Planck Radiation Is ELC-Native (No Trig Needed)

Tier: OBSERVATION (measurement; no new proofs)

Electromagnetic physics is a surprisingly mixed bag when you measure it in F16 nodes. The wave equation and the Poynting vector are boundary formulas — they live just outside $\mathrm{ELC}(\mathbb{R})$ because they call $\cos$, and $\cos$ is blocked on the reals by the Infinite Zeros Barrier. You pay for them by crossing to complex EML: one complex node per $\cos$ invocation via Euler’s formula. Thermal radiation is different. The Planck spectrum $B(\nu, T)$ uses only $\exp$, multiplication, division, and subtraction. It is an entirely ELC-native formula.

Six formulas, measured

Using v5.2 positive-domain F16 costs ($\mathrm{mul}=1n$, $\mathrm{exp}=1n$, $\mathrm{ln}=1n$, $\mathrm{div}=2n$, $\mathrm{sqrt}=1n$ via $\mathrm{EPL}(0.5,x)$, $\mathrm{sub}=2n$):

Formula	Nodes	Class
photon energy $E = hc/\lambda$	3	inside ELC
skin depth $\delta = \sqrt{2/(\omega\mu\sigma)}$	5	inside ELC
Planck radiation $B(\nu,T) = \dfrac{2h\nu^3/c^2}{\exp(h\nu/kT)-1}$	14	inside ELC
EM energy density $u = \tfrac{1}{2}\varepsilon_0 E^2 + \tfrac{1}{2\mu_0}B^2$	10	inside ELC
damped EM wave $E_0 e^{-\alpha x}\cos(kx-\omega t)$	8	boundary (needs $\cos$)
Poynting $S = E_0 B_0 \cos^2(kx-\omega t)/\mu_0$ (cos·cos form)	10	boundary
Poynting via $\cos^2 = (1+\cos 2\theta)/2$	14	boundary

(Inside-ELC subtotal: 32 nodes over 4 formulas. Boundary subtotal: 32 nodes over 3 formulas. Data: sE_em_costs.json.)

Planck is the surprise

Planck’s law calls $\exp(h\nu/kT)$ on positive arguments and then does algebra. No trig, no branch cuts, no complex bypass. By the Lean-verified T_EXP_LOG_DUALITY catalogue rule, any F16 tree with $\exp$ of a positive-real argument stays in the real component of $\mathrm{ELC}$. The 14-node tree computes $B(\nu,T)$ exactly at every physically meaningful $(\nu, T)$ — no approximation, no truncation.

What this buys you in practice: if you care about symbolic stability (e.g., automatic differentiation through the Planck kernel, or exact manipulation of blackbody integrals), Planck is on the easy side of the $\mathrm{ELC}$ frontier. Wave propagation is on the hard side.

The double-angle identity costs 4 extra nodes

The Poynting vector has two routes. Direct squaring of a single complex-bypass $\cos$ gives 10 nodes. Using the trigonometric identity $\cos^2\theta = (1+\cos 2\theta)/2$ to “simplify” gives 14 nodes — 4 more.

Why? The identity buys you one fewer $\cos$ call but introduces an addition (2n), a division by 2 (2n), and an extra multiplication for $2\theta$. In F16-node accounting, algebraic identities that look simpler on paper can be more expensive than direct computation. The lesson applies broadly: prefer the form with fewer arithmetic glue operations, not the form with fewer transcendental calls. $\cos$ is 1 node either way (complex bypass). The arithmetic around it is where cost accumulates.

Why “ELC-native” matters

$\mathrm{ELC}(\mathbb{R})$ is the closure of ${1, x, +, -, \cdot, /, \exp, \log}$ under composition. Every formula in that closure admits:

A finite F16 tree representation.
Exact symbolic manipulation (no series truncation).
A depth-bounded proof of equivalence via EQ_D (our bounded-depth F16 equality decision tool).

Thermal radiation, diffusion decay, skin depth, the photon-energy relation, relativistic energy-momentum — all of these are ELC-native. Wave propagation, by contrast, requires either (a) crossing to $\mathrm{ELC}(\mathbb{C})$ where $\cos$ is 1 complex node, or (b) accepting a Taylor approximation, which is 33+ nodes for a 4-term expansion and drops to 1 complex node only via Euler.

This is a structural feature of physics measured in F16: the thermal sector is cheap, the oscillatory sector is complex-only.

Reproduce

git clone https://github.com/agent-maestro/monogate
cd monogate
python exploration/blind-sessions/scripts/sE_em_costs.py

Expected output:

[E] EM wave cost catalog

  [boundary]  8 n   damped EM wave ...
  [inside  ] 10 n   EM energy density ...
  [boundary] 10 n   Poynting (cos·cos form)
  [boundary] 14 n   Poynting via cos²=(1+cos(2θ))/2
  [inside  ]  5 n   skin depth
  [inside  ] 14 n   Planck B(ν,T)
  [inside  ]  3 n   photon energy E=hc/λ

  Inside ELC total: 32 n (4 formulas)
  Boundary total  : 32 n (3 formulas)

Cite: Monogate Research (2026). “Planck Radiation Is ELC-Native (No Trig Needed).” monogate research blog. https://monogate.org/blog/planck-elc-native

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