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+# BioPhonon Translation Field Equations
+
+Status: EQUATION_UPDATE
+Claim state: FORMAL_SCAFFOLD / ANALOGY_BOUNDED
+
+This note defines the BioPhonon Translation Field: the class of evolved systems that convert mechanical wave propagation into actionable meaning. It includes percussive foragers, web/modal graph sensors, substrate vibration insects, seismic mammals, aquatic lateral-line systems, echolocators, and plant stress-ultrasound emission.
+
+Boundary: this is not proof that plants have subjective experience or intentional speech. The plant-stress sound source is treated as an informative biomechanical emission term in a broader phonon-mediated translation equation.
+
+---
+
+## 1. Base BioPhonon field
+
+Universal medium equation:
+
+```math
+\mathcal{M}\ddot{u}
++
+\mathcal{C}\dot{u}
++
+\mathcal{K}u
++
+\mathcal{N}(u,\nabla u)
+=
+S_a(x,t)
++
+S_w(x,t)
++
+N(x,t)
+```
+
+Observation equation:
+
+```math
+y_i(t)
+=
+\mathcal{R}_i[u(x_i,t)]
++
+\eta_i(t)
+```
+
+Translation equation:
+
+```math
+P(z\mid y_{1:n},a)
+\propto
+P(y_{1:n}\mid z,a)P(z)
+```
+
+Action policy:
+
+```math
+a^*
+=
+\arg\max_a
+\frac{
+I(z;y_{1:n}\mid a)
+}{
+E_{act}(a)+E_{move}(a)+C_{compute}(a)+C_{risk}(a)
+}
+```
+
+---
+
+## 2. Plant-stress ultrasonic emission term
+
+The plant-stress source term adds an emitter that is not a moving animal sensor, but a biomechanical state-to-sound translator.
+
+```math
+S_w(x,t)
+\rightarrow
+S_{world}(x,t)
++
+S_{plant}(x,t;\sigma_h,w,i,\chi)
+```
+
+where:
+
+```text
+σ_h = hydraulic / xylem tension stress
+w   = water deficit / dehydration state
+i   = injury or cutting state
+χ   = species / tissue / morphology parameters
+```
+
+The plant emission source can be modeled as a sparse ultrasonic pulse train:
+
+```math
+S_{plant}(x,t)
+=
+\sum_k
+A_k(\sigma_h,w,i,\chi)
+\,g(t-t_k;f_k,Q_k)
+\,\delta(x-x_{plant})
+```
+
+with event rate:
+
+```math
+\lambda_{plant}(t)
+=
+\lambda_0
++
+\lambda_w\,\Phi_w(w(t))
++
+\lambda_i\,\Phi_i(i(t))
++
+\lambda_h\,\Phi_h(\sigma_h(t)).
+```
+
+Interpretation:
+
+```text
+Plant state -> ultrasonic event statistics -> receiver inference.
+```
+
+---
+
+## 3. Cavitation-compatible source hypothesis
+
+A conservative biomechanical source model treats plant sounds as cavitation-compatible xylem events rather than intentional vocalization:
+
+```math
+\sigma_h(t) > \sigma_{cav}
+\Rightarrow
+\Delta P_{xylem}(t)
+\Rightarrow
+S_{plant}(t).
+```
+
+One possible event kernel:
+
+```math
+g(t;f,Q)
+=
+H(t)e^{-\pi f t/Q}\sin(2\pi f t).
+```
+
+This keeps the plant term compatible with mechanical emission:
+
+```text
+hydraulic stress / injury -> elastic release / cavitation-like pulse -> airborne ultrasound.
+```
+
+---
+
+## 4. Receiver inference: who listens?
+
+For any receiver organism r:
+
+```math
+y_r(t)
+=
+\mathcal{H}_{air,plant\to r}[S_{plant}](t)+\eta_r(t)
+```
+
+and:
+
+```math
+P(z_{plant}\mid y_r)
+\propto
+P(y_r\mid z_{plant})P(z_{plant}).
+```
+
+Receiver action:
+
+```math
+a_r^*
+=
+\arg\max_{a_r}
+\frac{
+I(z_{plant};y_r\mid a_r)
+}{
+E_{move}(a_r)+C_{compute}(a_r)+C_{risk}(a_r)
+}.
+```
+
+Examples:
+
+```text
+moth: avoid oviposition on stressed host plant
+bat/mouse/insect: detect ultrasonic environmental state
+neighbor plant: possible stress preconditioning route, currently research-bound
+farmer sensor: irrigation / stress monitoring
+```
+
+---
+
+## 5. Updated universal BioPhonon translation equation
+
+With plant-stress ultrasound included:
+
+```math
+\boxed{
+\mathcal{M}\ddot{u}
++
+\mathcal{C}\dot{u}
++
+\mathcal{K}u
++
+\mathcal{N}(u,\nabla u)
+=
+S_{self}(x,t)
++
+S_{prey}(x,t)
++
+S_{mate}(x,t)
++
+S_{predator}(x,t)
++
+S_{plant}(x,t)
++
+N_{env}(x,t)
+}
+```
+
+Observation:
+
+```math
+\boxed{
+y_i(t)
+=
+\mathcal{R}_i[u(x_i,t)]
++
+\eta_i(t)
+}
+```
+
+Translation:
+
+```math
+\boxed{
+P(z\mid y_{1:n},a)
+\propto
+P(y_{1:n}\mid z,a)P(z)
+}
+```
+
+Action:
+
+```math
+\boxed{
+a^*
+=
+\arg\max_a
+\frac{
+I(z;y_{1:n}\mid a)
+}{
+E_{act}(a)+E_{move}(a)+C_{compute}(a)+C_{risk}(a)+L_{FAMM}(a)
+}
+}
+```
+
+---
+
+## 6. Stack translation
+
+Plant-stress ultrasound adds a new category: passive biomechanical source emission.
+
+```text
+not only: active probe -> response -> meaning
+also: internal stress -> emitted phonon signature -> external listener inference
+```
+
+Compression analogy:
+
+```text
+internal state strain -> sparse acoustic/phonon events -> classifier route -> FAMM scar/update
+```
+
+So the compression/semantic equivalent is:
+
+```math
+route^*
+=
+\arg\max_{probe/listen}
+\frac{
+I(hidden\_state;response\mid probe/listen)
+}{
+C_{probe}+C_{listen}+C_{decode}+C_{interface}+C_{risk}+L_{FAMM}
+}.
+```
+
+---
+
+## 7. Warden boundary
+
+Allowed:
+
+```text
+Use plant-stress sounds as a biomechanical ultrasonic source term.
+Use the term to expand BioPhonon translation to include passive stress emissions.
+Use classifier/inference framing for animal, plant, or engineered receivers.
+```
+
+Blocked:
+
+```text
+Do not claim plants intentionally scream.
+Do not claim plant sentience or pain from ultrasonic emissions.
+Do not claim plant-to-animal communication unless receiver behavior is empirically shown.
+Do not promote BioPhonon claims outside the measured stress-sound/source-receiver boundary.
+```
+
+---
+
+## 8. Source anchor
+
+Primary source:
+
+```text
+Khait et al. 2023. Sounds emitted by plants under stress are airborne and informative. Cell 186(7):1328-1336.e10. DOI: 10.1016/j.cell.2023.03.009
+```