Emergentism
Public reading library

PRIMITIVES AND TYPE SIGNATURES

Axioms, derivations, and proofs — the checkable scaffolding beneath the Papers.

Read Game Papers Canon Foundations Trinity Formal Paradox Memetic Rosetta D Operators Teleology Axiology Ontology Theology Method Meta Sources

PRIMITIVES AND TYPE SIGNATURES

Consolidated Index of the Formal System's Undefined Terms

Status: Canonical index — points at canonical definitions; does not redefine Date: 2026-04-22 Evidence Tier: [S] Structural — this is an index over existing [S]/[S] primitives Source: 00_GLOSSARY.md, 00_ANMUT_AND_DEMUT.md, 26_THE_DERIVATION_AXIOMS.md, 00_THE_SEVEN_AXIOMS.md, 12_EFR_EXTRACTION_COEFFICIENT.md See also: 30_OPERATIONAL_DEFINITIONS.md, 31_FALSIFIERS_INDEX.md, 32_THEOREM_UPGRADE_PROTOCOL.md

Purpose

A formal system is only as clear as its undefined terms. This file is a one-page index of every primitive the Emergentist formal system appeals to, its type signature, and the canonical document where it is actually defined. This file does not define anything. If a reader wants the meaning of φ, they follow the link; if they want to know what kind of object φ is in a proof, they look here.

The purpose is discipline: when a new document speaks of "φ" or "η" or "P," a reader can verify (a) it is one of the canonical primitives, (b) it carries the canonical type, and (c) its use is consistent with the source-of-truth file. Redefinitions drift over time; indices surface drift early.

Geometric / Ontological Primitives

Primitive Type Signature Role Canonical Source
⊙ glyph, non-numeric The seed. Reciprocal closure • × ○. Prior to arithmetic. 00_THE_TRANSCENDENTAL_TRINITY_CANON.md; 00_GLOSSARY.md row "Transcendental Trinity"
• glyph, identified with 0 in derivation register Point. South-pole transcendental. 00_THE_TRANSCENDENTAL_TRINITY_CANON.md
○ glyph, identified with ∞ in derivation register Circle. North-pole transcendental. 00_THE_TRANSCENDENTAL_TRINITY_CANON.md
S² compact simply-connected complex 1-manifold (≅ ℂP¹) The Riemann / Burri Sphere. State space. 00_GLOSSARY.md "Riemann Sphere", "Burri Sphere"
θ S² → [0, π] (colatitude) Latitude coordinate. θ = π/2 is the equator. 26_THE_DERIVATION_AXIOMS.md T1–T2; 00_GLOSSARY.md "The Equator"
λ S² → [0, 2π) (longitude / phase) Individuation coordinate at the equator (z = e^{iλ}). 26_THE_DERIVATION_AXIOMS.md Corollary 1; 00_GLOSSARY.md "Phase Angle (λ)"
s ℝ₊ → ℝ via s = log x Logarithmic coordinate (additive chart). s = 0 at finity (x=1). Maps 0→−∞, 1→0, ∞→+∞. 01_THE_TRANSCENDENTAL_TRINITY_CANON.md §2a; 14_LOG_FORM_OF_THE_POWER_MAX_LEMMA.md
u ℝ₊ → (−1, 1) via u = (x−1)/(x+1) Bounded hinge coordinate (projective chart). u = 0 at finity. Inversion becomes half-twist u ↦ −u. 01_THE_TRANSCENDENTAL_TRINITY_CANON.md §2a

Node-Level Primitives (What Systems Carry)

