# Breakthrough Pass 1 — Quantitative Robustness and Dynamic Degradation Bounds

## Purpose

Advance the “breakthrough targets” (BP-1 and BP-3) from the planning layer into actual theorems with complete proofs in the canonical 1D setting \(\Sigma=S^1\).

This pass adds real mathematical value by connecting:

- topological winding (degree \(k\))
- analytic bounds on the lift derivative (\(\partial_\lambda F\))
- distributional diagnostics (pushforward densities, total-variation bounds)
- explicit degradation bounds under Syncre transport dynamics (STD)

The results below are formulated in Syncre language but written so they stand alone as standard theorems once the definitions are fixed.

---

## Notation

- Period \(P>0\), phase circle \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\)
- Spatial circle \(S^1=\mathbb{R}/2\pi\mathbb{Z}\) with coordinate \(\lambda\)
- Uniform measure on \(S^1\): \(d\nu(\lambda)=\frac{d\lambda}{2\pi}\)
- Haar (uniform) measure on \(\mathbb{T}_P\): \(d\mu_{\mathrm{H}}(y)=\frac{dy}{P}\)

Let \(F:\mathbb{R}\times S^1\to\mathbb{R}\) be a lift and define the phase map at time \(s\):
\[
\Phi_s(\lambda):=F(s,\lambda)\bmod P\in\mathbb{T}_P.
\]
Define the phase distribution:
\[
\mu_s := (\Phi_s)_\#\nu.
\]

---

## New diagnostic: Syncre Coverage Profile (SCP)

The earlier SCM (coverage modulus) becomes trivial (equal to \(P/2\)) as soon as \(\mu_s\) has a positive density everywhere. For quantitative work, we instead track the minimal mass of radius-\(\rho\) neighborhoods.

### Definition (SCP)

For \(\rho\in[0,P/2]\), define the **Syncre Coverage Profile**
\[
\boxed{\;\mathrm{CP}_s(\rho):=\inf_{y\in\mathbb{T}_P}\ \mu_s\big(B_\rho(y)\big),\;}
\]
where \(B_\rho(y)\) is the arc-neighborhood of radius \(\rho\) (length \(2\rho\)) in \(\mathbb{T}_P\).

- \(\mathrm{CP}_s(\rho)>0\) gives a quantitative “no holes and no near-holes” statement.
- Lower bounds on \(\mathrm{CP}_s(\rho)\) directly prevent Collapse (Pass C).

---

## BP-1A: Density bounds from spatial stretch

The central technical bridge is: derivative bounds in \(\lambda\) produce pointwise bounds on the pushforward density of \(\mu_s\).

### Assumption (Winding + stretch)

Fix a time \(s\). Assume:

- \(F(s,\cdot)\) is \(C^1\) in \(\lambda\)
- winding \(k\in\mathbb{Z}\setminus\{0\}\):
  \[
  F(s,\lambda+2\pi)=F(s,\lambda)+kP
  \]
- stretch bounds:
  \[
  0<a\le \partial_\lambda F(s,\lambda)\le b\quad\text{for all }\lambda
  \]

### Theorem 1 (Density sandwich)

Under the assumptions above, \(\mu_s\) is absolutely continuous with respect to Haar measure \(\mu_{\mathrm{H}}\), with Radon–Nikodym derivative \(f_s=\frac{d\mu_s}{d\mu_{\mathrm{H}}}\) satisfying
\[
\boxed{\;\frac{|k|P}{2\pi b}\ \le\ f_s(y)\ \le\ \frac{|k|P}{2\pi a}\quad\text{for a.e. }y\in\mathbb{T}_P.\;}
\]

#### Proof

Let \(Y=\Phi_s(\lambda)\) in the coordinate model \(y\in[0,P)\). For regular values \(y\), the preimage consists of \(|k|\) points \(\lambda_i(y)\) (counted with multiplicity). By the change-of-variables formula for a smooth covering map,
\[
\frac{d\mu_s}{dy}(y)=\sum_{i=1}^{|k|}\frac{1}{2\pi}\frac{1}{\partial_\lambda F(s,\lambda_i(y))}.
\]
Using \(a\le \partial_\lambda F\le b\),
\[
\frac{|k|}{2\pi b}\le \frac{d\mu_s}{dy}(y)\le \frac{|k|}{2\pi a}.
\]
Since \(d\mu_{\mathrm{H}}=\frac{dy}{P}\), we have \(f_s=\frac{d\mu_s}{d\mu_{\mathrm{H}}}=P\frac{d\mu_s}{dy}\), giving the stated bounds. ∎

---

## BP-1B: Explicit Coverage Profile and Uniformity Defect bounds

### Corollary 2 (Coverage profile lower bound)

Under Theorem 1, for every \(\rho\in[0,P/2]\),
\[
\boxed{\;\mathrm{CP}_s(\rho)\ \ge\ \frac{|k|}{2\pi b}\,(2\rho)\;}
\]
(where the right-hand side is the lower density bound w.r.t. Lebesgue \(dy\) times arc length \(2\rho\)).

In particular, \(\mathrm{CP}_s(\rho)\) scales at least linearly in \(\rho\), ruling out Collapse.

