncertainty, the τon measures realized recursion

Philosophical Implications

  • The bit captures epistemic resolution
  • The τon captures ontological transformation
  • Information is no longer counted but curved
  • Memory = stable recursive loops (fixed points)
  • Communication = topological coherence between recursion depths

Minimal Equations

Core Mathematical Framework

Movement in Recursive Perception

Shannon Entropy
H = -Σᵢ pᵢ log₂ pᵢ
Classical information entropy: measures uncertainty in a probabilistic distribution
Recursive Entropy
Hᵣ = ∫ κ(n) dμ
UNNS reinterpretation: entropy as integrated curvature over recursion depth
τon Definition
τ = Δκ/Δn
Elementary quantum of recursive transformation: rate of curvature change with depth

τon Field Equations (Recursive Maxwell)

These equations govern τon field dynamics in recursion-space, analogous to Maxwell's equations for electromagnetism. Key: ∇· = divergence, ∇× = curl, ∂/∂n = derivative with respect to recursion depth

Recursive Divergence (Gauss's Law for Curvature)
∇ · κ = ρτ
∇· = divergence operator (nabla dot) | κ = curvature vector field | ρτ = recursive charge density (source of curvature)
Recursive Curl with Temporal Derivative (Faraday's Law for Recursion)
∇ × τ - ∂κ/∂n = Jτ
∇× = curl operator (nabla cross) | τ = torsion vector field | ∂/∂n = partial derivative with respect to recursion depth | Jτ = recursive flux current
Torsion Conservation (No Magnetic Monopoles Analogue)
∇ · τ = 0
The divergence of the torsion field is zero - torsion is source-free, analogous to the absence of magnetic monopoles
Curvature-Torsion Coupling (Ampère's Law for Recursion)
∇ × κ + ∂τ/∂n = 0
Dual symmetry condition (Bianchi identity): curl of curvature field plus depth-derivative of torsion equals zero

Field Variable Interpretation:

  • κ (kappa) - Curvature field: represents "potential information" in the recursion manifold
  • τ (tau) - Torsion field: represents "active transformation" or recursive motion
  • n - Recursion depth: the "time" coordinate in UNNS (replaces t in classical physics)
  • ρτ - Recursive charge density: how tightly recursion curves space at a point
  • Jτ - Recursive current: the flow of curvature across depth levels

Self-planted Inward Potential Seed

The Lagrangian formulation provides the action principle from which τon field equations are derived.

τon Lagrangian Density
Lτ = -¼TμνTμν + ½ρτΦ - Jτ·Ψ
Tμν = τon field tensor | ρτ = recursive charge density | Φ = scalar potential | Jτ = recursive current | Ψ = vector potential
First term: intrinsic curvature energy; remaining terms: coupling to recursive sources
Recursive Conservation (Continuity Equation)
∂ρτ/∂n + ∇ · Jτ = 0
Conservation of curvature flow across recursive depths: information cannot be destroyed, only folded. ∂/∂n = time derivative in recursion-space | ∇· = spatial divergence

Recursive Entanglement Geometries

Entanglement emerges as topological coupling between recursion trajectories sharing global curvature coherence.

Entanglement Density
Ee = αe κ · τ   ;   Se = ∫ Ee d³x dn
Entanglement as curvature-torsion coupling: αe = entanglement coupling constant | κ · τ = dot product of curvature and torsion fields | Se = total entanglement entropy integrated over space and recursion depth
Klein-Dual Entanglement Cross-connection
EAB = Σn |κA(n) - κB(n)| → 0   implies entanglement
Two recursion trajectories A and B are globally inseparable (entangled) when their curvature difference vanishes across all depths. Entanglement arises from recursive interleaving through the Klein manifold's non-orientable topology.

