Defensive publication · heat–charge–information
One embedded electrolyte network, optimized jointly as coolant, power medium, and iontronic computer. Pricing any current alone is dominated whenever the couplings below are nonzero.
The Charlot Lab & The Hiner Lab, Institute for Physical AI @ BMI
CC0 1.0 Universal — dedicated to the public domain. Timestamp anchor: sha256://<fill>A single embedded electrolyte network is optimized jointly as coolant, localized power/regulation medium, and iontronic computational substrate. The three currents — heat, charge, information — share one flow and one dissipation budget. Under nonzero coupling (K1–K3, §4) the joint program below strictly dominates any pricing that optimizes cooling, power delivery, or computation in isolation. The disclosed object is the program itself: the objective, its ledger, the three couplings, and a determinism constraint that makes the readout bit-exact and safe. All constituent transfer functions are public physics; no term is enclosable.
Design (static) collects the channel diameters $d_i$ and lengths $\ell_i$, the manifold topology $\mathcal{T}$, electrode areas $A_{e,i}$, porosities $\varphi_i$, node geometries $(\mathbf{x}_k,a_k)$, and the per-node quantizer LSB $\delta_k$:
Operating (control, time-varying) — the flows $Q_i(t)$, inlet concentration $\sigma_{in}(t)$ and temperature $T_{in}$, drive current $j(t)$, and node gates $\mathbf{v}_k(t)$ — with state fields $\mathbf{s}$:
Momentum (Darcy–Brinkman; open channel and porous electrode):
Energy:
Species — Poisson–Nernst–Planck with the Soret term, per species $s$:
Poisson: $\;-\nabla\!\cdot(\varepsilon\nabla\phi)=F\sum_s z_s c_s.\;$ Electrode kinetics (Butler–Volmer, temperature-dependent exchange current):
Iontronic readout at node $k$: $\;y_k=\mathcal{G}_k(\{c_s\},T,\mathbf{v}_k)\;$ (local conductance/current functional, analog). Quantized readout: $\hat y_k=\mathsf{Q}(y_k)$ with $\mathsf{Q}$ a Q32.32 fixed-point quantizer of LSB $\delta$. The committed observable is the word $\hat y_k$, not the analog $y_k$; sub-LSB drift is absorbed by $\mathsf{Q}$. The trajectory digest is $H=\mathrm{SHA256}\big(\,\|_k\|_t\,\hat y_k(t)\,\big)$ over the canonical serialization of all node words.
| Term | Definition | Role |
|---|---|---|
| $P_{\text{comp}}=\sum_k v_k i_k$ | iontronic switching + CMOS | compute cost; floored by Landauer |
| $P_{\text{pump}}=\tfrac1{\eta_p}\sum_i Q_i\,\Delta p_i$ | hydraulic work | debit |
| $P_{\text{fc}}=\int_{A_e} j\,(E_{eq}-\eta_{act}-\eta_{ohm}-\eta_{conc})\,dA$ | flow-cell delivery | charge credit |
| $\dot X_{\text{th}}^{\text{self}}$ | outlet availability recirculated to upstream electrode | self-recovery credit — closes in the network, drives K1 |
| $\dot X_{\text{th}}^{\text{ext}}$ | outlet availability exported to a named external sink | export credit — weaker; defaults to $0$ |
| $\Lambda \le N_{node}/\tau_{ion}(T,\mathbf{g})$ | utility-weighted throughput | relevant outcomes / s |
with $\dot X_{\text{th}}^{\text{self}}+\dot X_{\text{th}}^{\text{ext}}=\dot m\,c_p\big[(T_{out}-T_0)-T_0\ln\tfrac{T_{out}}{T_0}\big]$. Only the self term is defensible without a counterparty; a reader cannot collapse export into self. Set $\eta_{\text{ex}}^{\text{ext}}=0$ for the standalone-die claim.
K1 — thermal → charge, positive (the exergy-recovery mechanism):
Removed heat upgrades delivered charge; dissipation becomes a credit, not pure loss.
K2 — flow → cooling vs. power vs. pumping, adversarial:
One scalar $Q$, three opposed partials ($\xi$ = residence-limited utilization). A second adversarial scalar is the per-node quantizer LSB $\delta_k$:
∂T_max/∂Q < 0 cooling improves with flow ∂ξ/∂Q < 0 residence time falls, utilization drops ∂P_pump/∂Q > 0 hydraulic work rises (∼Q²) J(Q) = P_pump(Q) + thermal_penalty(T_max(Q)) − credit·ξ(Q) → minimized in the interior
Constraint 5 lower-bounds it: $\delta_k\ge\delta_k^{\min}(\mathcal{F})=2\sup_{\mathcal{F}}|y_k-\bar y_k|$, and $\Lambda$ wants it small, so the optimum pins $\delta_k=\delta_k^{\min}(\mathcal{F})$ — quantization rides the invariance floor. Any widening of the feasible envelope $\mathcal{F}$ for K2 cooling/power freedom inflates $\sup_{\mathcal{F}}|y_k-\bar y_k|$, raises $\delta_k^{\min}$, and pays in $\Lambda$. Cooling headroom is charged, through invariance, against readout resolution — not through the energy ledger.
K3 — thermal ↔ information, bidirectional (Soret):
The thermal-management field is a computational input, and computation is a distributed heat source. $\nabla T$ is a controlled design input, not a nuisance to be uniformly suppressed. Two disjoint regimes are claimed: (a) write mode — $\nabla T$ shaped at node $k$ to program $c_s$ through the Soret flux (the cooling field doubles as the write bus); (b) invariance mode — $\nabla T$ held flat within tolerance so readout depends only on $\mathbf{v}_k$. Constraint 2 bounds $T_{\max}$ in both; it does not force $\nabla T\!\to\!0$ globally. (Secondary: $\partial\tau_{ion}/\partial T<0$ — warmth also accelerates $\Lambda$.)
The three currents share one network and one dissipation budget. Under $K1,K2,K3\neq 0$, any pricing that optimizes cooling, power delivery, or computation independently is provably dominated by the joint program above. The disclosed object is the program itself — objective, ledger, K1–K3, and the determinism constraint — priced in exergy rather than energy. All constituent transfer functions ($j_0(T)$, $S_{T,s}$, $\tau_{ion}(T)$, $E_{eq}(T,\sigma)$) are public physics; no term is enclosable.