PAPER II

TDH II — Eₙ Homeostasis, CLAH, and ORS Extinction Under an α-Floor

Abstract

This paper formalizes TDH’s phenotype-maintenance layer. Environmental envelopes Eₙ impose increasing adaptive demands dₙ. Homeostatic maintenance is feasible only when the thresholded coherence gate g(q) exceeds the relevant demand. This yields Eₙ homeostasis, a demand-specific critical coherence qₙ†, and a coherence-indexed capacity n_max(q). A probabilistic bridge enables empirical inference under noisy outcomes. The coherence-saturated limit q → 1 defines coherence-limited adaptive homeostasis (CLAH): within the system’s normal perturbation envelope, phenotype maintenance becomes effectively guaranteed. Finally, the paper proves a key TDH result: a positive α-floor does not prevent extinction when α < q* and nontrivial demands recur; long-run survival can still decay to zero.

Keywords

homeostasis; CLAH; environmental envelopes; extinction hazard; α-floor; threshold dynamics

---

1. Environmental envelopes and homeostatic margin

Let Eₙ denote the n-th environmental envelope with demand dₙ ∈ (0,1], indexed so that d₁ ≤ d₂ ≤ d₃ ≤ … .

Define the homeostatic margin:

mₙ(q) = g(q) − dₙ. (12)

Homeostasis under Eₙ is feasible when mₙ(q) ≥ 0.

---

2. Demand-specific critical coherence

Because g(q) is thresholded, the critical coherence required for envelope Eₙ is:

qₙ† = q* + (1 − q*)dₙ. (13)

Thus Eₙ can be maintained only when q ≥ qₙ†.

---

3. Largest maintainable envelope

Define the coherence-indexed capacity:

n_max(q) = max{n : dₙ ≤ g(q)}. (14)

As q increases, the largest maintainable environmental envelope increases.

---

4. Probabilistic bridge (for inference under noisy outcomes)

When maintenance is stochastic rather than deterministic, define:

Pₙ(maintain | q) = 1 / [1 + exp(−βmₙ(q))], (15)

with β > 0 controlling sharpness.

---

5. CLAH (coherence-limited adaptive homeostasis)

TDH defines coherence-limited adaptive homeostasis (CLAH) as the coherence-saturated limit in which, for perturbations within a system’s normal envelope, phenotype maintenance becomes effectively constant/guaranteed as q approaches 1.

Formally, for any fixed Eₙ with dₙ < 1:

lim(q→1) Pₙ(maintain | q) = 1. (16)

This is a homeostatic stability limit within the relevant envelope, not a moral claim and not an assertion that all change ceases.

---

6. ORS extinction with α-floor

If α < q*, then g(α) = 0. A positive floor can coexist with zero constructive gate output.

Define an extinction hazard:

λₑₓₜ(t) = λ₀ + λ₁ Σₙ wₙ[dₙ − g(q(t))]₊, (17)

where λ₀ ≥ 0 is baseline hazard, λ₁ > 0 scales coherence-sensitive hazard, wₙ ≥ 0 are envelope weights, and [x]₊ = max{x,0}.

Define survival over horizon T:

S(T) = exp[−∫₀ᵀ λₑₓₜ(u) du]. (18)

If q(t) → α < q*, then g(q(t)) → 0 and:

λₑₓₜ(t) → λ₀ + λ₁ Σₙ wₙdₙ. (19)

If recurring demands are nontrivial (Σₙ wₙdₙ > 0), then long-run survival satisfies:

S(T) → 0 as T → ∞. (20)

Bottom line: persistent ORS can lead to extinction even when coherence stabilizes above zero (α-floor), provided α < q* and demands recur.