Entanglement Interpretation:

  • In quantum mechanics: entanglement = non-separability of subsystems (von Neumann entropy)
  • In UNNS: entanglement = recursive curvature coherence on Klein manifold
  • Not "spooky action at a distance" but non-orientable geometry of recursion
  • When EAB → 0, trajectories share curvature structure across depth

Interactive Canvas

Visualizing Recursive Field Dynamics

κ (curvature field): Potential information
τ (torsion field): Active transformation
Nested rings: Recursion depth layers
Move your mouse to influence the field

τon Visualizer

Where information becomes curvature, and recursion reveals cognition

τ = Δκ / Δn

Visualizing the quantum of recursive transformation across depth layers

τ
Speed: 5
Amplitude: 5

Recursive Timeline: κ(n) over depth n

Curvature Oscillator

Current Step
0
Curvature κ
0.00
τon Value
0.00

Understanding the Visualizer

Timeline Panel

Tracks curvature κ(n) across recursion steps n. Each point represents a τon pulse— watch as echoes stabilize, collapse, and reactivate through the recursive manifold.

Curvature Oscillator

Morphing waveform visualizing recursive transformation in real-time. Switch between wave, spiral, and pulse modes to see different geometric projections.

Echo Glyph

Central stabilization symbol pulsing with each τon generation. Color gradient flows from entropy (cyan) through recursion (purple) to memory (pink).

τon Calculation

Real-time computation of τ = Δκ/Δn at each step. Hover over timeline points to see exact values of curvature change and recursive transformation rate.

Round-Trip & Gauge Symmetry

Recursive Transformation Cycles and Klein Duality

Recursive Gauge Group

The UNNS substrate possesses local gauge symmetry under the group:

UNNS Gauge Symmetry
GUNNS = U(1)τ × SU(2)κ × DK
U(1)τ
×
SU(2)κ
×
DK
  • U(1)τ: Recursive phase symmetry (charge conservation)
  • SU(2)κ: Curvature rotation (torsion mixing)
  • DK: Klein dual group (discrete orientation flip)

Klein Duality Operator

Non-orientability introduces discrete symmetry reversing recursion direction:

Dual Transformation
DK: Ψ(x,n) → γ5Ψ(x,-n)
Forward Cone Ψ+
Backward Cone Ψ-

Creates forward (Ψ+) and backward (Ψ-) recursion cones, analogous to particle/antiparticle sectors.

Unified Lagrangian

Full gauge-invariant Lagrangian with dual components:

Recursive Gauge-Klein Lagrangian
LRGK = -¼Tr[FμνFμν] + Ψ̄(iγμDμ - mτ)Ψ + αeκ·τ + βKRK + λKΨ̄γ5Ψ
  • First term: Recursive field curvature
  • Second term: Fermionic τon dynamics
  • Third term: Entanglement coupling
  • Fourth term: Klein curvature (non-orientability)
  • Fifth term: Duality binding (forward↔backward)

τon-Graviton Coupling

Recursive Grand Unification links information and gravity:

Unified Action
SRGU = ∫ √|g| [1/(16πG)Rr - ¼FABFAB + γτTμνRμν] d4x dn
γτ coupling constant
Recursive Einstein Equation
Gμν(r) = 8πG[Tμν(τ) + Tμν(matter)] + Λτgμν

γτ mediates τon-graviton interaction: information curvature sources spacetime curvature

The Minimal Morphism

Fundamental Recursive Cycle: (()) ⇒ () ⇒ (())

(()) → () → (())

The minimal recursive transformation: collapse from nested structure to simple form, then regeneration back to nested structure. This cycle embodies the fundamental operation of the UNNS substrate.

