# Pass F — Multi-Cycle Torus Upgrade and Coupling Laws

## Purpose

Upgrade Syncre from a single-cycle phase circle to a multi-cycle torus in a way that:

- preserves the canonical coverage law in a product setting
- encodes solar–lunar coupling as a global constraint (synodic offset)
- introduces torus-level analogues of degree/holonomy and diagnostics
- prepares the ground for convergence and sampling theorems (Pass G)

This pass introduces new program language for multi-cycle structure.

---

## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Syncre Torus Field | **STF** | A vector-valued phase field \(\Phi:M\to \mathbb{T}_{P_1}\times\cdots\times \mathbb{T}_{P_d}\) |
| Torus Coverage Law | **TCL** | Slice image covers the full torus: \(\Phi(\Sigma_s)=\prod_j\mathbb{T}_{P_j}\) |
| Coupling Constraint | **CPL** | A relation reducing the torus image to a subtorus (global offsets) |
| Synodic Offset Law | **SOL** | Solar–lunar difference is a global phase \(\Phi_{\leftmoon}-\Phi_\odot=\Psi(s)\) |
| Coupled Subtorus | **CST** | The subset of the torus defined by coupling constraints |
| Torus Holonomy Vector | **THV** | Loop integrals of the connection form as a vector in \(\mathbb{R}^d\) modulo periods |
| Resonance / Nonresonance Flag | **RNF** | Arithmetic condition on frequency vector for equidistribution |
| Stroboscopic Increment | **STI** | Discrete-time increment on torus under sampling |

---

## Multi-cycle codomain and fields

Fix periods \(P_1,\dots,P_d>0\). Define the torus
\[
\mathbb{T}_{\mathbf{P}} := \mathbb{T}_{P_1}\times\cdots\times \mathbb{T}_{P_d}.
\]

### Definition (Syncre Torus Field, STF)

A **Syncre Torus Field** is a map
\[
\Phi:M\to \mathbb{T}_{\mathbf{P}}.
\]
Write components \(\Phi=(\Phi^{(1)},\dots,\Phi^{(d)})\) with \(\Phi^{(j)}:M\to \mathbb{T}_{P_j}\).

A slicing \(M=\bigsqcup_s \Sigma_s\) induces slice restrictions \(\Phi_s:=\Phi|_{\Sigma_s}\).

---

## Torus Coverage Law (TCL)

### Definition (TCL)

The torus analogue of SCL is:
\[
\boxed{\;\forall s,\qquad \Phi(\Sigma_s)=\mathbb{T}_{\mathbf{P}}.\;}
\]

This states that each global instant contains the full set of multi-cycle phase combinations.

### Remark

TCL is stronger than requiring each component \(\Phi^{(j)}(\Sigma_s)=\mathbb{T}_{P_j}\). Component-wise coverage does not guarantee full torus coverage because components may be coupled.

This is where coupling constraints enter.

---

## Coupling constraints and coupled subtori

### Definition (Coupling constraint, CPL)

A **coupling constraint** is a relation of the form
\[
C\big(\Phi^{(1)},\dots,\Phi^{(d)},s\big)=0
\]
that holds on every slice and restricts \(\Phi(\Sigma_s)\) to a subset of the torus.

When the constraint is an affine relation among components, the image lies in a subtorus.

### Definition (Coupled subtorus, CST)

Given a set of constraints, the **Coupled Subtorus** at slice \(s\) is
\[
\mathrm{CST}_s := \{y\in\mathbb{T}_{\mathbf{P}}: C(y,s)=0\}.
\]
Then \(\Phi(\Sigma_s)\subseteq \mathrm{CST}_s\).

If \(\mathrm{CST}_s\neq \mathbb{T}_{\mathbf{P}}\), full torus coverage fails by design. In that case, the correct “coverage law” is coverage of the coupled subtorus:
\[
\Phi(\Sigma_s)=\mathrm{CST}_s.
\]

This is the proper multi-cycle generalization when physics imposes coupling.

