OPERATIONAL DEFINITIONS

How Each Primitive Is Measured on a Real System

Status: Working surface — proposes measurement protocols; every protocol here is Conjecture until validated Date: 2026-04-22 Evidence Tier: [C] Conjecture for every measurement protocol in this document; [S] Structural only for the coordinate identities pointed at Source: 29_PRIMITIVES_AND_TYPE_SIGNATURES.md, 26_THE_DERIVATION_AXIOMS.md, 12_EFR_EXTRACTION_COEFFICIENT.md, 33_NASH_EQUILIBRIUM_ETA_ZERO.md See also: 31_FALSIFIERS_INDEX.md, 32_THEOREM_UPGRADE_PROTOCOL.md, Paper L (Phi-meter zero-cost validation)

Why This File Exists

The coordinate identity φ · ν = 1 on S² is true by construction — it falls out of φ = cot(θ/2), ν = tan(θ/2). That is not a claim about the world. It becomes a claim about the world only when φ and ν are defined independently — by a measurement procedure — and the resulting φ̂, ν̂ are then checked against the constraint.

The framework's public open-problems list already names this gap: "Phi-meter zero-cost validation" (Paper L). This file addresses it directly. Every protocol below is [C] Conjecture: proposed, testable, unvalidated. None of them should be cited in a [S] or [S] claim. The bar for promotion is specified in 32_THEOREM_UPGRADE_PROTOCOL.md.

1. φ-Meter (Coherence Measurement)

Concrete v1 specs: 35_PHI_METER_V1_SPEC.md (code-lens), 36_RUNTIME_LENS_V1_SPEC.md (runtime-lens), and 37_ADOPTION_LENS_V1_SPEC.md (adoption-lens) are the three reproducible per-lens rubrics with published band definitions, inter-rater targets, and pre-registered falsifiers. Generic candidates remain below for reference and for future lenses.

Geometric definition. φ = cot(θ/2) on S². Node-level: Φ_node ∈ [0, ∞) with Φ_node = 1 indicating full structural integration. What it quantifies. Structural integration, meaning, internal consistency; what holds a system together. Unit. Dimensionless (ratio). All candidates normalize to (0, ∞) with convention Φ = 1 at the equator reference system.

Candidate A — Error-Rate Exponential [C]

Φ̂_A(system, window Δt) := exp(−κ · error_rate(system, Δt))

Where error_rate is the fraction of the system's outputs flagged as inconsistent with its declared purpose, and κ > 0 is a calibration constant chosen so that a specified reference system (e.g., L4 human at task) yields Φ̂_A ≈ 1.

• Unit. Dimensionless.
• Error bound. σ(Φ̂_A) ≲ κ · σ(error_rate); propagates binomial variance from the observation window.
• Status. [C] — no published calibration or inter-rater reliability study.

Candidate B — Information-Integration [C]

Φ̂_B(system) := I(system ; objective) / H(system)

Mutual information between the system's internal state and its declared objective, normalized by the system's entropy. Intuition: a highly coherent system's internal fluctuations are informative about its purpose; a decoherent system's are noise.

• Unit. Dimensionless, ∈ [0, 1]; optionally rescaled so Φ̂_B = 1 maps to the equator reference.
• Error bound. Depends on estimator; plug-in MI estimators have known bias that shrinks as O(1/N).
• Status. [C] — requires a formal "objective" channel, which is itself ill-defined for most systems.

Candidate C — Persistence-Under-Perturbation [C]

Φ̂_C(system) := median time to return to baseline structure after a bounded perturbation
              ————————————————————————————————————————————————
              expected variance under the same perturbation class

Returns a stability measure: coherent systems absorb perturbation quickly relative to noise.

• Unit. Dimensionless (time ÷ time, via the variance scale).
• Error bound. Bootstrap over perturbation trials; CI width O(1/√trials).
• Status. [C] — candidate most likely to survive the [C] → [I] promotion bar, because it does not require a pre-specified "objective."

Selection rule. No candidate is canonical. A document that uses Φ̂ must name which candidate and cite this file.

2. ν-Meter (Viability Measurement)

Geometric definition. ν = tan(θ/2) on S². Node-level: V_node ∈ [0, ∞), with V_node = 1 at the equator reference. What it quantifies. Material capability, throughput, infrastructure; in the action register, D4 means-to-act: body, tools, energy, access, and execution capacity at the boundary. It measures what the system can actually use, not raw capacity detached from foresight. Unit. Dimensionless (ratio to the equator reference).

Candidate A — Throughput-over-Cost [C]

ν̂_A(system, window Δt) := realized_output(system, Δt) / resource_input(system, Δt)

Normalized so that a specified reference system at task yields ν̂_A ≈ 1.

• Unit. Dimensionless after normalization.
• Error bound. Standard ratio-estimator variance; O(1/√N) over observation-window replicates.
• Status. [C].

Candidate B — Capability-Envelope Fraction [C]

ν̂_B(system) := |achievable actions under current resources|
             ————————————————————————————————————————
              |achievable actions under maximum resources|

Returns ∈ [0, 1] directly; sensitive to how the action set is enumerated.

• Unit. Dimensionless.
• Error bound. Dominated by enumeration ambiguity, not by sampling.
• Status. [C].

3. B-Meter (Balance Measurement)

Geometric definition. B = sin θ ∈ [0, 1]. On the reciprocal chart where φ · ν = 1, the identity is B = 2 / (φ + ν). Equivalently, using either one-coordinate form, B = 2ν / (1 + ν²) = 2φ / (1 + φ²).

