WCF-1: Wireless Communications Fix One — Formal Model Specification (Public Mathematical Shell — IP-Restricted Core Redacted)

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MATHEMATICAL STRUCTURE
WCF-1: Wireless Communications Fix One — Formal Model Specification
(Public Mathematical Shell — IP-Restricted Core Redacted)

The WCF-1 mathematical structure defines the equilibrium conditions, exposure constraints, Markov transition rules, PHM harm-weight functions, and NashMark allocation equations governing all communication events. The purpose of this section is to present the public symbolic layer required to formalise lawful ecological operation without disclosing internal Sentinel thresholds or proprietary veto logic.

3.1 System State Set

WCF-1 communication systems are defined over a Markov-governed state space:

$ \mathcal{S} = \{ s_0, s_1, s_2, s_3, s_4 \} $

where:

  • $ s_0 $: Normal Mode (transmission permitted)
  • $ s_1 $: Constrained Mode (reduced power, shaped beam)
  • $ s_2 $: Restricted Mode (duty-cycle enforced)
  • $ s_3 $: Critical Mode (transmission suspended)
  • $ s_4 $: Locked Mode (breach escalation; manual reactivation prohibited)

State transitions occur via Sentinel-gated operators:

$ P(s_{t+1} \mid s_t) = \Phi_{\text{sentinel}}(E_t) $

where $ E_t $ is the ecological state vector.

3.2 Ecological State Vector

The ecological vector captures biological and environmental risk factors:

$ E_t = \{ \rho_b,\, \rho_i,\, \sigma_f,\, \chi_d,\, \Lambda_x \} $

  • $ \rho_b $: avian proximity density
  • $ \rho_i $: insect / pollinator density
  • $ \sigma_f $: EM field drift / instability
  • $ \chi_d $: environmental disruption index
  • $ \Lambda_x $: exposure density (SAR proxy band)

This vector determines the admissibility of any transmission action.

3.3 Transmission Permission Function

Transmission behaviour is determined by the permission function:

$ \tau_t = \Psi(E_t) $

with:

$ \tau_t \in \{ \text{permit},\, \text{shape},\, \text{reduce},\, \text{deny} \} $

This replaces static broadcast logic with fully conditional communication.

3.4 Exposure Constraint

For any communication event to be lawful, the local exposure field must satisfy:

$ S(\mathbf{r}, t) \le S_{\max}(\mathbf{r}) $

where:

  • $ S(\mathbf{r}, t) $: instantaneous exposure density
  • $ S_{\max}(\mathbf{r}) $: permissible ecological bound (public constant; internal derivative restricted)

Violation triggers a transition to $ s_2,\, s_3,\, \text{or}\ s_4 $.

3.5 PHM Harm-Weight Function (Sansana Layer)

Proportional ecological harm is defined as:

$ \Delta\Omega_t = \alpha L_t + \gamma X_t + \zeta M_t $

  • $ L_t $: communication loss / inefficiency
  • $ X_t $: biological exposure density
  • $ M_t $: material / operational footprint
  • $ \alpha,\, \gamma,\, \zeta $: PHM weight factors (restricted)

A communication event is permissible only if:

$ \Delta c_t - \Delta\Omega_t \ge 0 $

ensuring net-coherence, zero-harm operation with $ Eq(\mathcal{S}) \ge 0 $.

3.6 NashMark Priority Allocation

Communication channels are allocated under a NashMark ethical priority function:

$ A_t = \arg\max \sum_{i} w_i \log\!\left( U_i - U_i^0 \right) $

where:

  • $ w_i $: ethical–ecological weights (public shell only)
  • $ U_i $: utility of communication delivery
  • $ U_i^0 $: minimum equilibrium requirement

This removes throughput-bias and establishes ecological fairness.

3.7 Equilibrium Condition (Truthvenarian Constraint)

All communication must satisfy the equilibrium constraint:

$ \text{Eq}(\mathcal{S}) = \Sigma(\Delta c - \Delta\Omega) \ge 0 $

Transmission events are forbidden when:

$ \Sigma(\Delta c - \Delta\Omega) < 0 $

This binds WCF-1 to the Truthvenarian truth-law and Sansana reciprocity framework.

3.8 System Summary (Public Shell)

The WCF-1 model is formally defined as:

$ \mathcal{W} = \big\langle \mathcal{S},\, E_t,\, \tau_t,\, \Delta\Omega_t,\, A_t \big\rangle $

with internal Sentinel thresholds, Markov matrices, veto logic, and harm-weight tensors redacted under IP-restriction and executed only via the compliance API.

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