Each card states a geometric structure, the theorem family it comes from, and the exact
quantity on our platform it governs. Formulas are standard; the mapping is ours.
01
The spectrum is an intrinsic curve
Nyquist plot = a curve \(\gamma(\omega)=(Z'(\omega),-Z''(\omega))\) in the right half-plane.
Its shape — not its parameterisation — is what the classifier reads. This is
intrinsic geometry: properties invariant under how you traverse it.
Geometry
$$\gamma^*\!g \;=\; \left|\tfrac{d\gamma}{d\omega}\right|^2 d\omega^2, \qquad \kappa_g(\omega)=\frac{\det(\gamma',\gamma'')}{\|\gamma'\|^3}$$
UCARETRON quantity
\(\kappa_g\) at the arc apex ↔ how fast \(R_{ct}\) separates from \(R_s\) — the single most
discriminative feature for binding detection. AI reads geometric invariants of \(\gamma\), never raw \(\omega\).
🗣 structural mapping Anchor structure: intrinsic vs extrinsic geometry — classification of
reductive-pair and length-space structures.
02
Diffusion is Ricci flow on the spectrum
Warburg diffusion adds a tail that flattens the low-frequency arc — geometrically, a
parabolic smoothing of \(\gamma\). The same PDE family that evolves hypersurfaces
toward canonical shapes (inverse curvature flow, Ricci flow) describes what drift does to our measurement.
Geometry (flow)
$$\partial_t \gamma = -\nabla_s K, \qquad Z_W = \frac{\sigma}{\sqrt{\omega}}(1-i)$$
UCARETRON quantity
Flow time \(t\) ↔ diffusion time constant. Our models undo the flow: de-convolving
the Warburg tail is a backward-parabolic regularisation whose stability is guaranteed by the
same maximum-principle estimates the literature uses for flow existence.
🗣 structural mapping Anchor structure: parabolic flow existence and maximum-principle estimates
(capillary curvature flow, fast-diffusion gradients).
03
The electrode is a Steklov domain
Potential satisfies Laplace's equation in the electrolyte with the boundary condition
living on the electrode. That is exactly the Steklov problem: spectrum determined by
boundary flux. Steklov eigenvalues encode the geometry of \(\partial\Omega\) — our electrode shape.
Geometry (spectrum)
$$\Delta u = 0 \ \text{in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \sigma_k u \ \text{on } \partial\Omega$$
UCARETRON quantity
Interdigitated vs disk electrodes ↔ different Steklov spectra ↔ different sensitivity to
surface binding. Electrode geometry design is eigenvalue engineering; non-negative
Ricci-type curvature assumptions control where eigenvalues can concentrate.
🗣 structural mapping Anchor structure: Steklov eigenvalue problems and Serrin-type
overdetermination under curvature conditions; Kazdan–Warner equations on networks.
04
Calibration is a connection
A measured spectrum depends on the cable, the temperature, the reference electrode — a
gauge. Formally: instrument drift is a connection \(\nabla\) on a bundle over
parameter space, and calibration is choosing a gauge where physics is separable from drift.
Geometry (connection)
$$\tilde{Z} = g(\theta)\cdot Z + b(\theta), \qquad F_\nabla = dA + A\wedge A \neq 0 \ \Rightarrow\ \text{holonomy} = \text{drift}$$
UCARETRON quantity
On-chip Cortex-M4 calibration ↔ flat-bundle section selection. Non-zero curvature \(F_\nabla\)
= uncompensated drift; our multi-frequency calibration collapses holonomy so that ΔRct
is gauge-invariant across lots, cables, and days.
🗣 structural mapping Anchor structure: Chern–Weil theory of connections, groupoid modular
classes, invariant density bundles.
Interactive
Watch binding as curvature flow.
The arc below evolves under a curvature-type flow: each frame redistributes the curve's
curvature while the enclosed diameter (ΔRct) grows — the geometric statement of
"binding blocks charge transfer". Drag to change the diffusion tail weight.
05
κ₀ = — · ΔRct = — · flow step = 0 (drag on the plot to adjust Warburg weight)
🔬 computed illustration
