# Pass A — Axioms, Objects, and Invariants for Syncre

## Purpose

Establish the minimal mathematical objects that define Syncre and prove the first general theorems that make “all times are present at once” a structural statement rather than an Earth-specific coincidence.

The goal is a foundation where the Earth model is a corollary, and where the same axioms can be verified in many other physical settings.

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## Foundations motivation: Syncre Form Theory (SFT)

Syncre Form Theory is motivated by a higher relational aim: a “point” does not possess form in isolation; its form is inseparable from the structure in which it is embedded. Because this is difficult to formalize directly, we begin with Syncre as a **principle-identical proxy**: a minimal observable structure that preserves the invariant/gauge logic of the higher principle.

Program language:

- **RFP** — Relational Form Principle: form is an invariant of embeddedness; points are studied as \((M,p)\).
- **PIP** — Principle-Identical Proxy: a minimal observable structure preserving invariant/gauge logic.
- **PIP statement:** Syncre is the PIP of RFP.

See `FOUNDATIONS_RFP_PIP.md` for the full foundations addendum and the method principle.


## Core codomain

Fix a period \(P>0\) and define the phase circle
\[
\mathbb{T}_P := \mathbb{R}/P\mathbb{Z}.
\]
Write \(\oplus\) for addition in \(\mathbb{T}_P\).

---

## Syncre frame

### Definition (Syncre frame)

A **Syncre frame** is a triple
\[
\boxed{\;\big(M,\{\Sigma_s\}_{s\in\mathbb{R}},\Phi\big)\;}
\]
where

- \(M\) is an ambient arena (spacetime region, configuration space, or state space)
- \(\{\Sigma_s\}\) is a slicing by “global instants”
  \[
  M=\bigsqcup_{s\in\mathbb{R}} \Sigma_s
  \]
- \(\Phi:M\to \mathbb{T}_P\) is a phase field

### Axiom (Syncre coverage)

A Syncre frame satisfies **coverage** if
\[
\boxed{\;\forall s\in\mathbb{R},\quad \Phi(\Sigma_s)=\mathbb{T}_P.\;}
\]

This is the abstract form of “at one global instant, the space contains points whose local phase labels span the full cycle.”

---

## Syncre witnesses and pullbacks

### Definition (Witness)

A **Syncre witness** is a Syncre frame \((M,\{\Sigma_s\},\Phi)\) satisfying coverage, together with an explicit mechanism producing \(\Phi\) from a known periodic structure (rotation, orbit, wave phase, or order-parameter phase).

### Lemma (Pullback witness)

Let \((M,\{\Sigma_s\},\Phi)\) be a Syncre frame. Let \(i_s:\mathcal{E}\hookrightarrow \Sigma_s\) be an embedding of a subsystem into the slice \(\Sigma_s\). Define the induced subsystem phase field
\[
\tau_s=\Phi\circ i_s:\mathcal{E}\to\mathbb{T}_P.
\]
If \(\Phi(\Sigma_s)=\mathbb{T}_P\) and \(i_s(\mathcal{E})\subseteq \Sigma_s\) meets every phase value, then \(\tau_s(\mathcal{E})=\mathbb{T}_P\).

In particular, when \(i_s(\mathcal{E})=\Sigma_s\), subsystem coverage is automatic.

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## The bridge statement

Now this is our theoretical model. Now given our assumption we aim to prove it by showing other similar models exist within the known physical universe.

Pass A makes this precise by isolating general hypotheses that imply coverage, then proving them as theorems. Later passes use those theorems to build a catalog of universe-scale witnesses.

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## General theorem family 1: phase coordinate implies coverage

### Definition (Phase coordinate form)

Fix a slice \(\Sigma_s\). A phase field \(\Phi|_{\Sigma_s}:\Sigma_s\to \mathbb{T}_P\) is in **phase coordinate form** if there exists a surjective map \(\vartheta:\Sigma_s\to \mathbb{T}_P\) and a constant \(c_s\in\mathbb{T}_P\) such that
\[
\Phi|_{\Sigma_s}(x)=\vartheta(x)\oplus c_s.
\]

### Theorem (Coverage from phase coordinate)

If \(\Phi|_{\Sigma_s}\) is in phase coordinate form with \(\vartheta(\Sigma_s)=\mathbb{T}_P\), then \(\Phi(\Sigma_s)=\mathbb{T}_P\).

