# Global AI / Embedded AI Physics MVP

## Physical system
2D isotropic harmonic oscillator:
- mass m = 1.7
- natural frequency omega0 = 1.3
- one physical period T = 4.83321946706
- rotating observer angular speed Omega = 0.481 = 0.37 omega0

Global state:
z=(x,y,vx,vy).

Conserved global quantities:
E = 1/2 m |v|^2 + 1/2 m omega0^2 |r|^2
Lz = m(x vy - y vx)

## Global AI
- Energy relative span over one period: 9.141e-16
- Angular-momentum relative span: 7.372e-16
- Full state return residual after one period: 2.107e-15

## Embedded A — position only
Instantaneous map rank = 2/4.
Two distinct global states can have the same observed position.

At t*=T/7:
- observed position = [0.9591992248 0.3503329219]
- compatible velocity #1 = [-1.000596458   1.1123425579]
- compatible velocity #2 = [-0.200596458   0.5123425579]
- energies: 3.400701250 vs 1.755300658
- angular momenta: 2.409750000 vs 0.954914017

So E and Lz are not identifiable from one position snapshot.

Using trajectory history and centered finite differences:
- relative energy RMSE = 4.266e-07
- relative angular-momentum RMSE = 4.112e-07

This is dynamic accessibility: history recovers information unavailable in one instant.

## Embedded B — x and vx only
Instantaneous map rank = 2/4.

At the same local observation [x,vx] = [ 0.9591992248 -1.000596458 ]:
- hidden state #1 [y,vy] = [0.3503329219 1.1123425579]
- hidden state #2 [y,vy] = [1.2503329219 0.7123425579]
- total energies: 3.400701250 vs 4.849729147
- total angular momenta: 2.409750000 vs 3.288407108

Again, the same local observation corresponds to different global invariants.

## Embedded C — rotating observer
The rotating observer has a full-rank observation transform:
rank = 4/4.

So this observer is information-complete if the frame transformation is known.

But if it naively applies the inertial equation a=-omega0^2 r in rotating coordinates:
- dynamics residual RMS = 1.133057e+00
- naive energy relative span = 5.327120e-02

Using the correct rotating-frame law:
a_R = -(omega0^2-Omega^2) r_R - 2 Omega J v_R

and covariant inertial velocity:
v_I(frame components)=v_R + Omega J r_R

we obtain:
- corrected dynamics residual RMS = 5.414710e-07
- covariant energy relative span = 1.044700e-15
- covariant angular-momentum relative span = 1.105731e-15

This is the key observer result:
apparent non-conservation can be generated by an observer-law mismatch even when the underlying global system exactly conserves E and Lz.

## Relative holonomy after one physical period
After T=2pi/omega0:
- Global physical state returns with residual 2.107e-15.
- Rotating observer coordinates do NOT return.
- Relative frame holonomy angle = -133.200000 deg.
- Rotating position return residual = 2.352390e+00
- Rotating velocity return residual = 9.017852e-01
- Predicted endpoint from H_rel=R(-Omega T) residual = 2.441e-15

So:
global state closure != observer-frame relational closure.

## Series-B interpretation
Global AI:
- full state
- direct E,Lz
- no observational ambiguity

Embedded A:
- rank-deficient observation
- instantaneously cannot identify E,Lz
- history partially restores accessibility

Embedded B:
- another rank-deficient quotient of the same world
- same local observation can correspond to different global invariants

Embedded C:
- full information but observer-dependent dynamics
- naive local law creates false anomaly
- correct covariance restores conservation
- one global physical cycle induces nontrivial relative holonomy

## MVP conclusion
The useful object is not one scalar "observer gap".
It is a structured gap:
Delta_G->i = (access, identifiability, dynamics-law defect, conservation defect, holonomy).

This classical MVP already demonstrates the exact architecture intended for a future quantum version without invoking quantum mechanics.