---

## Solar–lunar synodic offset as a coupling law

Take \(d=2\) with components:

- solar phase \(\Phi_\odot:M\to \mathbb{T}_{P_\odot}\)
- lunar phase \(\Phi_{\leftmoon}:M\to \mathbb{T}_{P_{\leftmoon}}\)

To compare them, fix an identification into a common circle when desired, or work with a mixed-period difference via chosen scaling. In the simplest shared-period model, set \(P_\odot=P_{\leftmoon}=P\).

### Definition (Synodic Offset Law, SOL)

The synodic offset coupling is:
\[
\boxed{\;\Phi_{\leftmoon}(p)\ominus \Phi_\odot(p) = \Psi(s)\quad\text{for all }p\in \Sigma_s,\;}
\]
where \(\Psi:\mathbb{R}\to\mathbb{T}_P\) is a global function of the slice parameter \(s\) only, and \(\ominus\) is subtraction in \(\mathbb{T}_P\).

This encodes the identity:
- the solar–lunar difference cancels position and depends only on the global moment

### Coupled subtorus for SOL

At each \(s\), the constraint defines a diagonal-like subtorus:
\[
\mathrm{CST}_s = \{(a,b)\in \mathbb{T}_P\times\mathbb{T}_P : b\ominus a = \Psi(s)\}.
\]

Thus \(\Phi(\Sigma_s)\subseteq \mathrm{CST}_s\), and the correct torus-coverage target is coverage of \(\mathrm{CST}_s\).

---

## Torus holonomy vector (THV)

When components admit local lifts \(F^{(j)}\), define local connection forms \(\alpha^{(j)}=dF^{(j)}\) and assemble
\[
\alpha = (\alpha^{(1)},\dots,\alpha^{(d)}).
\]

For a loop \(\gamma\), define the holonomy vector:
\[
\boxed{\;\mathrm{Hol}(\gamma) := \left(\int_\gamma \alpha^{(1)},\dots,\int_\gamma \alpha^{(d)}\right).\;}
\]
Consistency requires each component holonomy lie in \(P_j\mathbb{Z}\), and coupling constraints impose additional relations among these integers.

This is the multi-cycle analogue of HOL/HMR.

---

## A minimal theorem: coupled coverage from one free coordinate

Coupling reduces the dimension. Coverage then means “the free coordinate(s) cover.”

### Theorem (Coupled coverage under SOL)

Assume \(P_\odot=P_{\leftmoon}=P\) and the SOL constraint holds:
\[
\Phi_{\leftmoon}\ominus\Phi_\odot=\Psi(s)\ \text{ on }\Sigma_s.
\]
Assume further that \(\Phi_\odot(\Sigma_s)=\mathbb{T}_P\). Then
\[
\Phi(\Sigma_s)=\mathrm{CST}_s.
\]

### Proof

For any \(a\in\mathbb{T}_P\), there exists \(p\in\Sigma_s\) with \(\Phi_\odot(p)=a\). Then \(\Phi_{\leftmoon}(p)=a\oplus \Psi(s)\). Therefore \((a,a\oplus\Psi(s))\in\Phi(\Sigma_s)\). These are exactly the points of \(\mathrm{CST}_s\). ∎

This is the precise formal expression of “the solar–lunar offset is global, not positional.”

---

## Stroboscopic increments on the coupled subtorus

Sampling at discrete times \(s_n=s_0+n\Delta\) produces a sequence on \(\mathrm{CST}_{s_n}\). Under mild assumptions, this induces an effective circle rotation on the synodic phase \(\Psi(s_n)\).

### Program-language hooks for Pass G

- **STI**: \(\Delta\Psi := \Psi(s_{n+1})\ominus\Psi(s_n)\)
- **RNF**: arithmetic nonresonance of \(\Delta\Psi\) (or its lifted increment) drives equidistribution

Pass G will connect these to convergence of pushforward measures and time averages.

---

## Acceptance gates

- STF and TCL are defined and their relationship to component-wise coverage is made explicit
- CPL/SOL are defined and used to define CST
- A coupled coverage theorem is proved for SOL
- THV is defined and tied to lift/holonomy language from Pass D
- Stroboscopic increment terminology is introduced for convergence work in Pass G