### Corollary 3 (Total-variation bound to Haar)

Let \(\|\cdot\|_{\mathrm{TV}}\) denote total variation distance between probability measures. Under Theorem 1,
\[
\boxed{\;\|\mu_s-\mu_{\mathrm{H}}\|_{\mathrm{TV}}\ \le\ \frac{P}{2}\,\max\!\left(\frac{|k|}{2\pi a}-\frac{1}{P},\ \frac{1}{P}-\frac{|k|}{2\pi b}\right).\;}
\]

#### Proof

Let \(p_s(y)=\frac{d\mu_s}{dy}\). Then \(\mu_{\mathrm{H}}\) has density \(p_{\mathrm{H}}(y)=\frac{1}{P}\).
Total variation satisfies
\[
\|\mu_s-\mu_{\mathrm{H}}\|_{\mathrm{TV}}=\frac12\int_0^P |p_s(y)-1/P|\,dy
\le \frac12\cdot P\cdot \|p_s-1/P\|_{L^\infty}.
\]
Using the bounds on \(p_s\) from Theorem 1 (dividing by \(P\)) gives the result. ∎

### Optional upgrade (Wasserstein-1 bound)

Since \(\mathrm{diam}(\mathbb{T}_P)=P/2\), one has the crude estimate
\[
W_1(\mu_s,\mu_{\mathrm{H}})\le \frac{P}{2}\,\|\mu_s-\mu_{\mathrm{H}}\|_{\mathrm{TV}}.
\]
This converts Corollary 3 into a Wasserstein-1 control if desired.

---

## BP-3: Dynamic degradation bounds under Syncre Transport Dynamics

Now assume \(F\) evolves by STD on \(\Sigma=S^1\):
\[
\partial_s F + v(s,\lambda)\,\partial_\lambda F = \omega(s,\lambda),
\]
with \(v,\omega\) \(2\pi\)-periodic in \(\lambda\).

### Lemma 4 (Winding preservation)

If the winding relation holds at \(s=s_0\),
\[
F(s_0,\lambda+2\pi)=F(s_0,\lambda)+kP,
\]
then it holds for all \(s\) with the same integer \(k\), provided \(v,\omega\) are periodic in \(\lambda\).

This is the DPR result (Pass H); we use it as an input.

---

## BP-3A: Evolution of the spatial derivative

Let \(G:=\partial_\lambda F\). Differentiate STD in \(\lambda\) to obtain
\[
\partial_s G + v\,\partial_\lambda G + (\partial_\lambda v)\,G = \partial_\lambda \omega.
\]
Along a characteristic \(\lambda(s)\) solving \(\dot\lambda=v(s,\lambda)\),
\[
\frac{d}{ds}G(s,\lambda(s)) = -(\partial_\lambda v)(s,\lambda(s))\,G(s,\lambda(s)) + (\partial_\lambda\omega)(s,\lambda(s)).
\]

### Theorem 5 (Exponential stretch bounds under bounded shear and zero \(\partial_\lambda\omega\))

Assume:
- \(|\partial_\lambda v(s,\lambda)|\le C\) for all \((s,\lambda)\)
- \(\partial_\lambda\omega(s,\lambda)\equiv 0\) (i.e., \(\omega=\omega(s)\) depends only on time)
- initial stretch bounds \(0<a_0\le G(s_0,\lambda)\le b_0\)

Then for all \(s\ge s_0\),
\[
\boxed{\;a_0 e^{-C(s-s_0)}\ \le\ \partial_\lambda F(s,\lambda)\ \le\ b_0 e^{C(s-s_0)}\quad\text{for all }\lambda.\;}
\]

#### Proof

Along characteristics,
\[
\frac{d}{ds}G = -(\partial_\lambda v)\,G.
\]
So \(G(s)=G(s_0)\exp\!\left(-\int_{s_0}^s \partial_\lambda v(\sigma,\lambda(\sigma))\,d\sigma\right)\).
Using \(|\partial_\lambda v|\le C\) yields
\[
e^{-C(s-s_0)}\le \exp\!\left(-\int_{s_0}^s \partial_\lambda v\,d\sigma\right)\le e^{C(s-s_0)},
\]
and multiplying by the initial bounds gives the result. ∎

---

## BP-3B: Explicit degradation bounds for coverage profile and TV defect

Combine Theorem 5 with Theorem 1 and Corollaries 2–3.

### Corollary 6 (Time-degraded coverage profile)

Under the assumptions of Theorem 5, for every \(\rho\in[0,P/2]\) and \(s\ge s_0\),
\[
\boxed{\;\mathrm{CP}_s(\rho)\ \ge\ \frac{|k|}{2\pi\,b_0\,e^{C(s-s_0)}}\,(2\rho).\;}
\]

So the minimal neighborhood mass decays at most exponentially with rate \(C\), directly controlled by the shear bound on the flow.

### Corollary 7 (Time-degraded TV defect bound)

Under the assumptions of Theorem 5,
\[
\boxed{\;\|\mu_s-\mu_{\mathrm{H}}\|_{\mathrm{TV}}\ \le\ \frac{P}{2}\,\max\!\left(\frac{|k|}{2\pi\,a_0\,e^{-C(s-s_0)}}-\frac{1}{P},\ \frac{1}{P}-\frac{|k|}{2\pi\,b_0\,e^{C(s-s_0)}}\right).\;}
\]

This gives an explicit (conservative) control on uniformity degradation in time.

---

## What this completes

These results materially advance the breakthrough targets:

- BP-1: quantitative SCM-style information is upgraded into a usable SCP lower bound and an explicit SUD (TV) bound in the canonical nonlinear class under stretch constraints.
- BP-3: explicit exponential degradation bounds under STD connect dynamics control (\(|\partial_\lambda v|\le C\), \(\partial_\lambda\omega=0\)) to coverage-profile and uniformity-defect control.

The remaining gap for a stronger “breakthrough” is to weaken the assumptions (replace \(\partial_\lambda\omega=0\) by bounded forcing, incorporate diffusion or regularization, and upgrade the TV bound to a sharper Wasserstein or Fourier-based decay statement).

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## Acceptance gates

- A new quantitative diagnostic (SCP) is defined and used
- Density bounds are proved from stretch bounds and winding
- Explicit bounds are derived for SCP and TV-based SUD
- Under STD with bounded shear, explicit exponential degradation bounds are proved
