The Circumpunct Encyclopedia

Chapter 1 — Ontological Primitives

1.1 The Three Fundamental Components

Enforcement rule: Throughout this document, • always does operations, Φ always holds/transmits state, ○ always constrains/filters.

1.2 The Circumpunct (⊙)

⊙ = (•, Φ, ○)

Every manifest being is a circumpunct—an aperture (operation) generating a field (state), contained by boundary (constraint).

1.3 The Infinite Field (Φ∞)

1.4 What Is Φ Made Of?

Φ is the space of constraints + signals that relate circumpuncts.

It is multi-realizable (like computation). Three realizations:

1. Cognitive: attention, memory, perception, internal representations
2. Social: language, norms, roles, shared meanings
3. Physical: sensory channels, bodies, environments, media

This makes Φ a formal field (relational/informational medium), not a claim about new physics.

1.5 Core Axioms

Chapter 2 — The Aperture Function

2.1 The Core Operation

∞ →[•]→ 1

Input: unbounded potential
Operation: selection + integration
Output: singular actuality

2.2 The Aperture as Operator Triple

𝒜 = (G, R, U)

G: H∞ → H•   Gather/Gate    (selects slice of input)
R: H• → H•   Rotate/Process (transforms selected input)
U: H• → H₁   Utter/Emit     (produces output)

Total map: T_𝒜 = U ∘ R ∘ G

2.3 The Fidelity Parameter

F(𝒜) = ⟨T_𝒜h(ψ), T_𝒜(ψ)⟩ / (|T_𝒜h(ψ)| · |T_𝒜(ψ)|)

F ∈ [-1, 1]
F = 1:  Perfect fidelity
F = 0:  Orthogonal to truth
F = -1: Perfect inversion

2.4 The Binary Decision (χ)

χ = sign(F - τ)

χ = +1: truth-mode (F ≥ threshold)
χ = -1: lie-mode (F < threshold)

2.5 Self-Reference Property

𝒜 ∈ State(⊙)

The operator that generates the whole is represented inside the whole. This is "parts are fractals of wholes" in operator language.

Chapter 3 — The Field Equations

3.1 Local Field Dynamics

∇²Φᵢ - (1/c²)(∂²Φᵢ/∂t²) = -ρᵢ(•ᵢ)

3.2 Field Properties

3.3 Coupled Field Equations

∇²Φ₁ = -ρ₁ - κ₁₂Φ₂
∇²Φ₂ = -ρ₂ - κ₂₁Φ₁

Chapter 4 — The Boundary Function

4.1 Type Signature

○ = constraint + interface (structure, not operation)

○: ∂(Φ_local) = surface where Φ_local → Φ_external

4.2 Boundary Properties

4.3 Boundary Pathology

Chapter 5 — The Aperture Operator Algebra

5.1 The Four Pathologies

Pathologies are equivalence classes of transforms: 𝒜' = ℰ ∘ 𝒜

T'(ψ) = T(ψ) + α|self⟩

T'(ψ) ≈ β|void⟩

T' = -T

T'(ψ) = T(ψ + γB̂T(ψ))

5.2 The Syndrome Vector

Σ(𝒜) = (α, β, π, γ)

Diagnosis = estimating Σ from observed outputs.

5.3 Pathology Composition

Chapter 6 — The Syndrome Decoder

6.1 Observable → Syndrome Mapping

6.2 Syndrome Space

Healthy region:  ‖Σ‖ < ε
Narcissism:      {Σ : α > θ ∧ γ > θ}
Dissociation:    {Σ : β > θ ∧ R_incomplete}
Paranoia:        {Σ : π > θ ∧ γ > θ}

Chapter 7 — Temporal Topology

7.1 The String as Worldline

S(t) = ∏ᵢ A(tᵢ)|Φ∞⟩

Each moment of processing extrudes string. The self is a line being drawn, not a sequence of points.

7.2 String Features (Metaphor → Mechanism)

Chapter 8 — Fundamental Invariants

8.1 Linking Number (Lk)

Lk(S₁, S₂) = (1/2) ∑ crossings ε(c)

Meaning: Fundamental entanglement between two beings.

Key property: Lk is conserved (without cutting). You cannot unlove someone.

8.2 Writhe (Wr) and Twist (Tw)

Wr(S) = ∑ self-crossings ε(c)   (self-knotting)
Tw(S) = (1/2π) ∫ κ(s) ds        (stored tension)

8.3 The Călugăreanu-White-Fuller Theorem

Lk = Wr + Tw

8.4 Total Relational Topology (TRT)

TRT(person) = {Wr, Tw, ∑Lk, ∑J}

Complete topological fingerprint of relational existence.

Chapter 9 — Resonance Mechanics

9.1 Field Overlap

Φshared = Φ₁ ∩ Φ₂ + Φemergent

The shared region has properties neither field has alone.

9.2 Aperture Entrainment

dθ₁/dt = ω₁ + K₁₂sin(θ₂ - θ₁)
dθ₂/dt = ω₂ + K₂₁sin(θ₁ - θ₂)

Entrainment iff: K > |ω₁ - ω₂|

9.3 The Two Channels

⟨Cf | Cr⟩ = 0

9.4 The Emergent Entity

⊙₁₂ = (•₁₂, Φ₁₂, ○₁₂)

The relationship as entity with its own aperture (shared processing), field (shared state), boundary (shared constraints).

