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+# Bio-Optical Witness: Living-Light Material Equations
+
+Status: `DRAFT_RECEIPT_PROTOCOL`
+
+Claim boundary: this file extracts quantitative relations and stack-level equations from the public CU Boulder / Science Advances report on chemically sustained bioluminescence in living-light materials. It is not a wet-lab protocol, culture protocol, or claim that biological light is free energy. It treats living bioluminescent material as a slow optical witness surface: chemical/material state in, optical receipt out.
+
+Primary source spine:
+
+- CU Boulder Today, "Light without electricity? Glowing algae could make it possible" (2026-05-06).
+- EurekAlert release for the peer-reviewed Science Advances article, DOI `10.1126/sciadv.aee3907`.
+- Preprint/public abstract record for "Chemical Stimulation Sustains Bioluminescence of Living Light Materials", DOI `10.1101/2025.07.16.664986`.
+- IUPAC Gold Book definition of pH.
+
+## Extracted Source Facts
+
+```text
+organism: Pyrocystis lunula
+material: 3D-printed ionically crosslinked alginate / naturally derived hydrogel scaffold
+trigger_acidic: pH = 4
+trigger_basic: pH = 10
+acidic_response: bright, localized, persistent emission up to 25 minutes
+basic_response: diffuse, shorter-lived / biphasic stress-like emission
+longitudinal_retention: ~75% brightness after 4 weeks under acidic triggering
+application frame: living sensors, soft robotics, dark-environment autonomous systems, environmental monitoring
+```
+
+## 1. pH Gate
+
+Use the IUPAC activity definition as the canonical chemical input gate:
+
+```text
+pH = -log10(a_H+)
+```
+
+where `a_H+` is hydrogen-ion activity.
+
+For dilute/engineering approximations:
+
+```text
+[H+] ≈ 10^(-pH)
+```
+
+The reported acidic and basic triggers become:
+
+```text
+[H+]_acid ≈ 10^-4
+[H+]_base ≈ 10^-10
+[H+]_acid / [H+]_base ≈ 10^6
+```
+
+Receipt interpretation:
+
+```text
+pH 4 and pH 10 are not symmetric perturbations.
+They differ by approximately six orders of magnitude in hydrogen-ion activity.
+```
+
+## 2. Chemical-Light Response Gate
+
+Define a categorical pH response gate for living-light material:
+
+```text
+G_pH(pH) =
+  ACID_EMIT      if pH ≈ 4
+  BASE_STRESS    if pH ≈ 10
+  HOLD_UNKNOWN   otherwise
+```
+
+Stack interpretation:
+
+```text
+ACID_EMIT    -> ADMIT optical witness, localized persistent glow
+BASE_STRESS  -> HOLD/FAMM, diffuse or stress-like glow signature
+HOLD_UNKNOWN -> require calibration receipt before classification
+```
+
+## 3. Bio-Optical Witness Signal
+
+Let:
+
+```text
+L(x, t) = emitted optical intensity at surface coordinate x and time t
+C(x, t) = chemical trigger field, e.g. pH/stimulus concentration
+M(x, t) = material viability / living-cell reactivity field
+S(x, t) = mechanical stimulation field
+```
+
+Then a stack-level witness equation is:
+
+```text
+L(x,t) = M(x,t) · B( C(x,t), S(x,t), t ) + ε_opt(x,t)
+```
+
+where:
+
+```text
+B = bioluminescent response kernel
+ε_opt = optical noise / camera / scattering residual
+```
+
+For the chemistry-only gate:
+
+```text
+L_chem(x,t) = M(x,t) · B_pH(pH(x,t), t) + ε_opt(x,t)
+```
+
+## 4. Persistence Window
+
+The acidic response is reported as persistent up to 25 minutes. Treat this as a bounded emission horizon:
+
+```text
+T_acid ≤ 25 min
+```
+
+A minimal decay model for local optical witness intensity:
+
+```text
+L_acid(t) = L0 · exp(-t / τ_acid) · 1[0 ≤ t ≤ T_acid]
+```
+
+with receipt constraint:
+
+```text
+T_acid_observed ≤ 25 min
+```
+
+Do not assign `τ_acid` from the article alone; it must be fitted from time-series intensity data.
+
+## 5. Longitudinal Brightness Retention
+
+The public report states that acid-triggered 3D-printed living-light structures retained about 75% brightness after four weeks.
