# JMP0X1B Physics Conventions

Version: 2026-09-18-v1

These are the default conventions for the projection-electromagnetism papers. A
paper may override them only by stating the override next to the affected equation.
The cogravity papers use the same four-dimensional signature but remain a separate,
postulated flat-background vector model; shared notation does not identify the two
theories.

## Coordinates and geometry

- Four-dimensional signature: `(-,+,+,+)`.
- Five-dimensional coordinates: `x^A=(x^mu,chi)`, with `A,B=0,...,4` and
  `mu,nu=0,...,3`.
- `chi` is dimensionless unless a paper explicitly declares otherwise.
- For a block-diagonal chart,
  `ds_5^2=g_{mu nu} dx^mu dx^nu + epsilon Phi^2 dchi^2`, where
  `epsilon` is `+1` or `-1` and `Phi` has dimensions of length.
- Four-dimensional indices are raised with `g^{mu nu}` and five-dimensional
  indices with `G^{AB}`. Never raise a mixed component with an implicit Euclidean
  metric.
- The selected normal is `n^A=s delta^A_chi/Phi`, where `s=+1` fixes the default
  orientation and `n_A n^A=epsilon`. Reversing the normal sends `s -> -s`,
  `Pi_mu -> -Pi_mu`, and `K_mu nu -> -K_mu nu`; an observable cannot depend on
  that bookkeeping choice.
- Extrinsic curvature is
  `K_mu nu=e^A_mu e^B_nu nabla_A n_B`. It is spacetime embedding curvature,
  not the curvature of a metal edge in a spatial mesh. The normal acceleration is
  `a_mu=e^A_mu n^B nabla_B n_A`.

## Gauge field

The five-dimensional action is normalized as

```text
S_A = -1/(4 mu_5) integral d^4x dchi sqrt(|G|) F_AB F^AB
      - integral d^4x dchi sqrt(|G|) J^A A_A,
F_AB = partial_A A_B - partial_B A_A.
```

Variation gives

```text
nabla_A F^AB = mu_5 J^B,
nabla_[A F_BC] = 0.
```

Gauge invariance requires `nabla_A J^A=0`. With dimensionless `chi`, the units of
`A_chi`, `J^chi`, and `mu_5` follow from this action; they must not be inferred by
treating `chi` as a fifth SI length. For a constant compact radius `Phi_0` and period
`2 pi`, zero-mode normalization gives `1/mu_0 = 2 pi Phi_0/mu_5` before any additional
field rescaling.

Define

```text
bar F_mu nu = F_mu nu,
Pi_mu = e^A_mu n^B F_AB = s F_mu chi/Phi,
partial_perp = n^A nabla_A.
```

For the block-diagonal chart,

```text
F^chi nu = -s epsilon Pi^nu/Phi.
```

The orientation signs in `s` and `Pi` must transform together.

## Exact coordinate projection

For any antisymmetric tensor,

```text
nabla_A F^A nu = 1/(Phi sqrt(|g|)) [
  partial_mu(Phi sqrt(|g|) F^mu nu)
  + partial_chi(Phi sqrt(|g|) F^chi nu)
].
```

Therefore, in the stated block-diagonal chart,

```text
D_mu bar F^mu nu = mu_5 J^nu
  - (partial_mu ln Phi) bar F^mu nu
  - 1/(Phi sqrt(|g|)) partial_chi(Phi sqrt(|g|) F^chi nu).
```

This is the convention-complete source projection. More general foliations may rewrite
the same residual in terms of normal derivatives, extrinsic curvature, and normal
acceleration, but those terms must be derived rather than guessed. A safe coordinate-free
definition is

```text
R^nu = h^nu_B nabla_A F^AB - D_mu bar F^mu nu,
J_proj^nu = -R^nu/mu_0.
```

The Bianchi identity gives `D_[alpha bar F_beta gamma]=0`. Its mixed coordinate
component is

```text
partial_chi F_mu nu
  + s partial_mu(Phi Pi_nu)
  - s partial_nu(Phi Pi_mu) = 0.
```

Boundary and junction conditions must come from the varied action, including surface
terms and material constitutive laws. Electrode-edge weighting is only a design proxy
unless a separate action derives a map to spacetime embedding variables.

## Compact modes and Hodge dual

For dimensionless periodic `chi ~ chi+2 pi`, use
`X(x,chi)=sum_n X_n(x) exp(i n chi)`. With constant `Phi_0`, the geometric mode
scale is `|n|/Phi_0`. The five-dimensional orientation is chosen so that
`vol_5=vol_4 wedge n_flat` for `s=+1`. Hodge-star signs follow from this orientation
and the stated metric signature.

## Scalars, units, and force accounting

- SI units are the default for laboratory observables. Natural-unit formulae must be
  labeled and converted before numerical comparison.
- A dimensionless scalar `s` has kinetic term
  `-(f_s^2/2) partial_mu s partial^mu s`; omitting `f_s` is not dimensionally neutral.
- A scalar acceleration of matter requires a matter coupling, for example
  `S_m[A^2(s)g_mu nu,psi]`, and is controlled by
  `alpha_m=d ln A/ds` at the background.
- Forces are derivatives with respect to a declared mechanical generalized coordinate
  under a stated electrical boundary condition. Source, actuator, thermal, support,
  field, matter, and radiation work all belong in the same closed-system ledger.
- Negative gravitational binding or field energy is not a free-energy reservoir.

## Inference policy

- Exact equality of predictive distributions defines observational equivalence.
  Finite-tolerance closeness defines an indistinguishability graph, not a quotient.
- Candidate sensitivities are projected outside the nuisance tangent space before rank,
  singular-value, Fisher, or posterior-conditioning claims are made.
- Null contrasts are normalized only by a traceable nonzero calibration scale, never
  by an expected-null measurement.
- A signature proves integrity/authentication under a key policy, not physical truth.