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Unified Notation + Core Equations Page
Unified Notation + Core Equations Page
TDH Submission Package (Papers I–III) — Global Equation System Eq. (1)–(58) Unicode only. No underscore-style symbols. ORS floor parameter is α.
0. Reading map (how the three papers use the shared system)
Paper I (Core + CRS/ORS + completed CRS trajectory): Eq. (1)–(11) Paper II (Eₙ homeostasis + CLAH + ORS extinction with α-floor): Eq. (12)–(20) Paper III (Estimation + inference + diagnostics + extensions): Eq. (21)–(58)
A. Shared conceptual architecture (one sentence)
TDH separates: candidate generation → coherence-gated evolvability filter → observed outcomes (selection acts downstream of the filter).
B. Shared notation (copy-ready)
Indices and time
t: generational time (continuous unless stated; discrete updates use t, t+1) k ∈ {1,…,K}: active transposon family index n ∈ {1,2,…}: environmental envelope index, ordered by increasing demand
Coherence variables
qₖ(t) ∈ [0,1]: coherence of active family k at time t q(t) ∈ [0,1]: aggregate Active Transposon Family Coherence q*: coherence threshold for constructive throughput g(q): thresholded constructive gate (usable constructive throughput) h(q): incoherence penalty
Eᵢ(t): within-boundary reproductive edges (internal edges) Eᶜ(t): boundary-crossing reproductive edges (cross-boundary edges) Vₜ: set of reproducing nodes at generation t Eₜ: set of reproductive edges at generation t
Eₙ: n-th environmental envelope (nested perturbation class) dₙ ∈ (0,1]: demand associated with Eₙ, with d₁ ≤ d₂ ≤ d₃ ≤ … mₙ(q): homeostatic margin under Eₙ qₙ†: critical coherence required for envelope Eₙ n_max(q): largest maintainable envelope index β > 0: sharpness parameter for probabilistic homeostasis bridge
Extinction hazard and survival
λₑₓₜ(t): extinction hazard λ₀ ≥ 0: baseline hazard λ₁ > 0: coherence-sensitive hazard scaling wₙ ≥ 0: envelope weights [x]₊ = max{x,0} S(T): survival probability over horizon T
Estimation, measurement, and inference objects
N: number of sampled individuals (for estimation) sₖ(i,j,t) ∈ [0,1]: similarity score for individuals i and j in family k at time t q̂ₖ(t), q̂(t): estimators of qₖ(t), q(t) ε(t): dynamic noise/residual xₖ,ₘ(t): m-th observed indicator for family k at time t aₖ,ₘ, bₖ,ₘ: measurement coefficients ηₖ,ₘ(t): measurement noise yₙ(t) ∈ {0,1}: observed success (1) or failure (0) of maintenance under envelope Eₙ at time t θ: parameter collection for likelihood-based inference
Demography (your preferred symbols)
Nₘ(t): male population count at time t N𝒻(t): female population count at time t Nₑ(t): effective population size (sex-ratio adjusted)
Divergence extension (optional)
A, B: population labels D_AB(t) ∈ [0,1]: structured divergence between populations A and B M_AB(t) ∈ [0,1]: cross-boundary mixing between A and B E_AA(t), E_BB(t): within-population reproductive edges in A and B E_AB(t): cross-boundary reproductive edges between A and B κ > 0: divergence accumulation rate χ > 0: mixing homogenization rate D*: breakaway/speciation threshold (operational) Q_AB(t): closure-weighted coherence driver for divergence
C. Core equations (verbatim, global numbering)
C1. Coherence and the gate
q(t) = (1/K) Σₖ₌₁ᴷ qₖ(t). (1)
g(q) = 0, for q < q* g(q) = (q − q*)/(1 − q*), for q ≥ q*. (2)
h(q) = 1 − q. (3)
dq/dt = ρC(t)(1 − q) − [μ + ντ(t)](q − α). (4)
C2. Structure operationalization and CRS formation