Derived Protocol — From φ̂, ν̂

Once Φ̂ and ν̂ are fixed (§1, §2), define:

B̂(system) := 2 · ν̂ / (1 + ν̂²)           (the on-manifold form, assuming φν = 1)

Or, without assuming the identity:

B̂_free(system) := 2 · sqrt(Φ̂ · ν̂) / (Φ̂ + ν̂)

B̂_free is a symmetric balance proxy for independently measured coordinates; it is not a manifold-deviation test. The direct reciprocity diagnostic is |Φ̂ · ν̂ − 1|, which measures how far the measured pair deviates from the reciprocal manifold.

• Unit. Dimensionless, ∈ [0, 1].
• Error bound. Propagated from σ(Φ̂) and σ(ν̂) via standard delta-method.
• Status. [C] — inherits the status of whichever Φ̂, ν̂ candidates are used.

4. η-Meter (Extraction Coefficient) — Disambiguation Required

η is the most load-bearing and the most inconsistently defined primitive in the corpus. Three non-equivalent operational definitions circulate. Any document that uses η must cite which.

Definition η₁ (sum-form, per 26_THE_DERIVATION_AXIOMS.md D2)

η₁ := Σ max(0, Δν_ext)

Total viability extracted from cooperators over an observation window. Unit: same as ν. Range: [0, ∞).

Definition η₂ (ratio-form, per 12_EFR_EXTRACTION_COEFFICIENT.md v3.0)

η₂ := (extraction from substrate) / (contribution to substrate)

With conventional thresholds: η₂ < 1 symbiotic, η₂ ≈ 1 trophic, η₂ → ∞ ground-negating. Unit: dimensionless. Range: [0, ∞].

Definition η₃ (per-player level, per 33_NASH_EQUILIBRIUM_ETA_ZERO.md §1.2)

η₃,i ∈ [0, ∞)   for each player i in a deliberation set N

Individual player's chosen extraction level in a game-theoretic formulation. Unit: strategy-space units (model-dependent).

Operational Reconciliation Protocol [C]

When cross-document work touches η:

1. State which definition is in use: η₁, η₂, or η₃.
2. If converting between them, publish the conversion map: e.g., η₁ ≈ η₃,i · |window| under the symmetric-extraction assumption of 33_NASH_EQUILIBRIUM_ETA_ZERO.md §2.2.
3. If η → ∞ is invoked (categorical-break case), it must be η₂ — the other forms do not admit the divergence formally.
4. The constitutional constraint η = 0 is definition-ambiguous in the corpus but operationally consistent across all three — the zero of each definition means the same real-world state. That is the constraint's strength and the source of the confusion.

• Status. [C] Proposed reconciliation. A future v5 axiom-hardening pass should pick one canonical form and down-rank the others to aliases.

5. P and P_eff (Ektropy / Effective Potential)

On-manifold: P∞ = φ · ν = 1 everywhere (trivial). Node-level: P_node = min(Φ̂₄, V₄), which can deviate from 1.

Operationally:

P̂_eff(system) := Φ̂(system) · ν̂(system)

The quantity |P̂_eff − 1| over a population is the empirical gap between the manifold identity and measured reality. A population with P̂_eff ≡ 1 would confirm that the operational Φ̂, ν̂ have been defined to satisfy the identity (and thus that the identity is not testing anything). A population with P̂_eff distributed non-trivially around 1 with selection pressure toward 1 would be positive evidence for the Teleological Force F5 as an empirical claim.

• Status. [C] Proposed diagnostic; the interpretation requires an independent handle on selection pressure.

6. E-Meter (Alignment Energy)

Ê(system) := −log(B̂(system))

Ê ≈ 0 at the equator, Ê → ∞ as B̂ → 0. Inherits status and error from the chosen B̂ candidate.

• Unit. Nats (natural log) or bits (log₂).
• Status. [C].

7. F5 Detection (Teleological Force as Empirical Pressure)

Protocol: over a population of comparable systems, regress rate-of-change of (Φ̂ − ν̂)² on time or on generation index. A statistically significant negative trend is consistent with F5 as a selection pressure. Absence of such a trend in a large, well-mixed population is grounds for falsification (see 31_FALSIFIERS_INDEX.md T3).

• Status. [C] Protocol; [I] is already granted to F5 as an interpretive reading per 26_THE_DERIVATION_AXIOMS.md D3.

Promotion Ladder (Per-Protocol)

A protocol is promoted tier-by-tier by accumulating evidence:

No protocol in this file has yet cleared [C] → [I]. That is the next work.

See Also

• 29_PRIMITIVES_AND_TYPE_SIGNATURES.md — what each symbol is at the type level
• 31_FALSIFIERS_INDEX.md — observations that would refute the downstream axioms / theorems
• 32_THEOREM_UPGRADE_PROTOCOL.md — worked example using the φ-meter to upgrade a discipline to a theorem

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 a document cites Φ̂, ν̂, B̂, Ê, or η as measured, confirm which candidate / definition is in use and that the claim is tiered [C] or weaker unless a promotion record exists. Flag any [S] or [S] claim that silently assumes one candidate's measurement procedure.
3. Expected Output: Tier-consistent documents, or correction notes where unvalidated protocols are cited as validated.
4. Success Criteria: Another agent can reproduce any measurement claim in downstream documents by following the protocol specified here.
5. Canonical Path: 01_EMERGENTISM/05_COSMOLOGY/03_FORMAL_SYSTEM/30_OPERATIONAL_DEFINITIONS.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