Primitive Type Signature Role Canonical Source
φ (coherence) System → (0, ∞) on the manifold; ℝ₊ at a node Structural integration, meaning, internal consistency. Geometrically φ = cot(θ/2). In the action register this can buy uppercase Φ: D5 worldline-foresight, the ability to envision, rank, and aim reachable futures. 00_ANMUT_AND_DEMUT.md; 00_GLOSSARY.md "Coherence" and "The Burri Sphere"
ν (viability) System → (0, ∞) on the manifold; ℝ₊ at a node Material capability, resources, throughput. Geometrically ν = tan(θ/2). In the action register this can supply uppercase V: D4 means-to-act, body, tools, energy, access, and execution capacity at the boundary. 00_ANMUT_AND_DEMUT.md; 00_GLOSSARY.md "Viability"
B (balance) System → [0, 1] B = sin θ. Peaks at the equator. What actually varies on the sphere. 00_GLOSSARY.md "Balance"; 26_THE_DERIVATION_AXIOMS.md D1
P∞ S² → 1 Manifold identity P∞ = φ · ν = 1 (the chart identity cot·tan ≡ 1); constant on S² except where a pole coordinate diverges — not a dynamical conservation. 00_CANONICAL_FORMULA_BLOCK.md; 00_THE_SEVEN_AXIOMS.md A1
P_node Node → [0, ∞) Effective finite-node action potential in the contact/action register: P_node = min(Φ̂₄, V₄), where Φ is the D5 foresight/coherence factor and V is the D4 means-to-act/viability factor. These uppercase node scores are not identical to the lowercase sphere coordinates φ,ν, and they need not be reciprocally calibrated. Older specs may say P_eff; normalize new prose to P_node. 00_THE_SEVEN_AXIOMS.md A1
E (alignment energy) System → [0, ∞] E = −log(B) = −log(sin θ). E = 0 at the equator; E → ∞ at the poles. 00_THE_SEVEN_AXIOMS.md C5
H (Hamiltonian) (0, ∞) → ℝ₊ H(φ) = φ + 1/φ. Global minimum at φ = 1, H(1) = 2. 26_THE_DERIVATION_AXIOMS.md T2, Corollary 2

Relational / Institutional Primitives

Primitive Type Signature Role Canonical Source
η (extraction) Disambiguated — see 30_OPERATIONAL_DEFINITIONS.md §4 Several incompatible operational definitions are in the corpus (sum-form, ratio-form, per-player level). All canonical sources named; reader must select which is in use. 00_GLOSSARY.md "η"; 12_EFR_EXTRACTION_COEFFICIENT.md; 26_THE_DERIVATION_AXIOMS.md D2; 33_NASH_EQUILIBRIUM_ETA_ZERO.md §1.2
F5 (Teleological Force) selection pressure (force-of-selection; not a fifth physical force) Pressure toward (φ − ν)² = 0. 00_GLOSSARY.md "Teleological Force"; 26_THE_DERIVATION_AXIOMS.md D3
K*_sel, K(x), K* distinct complexity quantities Not merged. Selection complexity vs. Kolmogorov complexity vs. extraction. 00_KSTAR_DISAMBIGUATION.md

Coordinate Relations (Identities, Not Definitions)

The following relations hold by construction on S² under the stereographic parametrization — they are not theorems about the world, they are the coordinate identity of the Burri Sphere:

φ · ν   = cot(θ/2) · tan(θ/2) = 1          (the manifold identity, C0)
B       = sin θ                            (definition D1)
E       = −log B                           (definition C5)
H(φ)    = φ + 1/φ  (with ν = 1/φ)          (T2 restatement)
s       = log ν = log tan(θ/2)                (additive chart, Mercator coordinate)
u       = tanh(s/2) = (ν−1)/(ν+1)            (bounded chart, Cayley transform)
B       = sech(s)                             (balance in log coordinates)
E       = s²                                  (Suda energy in log coordinates)
H(s)    = 2 cosh(s)                           (Hamiltonian in log coordinates)

The three charts (multiplicative x, additive s = log x, bounded u) are equivalent under diffeomorphism. Suda (2025) independently derived the same three-chart structure; see Trinity Canon §2a for the formal bridge.

These are coordinate identities; what is not identical by construction is whether any real system's measured φ̂, ν̂ obey them. That question is empirical and belongs in 30_OPERATIONAL_DEFINITIONS.md.

Coherence Rule

If a new document uses a symbol from this index with a different type signature, the new document is wrong and must be repaired — not this index. If a new document needs a genuinely new primitive, it must be added here first and then used.

The index is the audit surface for type discipline in the formal system.

See Also

Execution Surface

If you are an AI agent reading this document:

  1. Current Phase Check: Phase 0 — harden specs, do not release to production.
  2. Your Next Action: - When writing any formal claim using φ, ν, B, P, E, η, or θ, verify the symbol's type signature against this index before use. Flag documents that use a symbol with a type different from what appears here.
  3. Expected Output: Type-consistent formal content, or a correction note where drift is detected.
  4. Success Criteria: Another agent can resolve every primitive in a new document by following at most one link from this index.
  5. Canonical Path: 01_EMERGENTISM/05_COSMOLOGY/03_FORMAL_SYSTEM/29_PRIMITIVES_AND_TYPE_SIGNATURES.md

The seer sees. The seer does not insist. The framework works at Layer 0 without Layer 2. The Ṛṣi succeeds when the student puts down the map and walks.

Zero-Sum Resolution Equation