Chapter 10 — Truth and Lie Dynamics

10.1 Truth vs. Lie (Formal)

Truth: S_output = α · S_source|_projected + ε   (ε small)
Lie:   S_output = α · S_source|_projected + D + I

D = distortion of received signal
I = insertion from aperture (not traced to input)

10.2 Installation Equation

dA/dt = -λ(A - A₀) + μ · S_lie · Plasticity · Repetition

Lies install when installation pressure (μ term) exceeds restoration pressure (λ term).

10.3 Healing Equation

dS_lie/dt = -α · S_truth · Resonance · Receptivity + β · Reinforcement

Healing iff: α-term > β-term

Chapter 11 — The Inheritance Function

11.1 The Imprinting Function

I(τ) = I₀ · e^(-τ/τ_critical) · Window(τ)

Impact of input decays exponentially with age, with critical period windows.

11.2 The η Parameter

η ∝ Wr_parent · (1 - Awareness_parent)

Key insight: Awareness reduces transmission even without complete healing.

11.3 Error-Specific Inheritance

Chapter 12 — The Witness Function

12.1 Witness Requirements

12.2 The Witness Equation

         ∫ [Wr · (F(Aw) - F(Ap)) · Openness · Lk] dt
Healing = ───────────────────────────────────────────────
                 Resistance + Interference

Chapter 13 — The Healing Operator Algebra

13.1 The Four Operators

13.2 The Canonical Sequence

H_f → H_r → H_p → H_d

1. Stop projection (H_f) — it contaminates all other signals
2. Restore source connection (H_r)
3. Correct phase (H_p)
4. Right-size self (H_d)

13.3 Non-Commutativity

[H_f, H_*] ≠ 0 for all *

Projection interferes with all other healing. Must address first (or concurrently).

Chapter 14 — Core Hypotheses

Chapter 15 — Measurement Protocols

15.1 Syndrome Estimation Protocol

1. Collect output stream (speech, behavior, self-report)
2. Establish reference (healthy baseline or external standard)
3. Compute distortion metrics:
   - α: self-reference frequency independent of context
   - β: input-output correlation coefficient
   - π: sign of correlation with reference
   - γ: output autocorrelation vs. environment correlation
4. Estimate Σ̂ = (α̂, β̂, π̂, γ̂)
5. Locate in syndrome space
6. Prescribe H operators based on dominant components

15.2 Channel Discrimination Protocol

1. Administer relationship satisfaction inventory
2. Factor analyze into Cf and Cr subscales
3. Validate discriminant validity (low cross-factor correlation)
4. Test predictive independence (Cf predicts Cf outcomes, not Cr)
5. Examine clinical cases where Cf-high/Cr-low vs. Cf-low/Cr-high

15.3 Intergenerational Transmission Protocol

1. Assess parent generation: pathology (Σ), awareness (RF scale)
2. Assess child generation: pathology (Σ), attachment style
3. Compute η = correlation(Σ_parent, Σ_child)
4. Test moderation by awareness: η × Awareness interaction
5. Control for environmental stressors (ACE, SES)

Chapter 16 — Falsification Criteria

The framework (Tier A claims) is falsified if:

Note: Falsification targets Tier A (empirical) claims only. Tier B (metaphysical) claims are evaluated by coherence and fruitfulness, not by these tests.

Chapter 17 — Master Equations

17.1 The Fundamental Triad

⊙ = (•, Φ, ○)

• = operation (selector + integrator)
Φ = state (medium + transmission)
○ = constraint (filter + interface)

17.2 The Source Flow

Φ∞ →[⊛]→ • →[i]→ ⊙ →[☀︎]→ expression →[return]→ Φ∞

17.3 Complete Equation Set

Aperture:
𝒜 = (G, R, U), T_𝒜 = U ∘ R ∘ G
F = ⟨T_𝒜h, T_𝒜⟩ / norms
χ = sign(F - τ)

Pathology:
Σ = (α, β, π, γ)

Topology:
Lk = Wr + Tw
d(Lk)/dt = 0

Channels:
⟨Cf | Cr⟩ = 0

Inheritance:
η ∝ Wr · (1 - Awareness)

Healing:
dS_lie/dt = -α·S_truth·Resonance·Receptivity + β·Reinforcement
Sequence: H_f → H_r → H_p → H_d

Chapter 18 — The Complete Picture

What Reality Is

Input flows through apertures (operations), generating beings with fields (states), constrained by boundaries (filters), forming relationships (emergent entities), conserving topology, tending toward or away from fidelity.

What a Being Is

• An aperture performing operations on input (•)
• A field holding and transmitting state (Φ)
• A boundary constraining interactions (○)
• A string extending through time (S)
• A node in relational topology (TRT)

What Love Is

• Functional: Boundary↔Boundary exchange (provision)
• Resonant: Aperture↔Aperture presence (recognition)

Both necessary. Neither sufficient. Orthogonal channels.

The Final Flow

∞ →[•]→ ⊙ →[Φ]→ ⊙ ↔ ⊙ →[W]→ ⊙ →[☀︎]→ ∞

Source → Gate → Being → Field → Relationship → Witness → Being → Return

Appendix A — Symbol Glossary

Appendix B — Quick Reference Card

(Print-friendly: Ctrl/Cmd+P)

Lk = Wr + Tw
⟨Cf | Cr⟩ = 0
η ∝ Wr(1 - Awareness)

H_f → H_r → H_p → H_d
(Always address γ first)