# Pass D — Lift Calculus and Nonlinear Time on General \(\Sigma\)

## Purpose

Move from “a lift exists” to a reusable calculus for lifts and their constraints:

- when a global lift exists (and what the obstruction is)
- how holonomy becomes the canonical nonlinear-time consistency datum
- how gauge transformations act
- a canonical admissible nonlinear class on \(S^1\) (refined) and its extension to general \(\Sigma\)
- closure properties needed for later dynamics/PDE passes

This pass introduces program language that makes later “fluid time” statements precise.

---

## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Syncre Lift Law | **SLL** | \(\Phi = F \bmod P\) for a real lift \(F\) |
| Local Lift Atlas | **LLA** | A family of local lifts \(F_i\) on an open cover |
| Transition Cocycle | **TCC** | Integer-valued transitions \(n_{ij}\) on overlaps |
| Syncre Obstruction Class | **SOC** | Cohomology class detecting failure of a global lift |
| Syncre Connection Form | **SCF** | The 1-form \(dF\) (defined locally) capturing spatial phase gradient |
| Syncre Holonomy Functional | **SHF** | Loop integral data induced by SCF |
| Admissible Future Direction | **AFD** | A chosen “time-like” direction along which monotonicity is enforced |
| Syncre Monotone Flow Condition | **SMF** | A monotonicity constraint along the AFD |
| Syncre Bounded-Rate Condition | **SBR** | A two-sided bound on phase-speed along the AFD |

---

## Syncre Lift Law (SLL) and local lifts

Fix \(P>0\), \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\), and an arena slice \(\Sigma\).

### Definition (SLL)

A phase field \(\Phi:\Sigma\to \mathbb{T}_P\) satisfies the **Syncre Lift Law** if there exists \(F:\Sigma\to\mathbb{R}\) such that
\[
\boxed{\;\Phi = F \bmod P.\;}
\]

When \(F\) exists, analysis can be done in \(\mathbb{R}\) and then projected mod \(P\).

### Local lift atlas (LLA)

Even when a global lift fails, local lifts exist whenever \(\Phi\) is regular enough.

Let \(\{U_i\}\) be an open cover of \(\Sigma\). A **Local Lift Atlas** is a family of functions \(F_i:U_i\to\mathbb{R}\) such that
\[
\Phi|_{U_i}=F_i\bmod P.
\]

On overlaps \(U_i\cap U_j\), two lifts differ by an integer multiple of \(P\):
\[
F_i - F_j = n_{ij}\,P
\quad\text{for some integer-valued function }n_{ij}:U_i\cap U_j\to \mathbb{Z}.
\]

When \(\Phi\) is continuous, each \(n_{ij}\) is locally constant, hence constant on connected overlap components.

---

## Transition cocycle and the obstruction to a global lift

### Definition (Transition cocycle, TCC)

The integers \(n_{ij}\) on overlaps define a Čech 1-cocycle with values in \(\mathbb{Z}\) (because on triple overlaps \(U_i\cap U_j\cap U_k\) one has)
\[
n_{ij}+n_{jk}+n_{ki}=0.
\]

This cocycle is the **Transition Cocycle** of the lift atlas.

### Definition (Syncre Obstruction Class, SOC)

The cohomology class \([n_{ij}]\in \check H^1(\Sigma;\mathbb{Z})\) determined by the TCC is the **Syncre Obstruction Class**.

- \(\mathrm{SOC}=0\) if and only if there exists a global lift \(F\) (SLL holds globally).
- \(\mathrm{SOC}\neq 0\) means there is a holonomy-type defect: lifts cannot be globally glued.

This is a formal version of the “Holonomy Defect” obstruction named in Pass C.

### Practical reading

- On \(\Sigma=S^1\), \(\check H^1(S^1;\mathbb{Z})\cong\mathbb{Z}\). The SOC corresponds to the degree/winding.
- On simply connected \(\Sigma\), \(\check H^1(\Sigma;\mathbb{Z})=0\), so continuous \(\Phi\) admits a global lift.

This is why holonomy becomes central precisely when \(\Sigma\) has nontrivial loops.

---

## Syncre connection form and holonomy functional

Assume \(\Phi\) is smooth enough so local lifts \(F_i\) are differentiable.

### Definition (Syncre connection form, SCF)

On each \(U_i\), define the 1-form
\[
\alpha_i := dF_i.
\]
On overlaps, \(F_i-F_j=n_{ij}P\) is locally constant, hence
\[
dF_i = dF_j.
\]
Therefore the \(\alpha_i\) glue to a globally defined 1-form \(\alpha\) on \(\Sigma\) even when a global \(F\) fails to exist.

We call
\[
\boxed{\;\alpha := dF\ \text{(locally)}\;}
\]
the **Syncre connection form**.

It captures the spatial phase gradient without requiring a globally single-valued lift.

