One substrate, every physical carrier
How does matter compute?
Every information carrier in nature is a wave. Every wave equation is chain order 0–2. One operator — eml(x, y) = exp(x) − log(y + 1) — generates the elementary functions that describe all of them. 21 demos across six tiers, 46 obligations closed against MachLib — no Mathlib dependency.
T2 — Electronics
3 demos1947 → presentelectron — charge density wave in doped silicon — compiles to Verilog → ASIC, C → embedded MCU
mosfet_iv
MOSFET drain-current saturation. The equation that built every chip you've ever owned.
cmos_inverter
Complementary inverter transfer characteristic at boundary points and the symmetric switching threshold.
pll_loop
Phase-locked loop gain positivity and zero-offset steady-state phase error.
T3 — Photonics
4 demos1960s → presentphoton — electromagnetic field oscillation — compiles to HLSL/GLSL → GPU shader, photonic IC layout
mach_zehnder
Two-arm interferometer transmission. The basic photonic logic gate.
ring_resonator
Whispering-gallery resonance — Lorentzian transmission peaks. Photonic filter / sensor.
poisson_spot
Fresnel diffraction's bright spot in the geometric shadow — the historic experiment that sealed wave optics.
optical_neuron
Saturable-absorber activation. The analog of GELU in a photonic neural net.
T4 — Spintronics
3 demos2000s → presentmagnon — spin-precession wave in magnetic order — compiles to Verilog → FPGA spin-wave logic
magnon_dispersion
Spin-wave dispersion in a uniformly-magnetised material. FMR base mode at k = 0.
spin_torque
Landau-Lifshitz-Gilbert precession. The equation that governs every spintronic device.
magnon_logic
Spin-wave interference gate. Same shape as the photonic Mach-Zehnder — different physics, same math.
T5 — Phononics
3 demos1950s → presentphonon — quantised lattice vibration — compiles to C → MEMS controller, acoustic metamaterial
phonon_bandgap
1-D Fabry-Perot transmission through a Bragg-spaced phononic crystal.
thermal_rectifier
Asymmetric thermal conductance — a phononic diode. Direction-dependent heat flow.
acoustic_cloak
Coordinate-transform metamaterial. Bends every ray inside the shell around a hidden core.
T6 — Ferronics
4 demos2025 → presentferron — coherent polarisation wave in a van der Waals ferroelectric — compiles to Verilog → ferroelectric controller, materials-integrated logic
ferron_propagation
Damped polarisation wave. Electric order itself carries coherent wave information.
ferron_emission
Laser-pulse-driven THz emission. Ultrafast polarisation switching radiates a narrow-band transient.
ferron_dispersion
Polarisation-wave dispersion from Landau-Ginzburg free energy. Soft-mode at k=0 measured by THz spectroscopy.
ferron_logic
Two-pulse interference gate. Byte-identical proof to the magnonic version — substrate independence in one diff.
T7 — Quantum
4 demos1980s → presentamplitude — complex-valued probability wave — compiles to Lean → proof certificate, quantum gate sequence
phase_gate
Single-qubit phase rotation. Unitarity proven via direct application of MachLib's Pythagorean axiom.
hadamard
Hadamard gate. Norm preservation on basis states, scaled to dodge the √2.
cnot
Controlled-NOT. Self-inverse on amplitudes — the property that makes CNOT reversible.
grover_oracle
Grover diffusion operator. Squared-amp witness closes; the algebra-identity proofs need MachLib's pending ring tactic.
Deep dive
Verified photonic computing →
Five interactive demos at /waves/photon: Mach-Zehnder, ring resonator, 4×4 MZI mesh, photonic attention, Poisson spot. Drag the parameters; see the equations move. Each backed by a closed Lean proof against MachLib (no Mathlib).
Carbon. Silicon. Monogate.
Carbon is biology's substrate (3.5 billion years of sensing optimisation — see /senses). Silicon is human engineering's substrate (eighty years of CMOS scaling). Monogate is the mathematical substrate that proves the other two correct and ports invariants between any two physical carriers without reimplementation.