# Univ Pass 8 — Physics Interpretability Notes (Mapping Hypotheses to Real Phenomena)

## Purpose

Make the universality argument legible to physicists and non-specialists without weakening rigor:

- translate theorem hypotheses into “what to check in a real system”
- clarify which assumptions are idealizations and which are structural
- connect SFT certificates to measurable quantities

This is not a literature survey; it is an interpretability map.

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## 1) What “phase observable” means physically

A phase observable is any quantity naturally defined modulo a period:

- rotational angle modulo \(2\pi\)
- time-of-day modulo \(P\)
- wave phase modulo \(2\pi\)
- order-parameter phase in complex scalar fields

SFT treats these as maps into \(\mathbb{T}_P\).

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## 2) What “\(S^1\)-action” means physically

An \(S^1\)-action is a one-parameter symmetry that cycles:

- rotations about an axis
- translations on a periodic medium
- motion around a ring orbit

In practice, check:

- does the system admit a continuous cyclic parameter that returns after one cycle?
- does moving along that parameter shift the phase in a consistent way?

That consistency is equivariance.

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## 3) What “surjective phase response” means

Surjective response means the symmetry parameter sweeps the full phase cycle. Physically:

- a full rotation sweeps a full day of local solar phase
- a full spatial period sweeps a full wave phase (if wavenumber nonzero)
- a loop around a vortex sweeps full order-parameter phase (if winding nonzero)

This is the checkable heart of witness existence.

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## 4) What “embedded witness” means in experiments

Many phase measurements are loop-based:

- interferometry around closed paths
- vortex circulation loops
- phase difference around ring resonators

SFT’s ESW tier matches this: you only need a subsystem loop carrying a nontrivial winding.

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## 5) What “robustness margins” correspond to

- LCM (<\(\pi\)) corresponds to: phase-noise amplitude is less than half a cycle on the witness loop, so degree cannot change.
- HMR (<\(P/2\)) corresponds to: holonomy drift on a witness loop is less than half a cycle, so the integer class is stable.

These margins are measurable as bounds on phase error or loop-integral error.

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## 6) What “diffusion” corresponds to

Diffusion in phase is a model for:

- phase jitter/noise
- unresolved small-scale variability
- random sampling increments

SFT calls this out explicitly as NDB (noise-to-diffusion bridge). Once adopted, mixing statements become theorem consequences.

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## Acceptance gates

- each major hypothesis from Univ Pass 2–7 is translated into a practical “what to check”
- idealizations are labeled as such
- robustness margins are tied to measurable phase error scales
- diffusion is framed as a modeling bridge (NDB), not asserted as universal fact