### Definition (Syncre holonomy functional, SHF)

For a closed loop \(\gamma\) in \(\Sigma\), define
\[
\boxed{\;\mathrm{Hol}(\gamma) := \int_{\gamma}\alpha.\;}
\]

- If a global lift exists, \(\alpha=dF\) globally and \(\mathrm{Hol}(\gamma)=0\) for all closed loops.
- In general, \(\mathrm{Hol}(\gamma)\in P\mathbb{Z}\) is the loop-consistency condition for \(\Phi=F\bmod P\).
- Nonzero holonomy detects nontrivial winding/degree behavior.

This matches the Pass C holonomy language and makes it analytic.

---

## Gauge transformations

A global gauge shift is \(\Phi\mapsto \Phi\oplus c\) with constant \(c\in\mathbb{T}_P\). On lifts, this is \(F_i\mapsto F_i + c'\) for any real representative \(c'\) of \(c\).

### Lemma (Gauge invariance)

- The TCC cocycle \([n_{ij}]\) is unchanged by gauge.
- The connection form \(\alpha\) is unchanged by gauge.
- Holonomy values \(\mathrm{Hol}(\gamma)\) are unchanged by gauge.

So SOC/SCF/SHF are stable structural features, not coordinate artifacts.

---

## Nonlinear time constraints via an admissible future direction

To express “time is nonlinear” while keeping a coherent direction of progression, specify a future direction.

### Definition (Admissible future direction, AFD)

On an arena \(M\) with slicing \(\{\Sigma_s\}\), an **AFD** is a vector field \(T\) (or flow direction) that is transverse to the slices and points from \(\Sigma_s\) to later slices \(\Sigma_{s+\varepsilon}\).

In the simplest model \(M=\mathbb{R}\times \Sigma\), the canonical AFD is \(T=\partial_t\).

### Definition (Syncre monotone flow condition, SMF)

Given a lift \(F\) on \(M\), require
\[
\boxed{\;T(F) > 0\;}
\]
or, in a non-smooth setting, monotonicity along the flow lines of \(T\).

This prevents local time reversal.

### Definition (Syncre bounded-rate condition, SBR)

Require two-sided bounds along \(T\):
\[
\boxed{\;0<m\le T(F)\le M\;}
\]
for constants \(m,M\). This prevents freezing or blow-up and enables compactness arguments in later convergence results.

---

## Canonical normal form (refinement) on \(S^1\)

On \(\Sigma=S^1\) with coordinate \(\lambda\), Pass C introduced:
\[
F(t,\lambda)=\psi(t)+\frac{kP}{2\pi}\lambda + u(t,\lambda).
\]

We now formalize the normal-form role of this decomposition.

### Proposition (Normal form up to gauge and periodic remainder)

Let \(F(t,\cdot)\) be continuous and satisfy the winding relation
\[
F(t,\lambda+2\pi)=F(t,\lambda)+kP.
\]
Then for each fixed \(t\), there exists a continuous periodic \(u(t,\cdot)\) such that
\[
F(t,\lambda)=a(t)+\frac{kP}{2\pi}\lambda + u(t,\lambda),
\quad u(t,\lambda+2\pi)=u(t,\lambda).
\]

Here \(a(t)\) is a gauge-dependent offset. The integer \(k\) is the degree/winding, and it is invariant under continuous deformation that preserves continuity and winding.

### Value

This normal form isolates:

- the topological winding (responsible for coverage)
- the nonlinear remainder \(u\) (responsible for distortion and dynamics)

Later passes will impose dynamical laws on \(u\) and measure how \(\mu_t\) evolves.

---

## Extension to general \(\Sigma\): decomposition into cohomology class + remainder

On general \(\Sigma\), the connection form \(\alpha\) is globally defined. The obstruction class SOC corresponds to the cohomology class of \(\alpha\) in de Rham/Čech language, depending on regularity.

A practical program decomposition is:

- a chosen reference connection \(\alpha_0\) representing the SOC class
- a remainder exact form \(dU\) representing nonlinear deformation within the same SOC class

Symbolically:
\[
\alpha = \alpha_0 + dU.
\]

When \(\Sigma\) has nontrivial \(H^1\), \(\alpha_0\) carries the topological information (winding/holonomy), and \(U\) carries the nonlinear distortion.

This is the general-manifold analogue of \(k\lambda + u\) on \(S^1\).

---

## Acceptance gates

- SLL, LLA, TCC, and SOC are defined, and SOC is stated as the global-lift obstruction
- SCF and SHF are defined and shown to be gauge invariant
- AFD, SMF, and SBR are defined to formalize nonlinear time progression
- The \(S^1\) normal form is proved and linked to the Syncre winding parameter \(k\)
- A general-\(\Sigma\) decomposition is described using the connection form viewpoint
