TDH III — Computational Feasibility, Inference, and Extended Formalism -

PAPER III

TDH III — Computational Feasibility, Inference, and Extended Formalism (Eq. 21–58)

Abstract

This paper provides the computational and inferential program for TDH and extends the formalism to reach a complete Eq. (1)–(58) package. Family-level coherence is estimated via similarity scoring and aggregated into q̂(t); the structure-driven dynamic is fit in discrete time; and homeostatic outcomes are linked to coherence via the thresholded gate. Extensions include activity-weighted coherence, coherence dispersion across families, effective-family-number diagnostics, family-heterogeneous dynamics, persistence-adjusted closure, node/edge turnover decomposition, time-to-threshold measures, envelope deficit summaries, hazard summaries and timescales, a latent measurement model, likelihood-based inference, effective population size scaling using Nₘ and N𝒻, and a divergence extension showing how sustained CRS plus low mixing can generate long-run breakaway.

Keywords

computational biology; inference; state-space modeling; effective population size; divergence; breakaway; measurement

1. Estimating family-level coherence and aggregate coherence

A practical family-level estimator is:

q̂ₖ(t) = [1 / N(N − 1)] Σᵢ₌₁ᴺ Σⱼ₌₁ᴺ,ⱼ≠ᵢ sₖ(i,j,t), (21)

where sₖ(i,j,t) ∈ [0,1] is a similarity score for individuals i and j with respect to family k (insertion states, expression, chromatin, methylation, or other family observables).

Aggregate coherence estimator:

q̂(t) = (1/K) Σₖ₌₁ᴷ q̂ₖ(t). (22)

2. Inference-ready discrete dynamic fit

An empirical discrete form of Eq. (4) is:

q̂(t + 1) − q̂(t) = ρC(t)[1 − q̂(t)] − [μ + ντ(t)][q̂(t) − α] + ε(t), (23)

with ε(t) capturing noise and unmodeled structure. This can be fit with nonlinear regression, state-space models, Bayesian hierarchical models, or particle filtering.

3. Activity-weighted coherence and family heterogeneity

Families differ in current activity or importance. Let Aₖ(t) ≥ 0 denote family activity. Define normalized activity weights:

ωₖ(t) = Aₖ(t) / Σⱼ₌₁ᴷ Aⱼ(t), with Σₖ₌₁ᴷ ωₖ(t) = 1. (24)

Define activity-weighted coherence:

q_w(t) = Σₖ₌₁ᴷ ωₖ(t)qₖ(t). (25)

Estimator:

q̂_w(t) = Σₖ₌₁ᴷ ω̂ₖ(t)q̂ₖ(t). (26)

Family dispersion (heterogeneity diagnostic):

σ_q²(t) = (1/K) Σₖ₌₁ᴷ [qₖ(t) − q(t)]². (27)

Effective number of active families (concentration diagnostic):

K_eff(t) = 1 / Σₖ₌₁ᴷ ωₖ(t)². (28)

A family-heterogeneous dynamic (optional refinement) is:

dqₖ/dt = ρₖC(t)(1 − qₖ) − [μₖ + νₖτ(t)](qₖ − αₖ) + ξₖ(t). (29)

Consistency relation linking family drift to aggregate drift:

dq/dt = (1/K) Σₖ₌₁ᴷ dqₖ/dt. (30)

4. Persistence-adjusted structure and turnover decomposition

A persistence-adjusted structure score (useful when closure and low-turnover both matter) is:

C̃(t) = ηC(t) + (1 − η)[1 − τ(t)], with η ∈ [0,1]. (31)

Edge persistence:

P(t) = |Eₜ ∩ Eₜ₊₁| / |Eₜ ∪ Eₜ₊₁|. (32)

Persistence-adjusted closure:

Ĉ(t) = C(t)P(t). (33)

Node turnover (explicitly named):

τ_V(t) = 1 − |Vₜ ∩ Vₜ₊₁| / |Vₜ ∪ Vₜ₊₁|. (34)

Edge turnover:

τ_E(t) = 1 − |Eₜ ∩ Eₜ₊₁| / |Eₜ ∪ Eₜ₊₁|. (35)

Composite turnover (if both node and edge turnover matter):

τ(t) = γ_Vτ_V(t) + γ_Eτ_E(t), with γ_V ≥ 0, γ_E ≥ 0, γ_V + γ_E = 1. (36)

A simple openness index is:

O(t) = 1 − C(t). (37)

