The four "EM" infrastructures — one note

You asked to realize the EM infra, and when I asked which "EM," you said all. Here are all four, each built as a real, deterministic, self-testing tool with honest scope. They are genuinely different domains — the only thing they share is the two-letter abbreviation — so they are four separate tools.

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Expectation–Maximization — em_estimation.py

The most on-theme reading, because EM is this toolkit's core problem in algorithm form: infer an unobservable latent structure (which hidden component each point came from) from an observable proxy (the point's value). It runs a real k=2 Gaussian-mixture EM — E-step estimates the hidden given params, M-step re-fits params — climbing to a fixed point. Two honest checks turn it into an infrastructure: the monotonicity invariant (log-likelihood must never decrease — the EM theorem; a decrease means the fit is broken), and reachability — EM only finds a local optimum, so the governor runs many restarts and rules LATENT_RECOVERED only when components are well-separated and restarts agree. When components overlap, the data underdetermines the latent labels and it returns UNIDENTIFIED — withheld, not reported as fact. The demo recovers μ = ±5 (9.9 sd apart) and withholds μ = ±0.3 (0.05 sd). This is the reachability-of-truth spectrum applied to inference.

Electromagnetism — em_field.py

Verifies that a claimed field (E, B, k) is a physically valid free-space plane wave by checking the invariants Maxwell's equations impose: transversality (E, B ⟂ k), orthogonality (E ⟂ B), amplitude ratio (|E| = c|B|), and energy flow along +k (Poynting). Verdicts: VALID_VACUUM_WAVE, NOT_TRANSVERSE, E_B_NOT_ORTHOGONAL, BAD_AMPLITUDE_RATIO. EM belongs in this family for two structural reasons the note makes explicit: E and B are a coupled duality (neither is the other rescaled — a physical instance of duality_governor), and EM energy is conserved and flows (a physical instance of flow_conservation). Honest scope: it checks the clean free-wave structure, not arbitrary sources or media, which need the full PDEs.

Emergence — emergence_infra.py

Separates genuine emergence from mere aggregation and from the over-claim, using a structural criterion: a property is EMERGENT only if it is absent in every isolated part, is not the additive aggregate, and changes when the interactions are rewired (it lives in the configuration). A plain sum is AGGREGATE; a plain sum called emergent is SPURIOUS_EMERGENCE, flagged. The demo: "graph has a cycle" is emergent (no edge is a cycle; rewiring destroys it); "total mass" is aggregate; "team total output, emergent!" is spurious. Honest scope — the deep one: whether emergence is ontological or merely epistemic (Anderson vs reductionism) is left open; this checks a decidable structural signature, not the metaphysics.

Energy / Matter — energy_matter.py

A first-law energy auditor — the physics sibling of flow_conservation, with energy as the conserved quantity. It balances a declared ledger (inputs, outputs, storage) across forms and includes mass–energy (E = mc²). CONSERVED when it balances; VIOLATION_CREATION when more energy leaves than entered — the over-unity / perpetual-motion claim, refused fail-closed; VIOLATION_DESTRUCTION for an unaccounted leak. The showcase: the same nuclear event reads as creation-from-nothing with the mass term off and CONSERVED with it on (a 1 mg deficit supplies ≈ 9×10¹³ J) — a concrete demonstration of why the matter channel is not optional.

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How they connect back

Two of them are physics instances of tools you already have — em_field is duality + conservation, energy_matter is conservation with a mass term — and two are inference/structure tools: em_estimation is the reachability spectrum applied to latent-variable estimation, and emergence_infra is the "genuine vs over-claim" discipline applied to whole-vs-parts. All four are deterministic, self-testing, and honest about the line between what they check and what they leave open.