heat.md

---
alias: heat kernel, multi-scale smoothing, adaptation, thermostat, heat
tags: cyber
crystal-type: entity
crystal-domain: biology
---
third operator of the tri-kernel

`∂H/∂τ = -LH`, `H₀ = I`, so `H_τ = exp(-τL)`

temperature τ controls scale

answers: "what does the graph look like at scale τ?"

high τ explores (annealing), low τ commits (crystallization)

provides adaptive context — the thermostat of collective attention

positivity-preserving, semigroup: `H_{τ₁} H_{τ₂} = H_{τ₁+τ₂}`

Chebyshev polynomial approximation gives h-local computation with bounded error

locality: Gaussian tail decay, `h = O(log(1/ε))` hops

the adaptation force — metabolism, phase changes, the ability to shift

universal pattern

- physics: thermostat, phase changes
- biology: metabolism, immune plasticity
- ecology: seasons, succession, disturbance
- economics: booms, busts, revolutions

together with diffusion and springs forms the tri-kernel that computes cyberank

see tri-kernel for completeness proof

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