When Selection Is Silent: Observability-Gated Structure in the UNNS Substrate

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Category: Labs

UNNS κ-Series · Chamber κ₂ · Observability Layer

κ₂ Dormancy: Selection Exists — But May Be Unobservable

κ₂ is the first κ-operator in UNNS that is not universally active. It runs only when an observability gate (Ω₂) detects a real, non-null parity distinction. When Ω₂ is inactive, κ₂ is silent by design — and that silence is a scientific result.

Key finding

Real κ₁-selected ensembles often collapse Σ₂ᵖ (parity) variance. As a result, Ω₂ remains inactive and κ₂ performs no selection. Dormancy is the dominant regime, not an exception.

What this changes

UNNS separates (i) structure, (ii) ranking, and (iii) observability. A distinction can exist yet be operationally invisible to selection.

Where κ₂ Lives in the Operator Stack

κ₁ selects directly on ensembles (universal selection). κ₂ is conditional: it lives inside Ω. If Ω₂ is inactive, κ₂ is skipped and the ensemble passes through unchanged.

Σ E Ω layer Ω₂ gate decides if κ₂ can run If Ω₂ inactive → κ₂ skipped → E passes through If Ω₂ active → κ₂ executes (policy a/b/c) κ₂ τ κ₂ is conditional selection; τ is optional validation (non-interference) in the chamber.
κ₂ is the first observability-gated κ-operator: selection is permitted only when Ω₂ detects meaningful parity structure.

Definitions (Public-facing)

Σ2p: parity classifier (not a magnitude)

Each state receives a discrete parity label: Σ2p(s) ∈ {EVEN, ODD, NULL}. This is not a real-valued score and does not “rank” states. NULL means “absence of parity-bearing structure”.

Ω₂: observability gate (empirical)

Ω₂ activates only if BOTH parity classes are present and the parity variance exceeds ε. In Chamber κ₂, variance is computed over parity-bearing states only (EVEN/ODD); NULL is excluded.

activeCounts = count(EVEN) + count(ODD)

peven = count(EVEN) / activeCounts

podd = count(ODD) / activeCounts

Var(Σ2p) = peven(1 − peven) + podd(1 − podd)

Ω₂ is ACTIVE iff Var(Σ2p) > ε and activeCounts ≥ 2.

Ω₂ Gate: Why κ₂ Often Does Nothing

κ₂ dormancy is not a failure state. It is the correct outcome when parity structure is absent or collapsed. The chamber explicitly treats “Ω₂ inactive” as meaningful: κ₂ is skipped and validation is unavailable for that run. :contentReference[oaicite:1]{index=1}

Input: κ₁-selected ensemble E Compute Σ₂ᵖ for each state (EVEN / ODD / NULL) Do BOTH parity classes exist? Require count(EVEN) ≥ 1 and count(ODD) ≥ 1 Ω₂ INACTIVE κ₂ skipped · E passes unchanged No κ₂ validation (nothing executed) Compute variance Var(Σ₂ᵖ) over EVEN/ODD only If Var > ε → Ω₂ ACTIVE → κ₂ executes NO YES
Ω₂ prevents “forced selection”: if parity structure is not empirically present, κ₂ is silent.

What We Verified Experimentally

Dormancy regime (dominant)

On real κ₁ outputs, Σ₂ᵖ often collapses to a single parity class (e.g., all EVEN). Then Var(Σ₂ᵖ) = 0 ⇒ Ω₂ inactive ⇒ κ₂ does nothing.

If Var(Σ₂ᵖ) = 0, then κ₂(E) = E.

Forced activation (control)

If the ensemble is constructed to include at least one EVEN and one ODD state, then Var(Σ₂ᵖ) > 0 and Ω₂ can activate. κ₂ then executes deterministically.

If count(EVEN) ≥ 1 and count(ODD) ≥ 1 and Var(Σ₂ᵖ) > ε, then κ₂ selects.

κ₁ → Ω₂ Collapse: How Observability Is Lost

This figure shows the empirical mechanism behind κ₂ dormancy. Before κ₁ selection, parity structure may exist. After κ₁ symmetry selection, parity variance typically collapses, rendering Ω₂ inactive.

Before κ₁ (structure exists) Parity contrast present EVEN ODD Var(Σ₂ᵖ) > 0 → Ω₂ can activate κ₁ selection After κ₁ (projection) Parity collapsed EVEN ODD = 0 Var(Σ₂ᵖ) = 0 → Ω₂ inactive → κ₂ dormant
κ₁ does not merely rank states — it projects ensembles into an Ω₂-silent subspace. κ₂ dormancy emerges because parity distinctions no longer survive selection.

This pair of outcomes matters: it proves κ₂ is neither “always on” nor “vacuous”. Dormancy is structural, not a bug.

The κ₂ Bifurcation

κ₂ introduces a new kind of experimental statement: whether selection is even observable is itself measurable (via Ω₂), and can remain false under symmetry-selected ensembles.

Compute Σ₂ᵖ distribution on κ₁ ensemble EVEN / ODD / NULL counts Dormancy single parity class ⇒ Var = 0 ⇒ Ω₂ inactive Activation EVEN and ODD present ⇒ Var > ε ⇒ Ω₂ active ⇒ κ₂ runs κ₂(E) = E κ₂(E) ≠ E (policy-dependent)
κ₂ adds a measurable “on/off” condition for selection itself.

Significance and Implications

1) Observability is structural

Ω₂ is not a narrative device; it is a computed condition. A distinction may exist yet remain operationally unobservable to selection. This breaks the common assumption “if a distinction exists, it must act”.

2) Dormancy is not failure

“Nothing happened” becomes a legitimate experimental outcome: the system refuses to hallucinate selection when structure does not support it.

3) Why κ₃ is logically forced

The forced-activation control shows κ₂ can act when observability exists. But κ₁ often projects away parity variance. The next question is therefore: what governs persistence or re-entry of observability across layers? This motivates κ₃ as nested observability / higher-order selection of gates, not “stronger κ₂”.

How to Reproduce (No Hidden Steps)

  1. Open the κ-Series Selection Laboratory and run κ₀ → κ₁ to produce a κ₁ results JSON (input B).

    Open in Fullscreen!

  2. Load the κ₁ JSON into Chamber κ₂. Choose parity mode (domain-wall recommended) and ε (default 0.10).
    Chamber κ₂ (v1.2.0)
  3. Observe Ω₂ status:
    • If Ω₂ is INACTIVE: κ₂ is skipped and the ensemble passes unchanged.
    • If Ω₂ is ACTIVE: choose κ₂ policy a/b/c and run selection.
  4. Optional: enable τ validation (CK2.3) to test non-interference when Ω₂ is inactive.
  5. Export κ₂ results JSON for reporting / comparison.
Tip: Minimal Ω₂-activating test

Ω₂ requires at least one EVEN and one ODD parity-bearing state. A minimal control ensemble with two states (one EVEN, one ODD) will activate Ω₂ immediately (Var = 0.5), enabling κ₂ execution and validation.

References

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