+
+```text
+R_B(4 weeks) = L_week4 / L_initial ≈ 0.75
+```
+
+A conservative viability/brightness decay model:
+
+```text
+R_B(t) = exp(-k_B t)
+```
+
+Solving from the reported four-week retention:
+
+```text
+k_B ≈ -ln(0.75) / 4 weeks
+k_B ≈ 0.0719 week^-1
+```
+
+Half-brightness extrapolation under the same simple model:
+
+```text
+t_1/2 = ln(2) / k_B ≈ 9.64 weeks
+```
+
+Claim boundary: this half-life is a model extrapolation, not a source claim.
+
+## 6. Integrated Optical Receipt
+
+For camera-readable witness telemetry, the useful value is often total emitted light over a bounded read window:
+
+```text
+I_emit(x; T) = ∫_0^T L(x,t) dt
+```
+
+For a full witness tile region `Ω_tile`:
+
+```text
+I_tile(T) = ∫_{Ω_tile} ∫_0^T L(x,t) dt dx
+```
+
+For digital receipt extraction:
+
+```text
+packet = DecodeCamera( L_frame(x,t), calibration, threshold )
+```
+
+## 7. Synergy Gate: Chemical + Mechanical Stimulation
+
+The public abstract reports that coupling chemical and mechanical stimulation yields synergistic enhancement of amplitude and duration.
+
+Define:
+
+```text
+L_chem = response to chemical stimulus alone
+L_mech = response to mechanical stimulus alone
+L_combo = response to combined chemical + mechanical stimulus
+```
+
+Synergy witness:
+
+```text
+Σ_syn = L_combo - (L_chem + L_mech)
+```
+
+or for integrated light:
+
+```text
+Σ_syn(T) = I_combo(T) - [ I_chem(T) + I_mech(T) ]
+```
+
+Classification:
+
+```text
+Σ_syn > 0  -> synergistic enhancement
+Σ_syn = 0  -> additive response
+Σ_syn < 0  -> antagonistic / stress / inhibition response
+```
+
+Do not assign numeric synergy without source time-series or amplitude data.
+
+## 8. Living-Light Receipt Surface
+
+Canonical receipt transform:
+
+```text
+chemical/material event
+  -> living-light emission
+  -> camera observation
+  -> optical packet
+  -> Sniffer classification
+  -> BVMR gate
+  -> CMR receipt
+  -> FAMM if abnormal / scarred
+  -> Underverse if excluded, failed, unpaid, or residual
+  -> Warden replay check
+```
+
+Minimum event vector:
+
+```json
+{
+  "protocol": "bio_optical_witness_tile_v0",
+  "source": "chemical_stimulation_sustains_bioluminescence_living_light_materials",
+  "organism": "Pyrocystis_lunula",
+  "substrate": "3D_printed_alginate_hydrogel",
+  "stimulus": {
+    "pH": "assigned_or_measured",
+    "mechanical_stimulation": "present_or_absent"
+  },
+  "observed": {
+    "emission_pattern": "localized | diffuse | biphasic | none",
+    "emission_duration": "measured_seconds",
+    "integrated_intensity": "I_tile(T)",
+    "brightness_retention": "optional_longitudinal_R_B"
+  },
+  "decision": "ADMIT | HOLD | FAMM | QUARANTINE",
+  "claim_boundary": "slow_living_sensor_not_general_lighting"
+}
+```
+
+## 9. No-Free-Light Accounting
+
+Biological light is not free energy. The electrical bill may be reduced at the emitting surface, but the payment moves into metabolic, photosynthetic, chemical, material-maintenance, and readout costs.
+
+```text
+E_total = E_photo_input + E_metabolic + E_chemical_trigger + E_material_maintenance + E_camera_readout + E_decode
+```
+
+For stack accounting:
+
+```text
+optical_output <= paid_biochemical_energy + stored_material_state - losses
+```
+
+Receipt rule:
+
+```text
+No emitted-light claim without energy/payment lane.
+No optical packet without calibration/residual lane.
+No living material claim without viability/retention lane.
+```
+
+## 10. Stack Integration
+
+Name:
+
+```text
+Bio-Optical Witness Material
+Living-Light Receipt Surface
+```
+
+Keeper phrase:
+
+```text
+This is not free light; it is biology paying the optical bill.
+Chemical state enters, living material emits, camera reads, receipts classify.
+```
+
+Connection to existing semiautonomous stack:
+
+```text
+Bio-optical tile = slow optical witness surface
+Equation Sniffers = classify optical/chemical scent trail
+BVMR = gate event vector
+AVMR = combine surviving witness vectors
+CMR = receipt combined optical event
+FAMM = inspect abnormal scars
+Underverse = account excluded or failed material/light response
+Warden = replay/calibration check
+```