### Proof

Addition by a fixed constant \(c_s\) is a bijection of \(\mathbb{T}_P\). Therefore
\[
\Phi(\Sigma_s)=\big(\vartheta(\Sigma_s)\big)\oplus c_s=\mathbb{T}_P\oplus c_s=\mathbb{T}_P.
\]
∎

This is the simplest abstract reason the Earth longitude model works.

---

## General theorem family 2: circle symmetry implies coverage

### Definition (Equivariant circle action)

Fix a slice \(\Sigma_s\). Assume there is a continuous action
\[
A:S^1\times \Sigma_s\to \Sigma_s
\]
and a phase field \(\Phi|_{\Sigma_s}:\Sigma_s\to \mathbb{T}_P\) such that there exists a nonzero “phase response” homomorphism \(\chi:S^1\to \mathbb{T}_P\) satisfying equivariance:
\[
\Phi\big(A(\theta,x)\big)=\Phi(x)\oplus \chi(\theta).
\]

### Theorem (Coverage from an equivariant action)

If there exists \(x_0\in\Sigma_s\) whose orbit \(\{A(\theta,x_0):\theta\in S^1\}\) is a circle and \(\chi\) is surjective, then \(\Phi(\Sigma_s)=\mathbb{T}_P\).

### Proof

Restrict \(\Phi\) to the orbit:
\[
\Phi(A(\theta,x_0))=\Phi(x_0)\oplus \chi(\theta).
\]
Since \(\chi(S^1)=\mathbb{T}_P\), the orbit image is \(\Phi(x_0)\oplus \mathbb{T}_P=\mathbb{T}_P\). The orbit lies in \(\Sigma_s\), hence \(\mathbb{T}_P\subseteq \Phi(\Sigma_s)\). ∎

This is the symmetry mechanism behind rotating-body, ring/orbit, and traveling-wave witnesses.

---

## Invariant on \(S^1\): degree and surjectivity

To make coverage stable and classifiable, begin on \(\Sigma=S^1\). Continuous circle maps have an integer degree (winding number).

### Definition (Degree via lifts)

Let \(f:S^1\to S^1\). Write points as \(e^{i\theta}\) with \(\theta\in\mathbb{R}\).  
A **lift** of \(f\) is a continuous function \(g:\mathbb{R}\to\mathbb{R}\) such that
\[
f(e^{i\theta}) = e^{ig(\theta)}.
\]
A lift satisfies
\[
g(\theta+2\pi)=g(\theta)+2\pi k
\]
for a unique integer \(k\). This integer is the **degree** of \(f\), written \(\deg(f)=k\).

### Lemma (Nonzero degree implies coverage on the circle)

If \(f:S^1\to S^1\) is continuous and \(\deg(f)\neq 0\), then \(f\) is surjective.

### Proof

Let \(g\) be a lift with \(g(\theta+2\pi)=g(\theta)+2\pi k\) and \(k\neq 0\).  
By continuity, \(g([0,2\pi])\) is an interval \([m,M]\). The endpoint relation implies
\[
g(2\pi)-g(0)=2\pi k,
\]
so \(M-m\ge 2\pi|k|\). Modulo \(2\pi\), the interval of length at least \(2\pi\) covers every residue class, hence \(e^{ig(\theta)}\) attains every point of \(S^1\). Therefore \(f\) is surjective. ∎

This lemma is the first robustness lever: degree is homotopy-invariant, and within a fixed degree class, coverage persists.

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## Earth-longitude as a canonical witness

Fix a global instant \(t_0\). Let longitude be \(\lambda\in S^1\). In the idealized solar-time model:
\[
\tau(t_0,\lambda)=\Bigl(t_0+\frac{P}{2\pi}\lambda\Bigr)\bmod P.
\]
Equivalently as a circle map \(f:S^1\to \mathbb{T}_P\cong S^1\), it is a degree-\(1\) wrap. By the nonzero-degree lemma, it is surjective, so at that instant the globe spans the full cycle of local times.

This is the precise mathematical core of the Earth statement.

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## Pass A outputs and what they enable

- Syncre frame definition, coverage axiom, witness and pullback notions
- Two general theorem templates guaranteeing coverage
- Degree invariant on \(S^1\) and a surjectivity lemma

These are the components needed to make the universe-scale catalog in later passes rigorous: each new physical model is proved to be a witness by checking a theorem template’s hypotheses, not by analogy.

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## Acceptance gates

- All objects \((M,\{\Sigma_s\},\Phi)\), coverage, and witness notions are stated without ambiguity
- At least two general theorems include complete proofs
- Degree is defined via lifts and used to prove surjectivity on \(S^1\)
- The Earth-longitude case is explicitly shown to have nonzero degree and therefore coverage
