# Breakthrough Push: Closest Real-Novel Targets Inside Syncre

## Purpose

Identify the smallest set of “referee-hard” results that, once proved, make the Syncre program clearly additive rather than a repackaging of known degree/torus facts.

This document is intentionally scoped and operational: it names targets, required hypotheses, and what would count as completion.

---

## Target family BP-1: Quantitative robustness for SCM and SUD in the canonical nonlinear class

### Setup

On \(\Sigma=S^1\), take the SADC class (Pass C):
\[
F(s,\lambda)=\psi(s)+\frac{kP}{2\pi}\lambda+u(s,\lambda),
\qquad \Phi=F\bmod P.
\]

Assume:
- \(k\neq 0\)
- \(u(s,\cdot)\) is Lipschitz in \(\lambda\) with constant \(L\) uniformly in \(s\)
- the lift is time-monotone and bounded-rate along the chosen future direction

### Deliverable theorem

Prove an explicit lower bound of the form:
\[
\mathrm{SCM}(s)\ \ge\ c_0(k,P,L)
\]
and a stability bound:
\[
\mathrm{UD}(s)\ \le\ \mathrm{UD}(s_0) + \int_{s_0}^s \mathcal{E}(\sigma)\,d\sigma
\]
for a computable error term \(\mathcal{E}\) depending on derivative bounds of \(u\) and the flow.

Why it is novel in project terms:
- it connects a canonical deformation class to a quantitative anti-collapse statement and a uniformity-control statement

---

## Target family BP-2: Coupled-subtorus diagnostics and convergence

### Setup

Under SOL coupling (Pass F), the image lies in \(\mathrm{CST}_s\). Define the correct Haar target \(\mu_{\mathrm{H},\mathrm{CST}_s}\).

### Deliverable theorem

Prove that if the free coordinate phase is equidistributed under sampling (RNF nonresonance), then the induced coupled measure sequence \(\mu^{(n)}\) converges to \(\mu_{\mathrm{H},\mathrm{CST}}\) and SUD decays in a specified metric.

This turns “synodic phase equidistribution” into a measurable convergence statement in Syncre diagnostics.

---

## Target family BP-3: Preservation of SCM under Syncre transport dynamics

### Setup

Under STD (Pass H):
\[
\partial_s F + v\cdot\nabla F = \omega.
\]

Assume:
- \(v\) is Lipschitz with controlled compression/expansion along the flow
- \(\partial_\lambda F\) obeys a maximum principle or a differential inequality preventing gradient blow-up

### Deliverable theorem

Prove an inequality of the form:
\[
\mathrm{SCM}(s)\ \ge\ e^{-C(s-s_0)}\,\mathrm{SCM}(s_0)
\]
or another explicit degradation bound depending on flow distortion constants.

This makes the “fluid” aspect real: it translates dynamical control into quantitative coverage preservation.

---

## How these targets fit the pass plan

- Pass G provides the diagnostics and convergence notion
- Pass H provides dynamics structure
- Pass I consolidates the chain
- