PAPER I

TDH I — Coherence-Gated Evolvability and Structure-Driven Coherence Dynamics (CRS vs ORS)

Abstract

This paper states the core formal claim of the Transposon Dial Hypothesis (TDH): constructive evolvability depends not only on candidate-variant generation and downstream natural selection, but also on a thresholded coherence state that gates whether potentially constructive novelties can be retained, integrated, and expressed. The state variable is Active Transposon Family Coherence q(t) ∈ [0,1], defined as the aggregate coherence of active transposon families across a reproducing population. TDH distinguishes closed reproductive structure (CRS)—a bounded, persistent reproductive network—from open reproductive structure (ORS)—a persistently rewired, high-turnover network. A unified structure-driven dynamic is given for q(t), and the completed CRS trajectory is shown to be generically U-shaped during CRS formation: coherence may decline initially, then “relock,” recover, and rise toward a high-coherence plateau. Persistent ORS is shown to approach a low positive α-floor, preparing the ground for the homeostatic and extinction results developed in Paper II.

Keywords

Transposon Dial Hypothesis; coherence gating; evolvability; CRS; ORS; reproductive structure; turnover; relocking

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1. The TDH causal ordering

TDH enforces a three-layer architecture:

generation → coherence-gated evolvability filter → observed outcome (then selection acts downstream).

This ordering matters: a population may generate abundant candidate novelty and still fail to exhibit constructive long-horizon change if constructive candidates are filtered out prior to stable phenotypic retention.

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2. Core state variable and gate

Let qₖ(t) ∈ [0,1] denote coherence of active transposon family k at time t, for k = 1,2,…,K. Define aggregate coherence:

q(t) = (1/K) Σₖ₌₁ᴷ qₖ(t). (1)

Let q* ∈ (0,1) be the coherence threshold required for constructive outcomes to begin passing the filter. Define the thresholded-linear constructive gate:

g(q) = 0, for q < q*
g(q) = (q − q*)/(1 − q*), for q ≥ q*. (2)

Define the incoherence penalty:

h(q) = 1 − q. (3)

Interpretation: g(q) is usable constructive throughput (filter output). h(q) is residual incoherence burden.

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3. CRS and ORS (structural definitions)

CRS (closed reproductive structure): a bounded and persistent reproductive network in which most reproduction occurs within the same population across generations.

ORS (open reproductive structure): a more persistently rewired, less bounded, higher-turnover reproductive network.

CRS is not inbreeding (close-kin mating). A CRS can be large enough to avoid close-kin mating entirely.
CRS is not assortative reproduction (mate choice based on similarity). CRS is about bounded persistence of the reproductive graph.

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4. A unified coherence dynamic driven by structure

TDH uses a minimal structure-driven dynamic:

dq/dt = ρC(t)(1 − q) − [μ + ντ(t)](q − α). (4)

Parameters: ρ > 0 (closure-driven coherence building), μ > 0 (baseline erosion), ν > 0 (turnover-sensitive erosion).
α ∈ (0,q*) is the low positive coherence floor approached under persistent ORS.

Interpretation: closure builds coherence toward saturation; turnover (plus baseline erosion) pulls coherence toward α.

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5. Operational structure measures (no underscore-style names)

Let Eᵢ(t) be within-boundary reproductive edges and Eᶜ(t) be boundary-crossing reproductive edges. Define closure:

C(t) = Eᵢ(t) / [Eᵢ(t) + Eᶜ(t)]. (5)

Let Vₜ be the set of reproducing nodes at generation t. A practical turnover measure is:

τ(t) = 1 − |Vₜ ∩ Vₜ₊₁| / |Vₜ ∪ Vₜ₊₁|. (6)

(Other empirically appropriate turnover measures can be substituted without changing TDH.)

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6. The completed CRS trajectory (U-shape)

A simple CRS formation parameterization is:

C(t) = C∞(1 − e^(−λt)), (7)

τ(t) = τ∞ + (τ₀ − τ∞)e^(−δt), (8)

with C∞ near 1 in mature CRS, τ₀ > τ∞, and λ, δ > 0.

Turning point (“relocking”) occurs at time t = t_c defined by dq/dt = 0:

ρC(t_c)(1 − q(t_c)) = [μ + ντ(t_c)](q(t_c) − α). (9)

For t < t_c, coherence can decline; for t > t_c, coherence rises. This yields the predicted U-shaped CRS relocking trajectory.

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7. Long-run equilibria under stable regimes

If C(t) → C∞ and τ(t) → τ∞ (mature CRS), Eq. (4) yields:

q̄_C = [ρC∞ + (μ + ντ∞)α] / [ρC∞ + μ + ντ∞]. (10)

If C(t) → Cₒ (small) and τ(t) → τₒ (positive) under persistent ORS, then:

q̄_O = [ρCₒ + (μ + ντₒ)α] / [ρCₒ + μ + ντₒ]. (11)

When Cₒ is small, q̄_O ≈ α: ORS does not require q → 0; it can settle near a low positive floor.

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8. Two asymptotic cases (kept for completeness)

Case A (unbounded decline): q(t) → 0 under sufficiently strong disordering forces.
Case B (low equilibrium with floor): q(t) → α with α > 0.

One-line summary: TDH proceeds with Case B (α-floor) while retaining Case A as a limiting failure mode.