5. Threshold-crossing times and envelope deficit summaries

Time to constructive throughput (gate opening):

t* = inf{t ≥ 0 : q(t) ≥ q*}. (38)

Time to maintain a specific envelope Eₙ:

tₙ† = inf{t ≥ 0 : q(t) ≥ qₙ†}. (39)

A time-varying maintainable-envelope index:

n̂(t) = max{n : dₙ ≤ g(q(t))}. (40)

Weighted envelope deficit:

Δ_E(t) = Σₙ wₙ[dₙ − g(q(t))]₊. (41)

Mean deficit over a horizon T:

Δ̄_E(T) = (1/T)∫₀ᵀ Δ_E(u) du. (42)

6. Hazard summaries and survival timescales

Mean hazard over horizon T:

λ̄ₑₓₜ(T) = (1/T)∫₀ᵀ λₑₓₜ(u) du. (43)

Equivalent survival representation (from Eq. 18):

S(T) = exp[−Tλ̄ₑₓₜ(T)]. (44)

Define the asymptotic hazard constant under the ORS α-floor regime (Eq. 19):

λ∞ = λ₀ + λ₁ Σₙ wₙdₙ. (45)

Survival half-life:

T₁/₂ = (ln 2) / λ∞. (46)

Mean survival time:

E[T_surv] = 1 / λ∞. (47)

These provide interpretable timescales without changing the core TDH extinction logic.

7. Latent measurement model and likelihood-based inference

A general latent measurement model for family coherence uses multiple indicators xₖ,ₘ(t):

xₖ,ₘ(t) = aₖ,ₘ + bₖ,ₘqₖ(t) + ηₖ,ₘ(t). (48)

Homeostatic outcomes can be modeled as Bernoulli draws:

yₙ(t) ∼ Bernoulli(Pₙ(maintain | q(t))). (49)

A standard log-likelihood for parameters θ (collecting dynamic, measurement, and demand parameters) is:

ℓ(θ) = Σₜ Σₙ { yₙ(t) ln Pₙ(maintain | q(t)) + [1 − yₙ(t)] ln[1 − Pₙ(maintain | q(t))] }. (50)

This likelihood can be optimized (MLE) or used in Bayesian inference. TDH’s distinctive prediction is that the dependence on q enters through g(q) and thus is thresholded.

8. Effective population size scaling (Nₘ and N𝒻)

Define effective population size using your preferred male/female counts:

Nₑ(t) = [4Nₘ(t)N𝒻(t)] / [Nₘ(t) + N𝒻(t)]. (51)

A common modeling choice is that dynamic noise shrinks as effective sample size grows:

Var(ε(t)) = σ² / Nₑ(t). (52)

This links demographic structure to estimation noise and (optionally) to expected coherence volatility.

9. Structural adjustment (generic control law)

Beyond the specific CRS formation parameterization (Eq. 7–8), one may model gradual structural adjustment toward a target closure level:

C(t + 1) − C(t) = κ_C [C_target − C(t)], with κ_C ∈ (0,1]. (53)

This is useful in simulations where closure is treated as an evolving macro-structural state rather than an imposed function of time.

10. Divergence extension (CRS breakaway under low mixing)

This extension formalizes the idea that sustained high closure and coherence can support long-run structured divergence when cross-boundary mixing is low.

Let D_AB(t) ∈ [0,1] denote structured divergence between populations A and B. Let M_AB(t) ∈ [0,1] denote cross-boundary reproductive mixing. Define:

D_AB(t + 1) − D_AB(t) = κ[1 − D_AB(t)]Q_AB(t) − χM_AB(t)D_AB(t). (54)

Define the mean closure-weighted coherence driver:

Q_AB(t) = [C_A(t)q_A(t) + C_B(t)q_B(t)] / 2. (55)

Define a mixing measure from reproductive edge counts:

M_AB(t) = E_AB(t) / [E_AA(t) + E_BB(t) + E_AB(t)]. (56)

If Q_AB(t) → Q̄_AB and M_AB(t) → M̄_AB, the implied equilibrium divergence is:

D̄_AB = κQ̄_AB / [κQ̄_AB + χM̄_AB]. (57)

Let D* ∈ (0,1) denote a breakaway/speciation threshold. The model predicts threshold crossing when:

D̄_AB > D*. (58)

Scope note (important): this is a conditional dynamical extension. It does not claim CRS automatically implies speciation, and it must not be used to justify coercion or discrimination in human contexts. TDH’s core claims (coherence gating, CRS relocking, Eₙ homeostasis, CLAH, ORS fragility) do not require this extension.

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