Math to reality — governing the correspondence, not asserting it

A short note accompanying math_to_reality.py. It formalizes "math to reality" as an infrastructure that checks whether a mathematical model corresponds to measured reality — and, more importantly, where it stops — rather than as a claim that reality is mathematical.

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What "math to reality" is, made precise

dependency_graph.py formalized math as the foundational dependency (the well-founded root). sciences_layers.py formalized physics as the interface between math and biology. Both describe the layers. This tool governs the seam: the map from a mathematical model's predictions to independent measurement of the world.

The one honest correction, stated up front — and it is the mirror image of the correction in sciences_layers. There, "physics as the proxy between math and biology" was corrected to interface, because a layer is not a stand-in that can be gamed. Here the opposite holds and must be said plainly: the model→world relation is a proxy relation. A mathematical model is a proxy for reality — a measurable stand-in — and it fails the way proxies fail, by decoupling from the reality it represents when pushed outside its validated regime (over-idealization). So this tool is decoupling_monitor lifted to the model-world seam.

What it certifies — and refuses to

Given a model and a set of regimes, each carrying the model's prediction and (where available) an independent measurement, govern returns:

• VALIDATED_IN_REGIME — in every regime actually measured, prediction matched measurement within tolerance. The map holds — but only across the measured envelope, and any untested regime is still flagged.
• IDEALIZED_DECOUPLED — some measured regime diverged beyond tolerance; the model idealizes away something real there. Reported fail-closed as broken from that control value on.
• UNVERIFIED_EXTRAPOLATION — asked about a regime with no measurement; the map cannot be certified there, however elegant the math.

This is the same discipline the temporal governor applies to the future and the qualia governor to a mind: refuse to certify beyond the evidence. A model validated at every speed you have measured is still UNVERIFIED at the speed you have not.

Two worked cases (real physics)

• Newtonian momentum p = m·v vs relativistic reality p = γ·m·v (m = c = 1). Faithful at low speed — 0.5% error at v/c = 0.1, 4.6% at 0.3 — then decouples: 13% at 0.5, 69% at 0.95. The regime v/c = 0.99 is left unmeasured on purpose and comes back UNVERIFIED. Verdict: IDEALIZED_DECOUPLED, validated up to ≈ 0.3 under a 5% tolerance.
• Hooke's law F = k·x vs a material that yields past its elastic limit. Exact below the limit, decoupled (40%, then 104%) in the plastic regime, UNVERIFIED past the last measurement.

Both are cases of a beautiful, correct model that is nonetheless a proxy — trustworthy in its regime, decoupling outside it.

The deep honest scope

The tool does not explain why mathematics corresponds to the world at all — Wigner's "unreasonable effectiveness of mathematics" stays an open puzzle — and it does not license the slogan "reality is mathematical." It replaces that unfalsifiable claim with a checkable one: this model is validated in this regime, and unverified outside it. That downgrade — from metaphysics to a defeasible, regime-bounded correspondence — is the entire contribution. Stdlib-only, deterministic, self-testing; non-safety-critical scope.