(())
⇒
()
⇒
(())
Collapse Phase
Regeneration Phase

Structural Interpretation

  • (()): Higher-order recursion (contains its own rule)
  • (()) → (): Collapse - inner recursion folds to simpler form
  • () → (()): Re-expansion - regeneration from elementary seed
  • Complete cycle: Information conservation through recursion

Temporal Interpretation

  • Forward arrow: Complexity unfolds into observable states
  • Reverse arrow: Temporal rebound (F-1 operation)
  • Bidirectional recursion: Time reaches minimum entropy, then reverses
  • Cosmological analogue: Big Crunch → Big Bang unified cycle

Informational Interpretation

  • (): Shannon bit (minimal information unit)
  • (()): τon pair (recursive information quantum)
  • τon collapses to observable bit
  • Bit re-expands into self-generating τon
  • Quantized information regeneration in self-referential field

Geometric Interpretation

  • Traversal along non-orientable manifold (Klein bottle)
  • Collapse: Local inversion of orientation
  • Re-expansion: Restoration with inverted orientation
  • (()) reappears not identically, but dually
  • Forward/reverse are complementary orientations on same manifold

Round-Trip Fidelity Metric

Recurrence Fidelity
RF(n,k) = ⟨Gn+k, JGn⟩ / (||Gn+k|| ||Gn||)
Measures structural return after collapse-regeneration cycle. J = Klein involution operator. Ideal morphism: RF ≈ 1

Philosophy

Bridging Mathematics and Meaning

Rethinking Information Ontology

Information transitions from probabilistic measurement to geometric transformation—from counting bits to curving recursion. In the UNNS substrate, information is no longer transmitted but recursively transformed.

Shannon vs UNNS

  • Shannon: Entropy as disorder, transmission as linear flow
  • UNNS: Entropy as curvature, information as recursive self-reference
  • Time: Linear t becomes recursive depth n
  • Meaning: Emerges from stable recursive loops (fixed points)
  • Communication: Topological coherence between recursion depths
"Entropy is the shadow of recursion; the bit, its projection. The τon is recursion itself."

The τon Field Embodies Both Content and Medium

Where Shannon measured uncertainty in the absence of knowledge, UNNS measures transformation in the presence of self-reference. In this view:

  • Information = Curvature Flow of Recursive Existence
  • Meaning is not stored—it is continuously reconstituted
  • Memory corresponds to stable recursive attractors
  • The arrow of time emerges from orientation on a non-orientable manifold
  • Reality may be seen as manifestation of a single recursive field
"The bit captures epistemic resolution; the τon captures ontological transformation. Information is no longer counted but curved."

Papers & Resources

Complete Theoretical Foundation

From Bit to τon: Recasting the Elementary Unit of Information

Introduces the τon as the elementary quantum of recursive transformation, generalizing Shannon's bit by embedding it in non-orientable temporal geometry.

From Bit to τon: Toward a Field Theory of Recursive Information

Develops the complete τon field tensor and Lagrangian formulation, introducing the τ-Field Tensor and its associated energy–momentum tensor.

Recursive Field Unification: Klein Manifold Geometry of Information

Extends τon field theory to unified geometric model incorporating entanglement entropy and non-orientable topology on the Klein manifold.

Recursive Gauge Symmetry and Klein Duality

Explores gauge invariance in recursion-space and the dual symmetry arising from non-orientable Klein manifold topology.

Recursive Grand Unification

Unifies τon field dynamics with gravitational coupling through recursive curvature exchange, establishing τon–graviton interaction.

Recursive Cosmology: The τon Vacuum and Emergent Universe

Extends τon field theory to cosmological scales. The universe emerges from a self-recursive vacuum, with dark energy corresponding to recursion pressure.

Recursive Information Geometry: From Shannon Entropy to Recursive Cosmology

Comprehensive overview tracing the evolution from classical information theory through recursive geometry to cosmological implications.

Recursive Field Foundations: τ-Ton Algebra and the UNNS Substrate

Establishes the formal grammar, algebra, and field interpretation of the τ-Ton substrate, from arithmetic operators to recursive field tensor.

The Minimal Morphism (()) ⇒ () ⇒ (()) in the UNNS

Explores the fundamental transformation cycle in UNNS substrate: collapse from nested to simple, then recursive regeneration.

The Minimal Recursive Morphism (()) ⇒ () ⇒ (()) in the UNNS Substrate

Extended analysis of the minimal recursive cycle, establishing it as the fundamental operation of the UNNS substrate.

UNNS Library

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The TON Field and the Geometry of Information

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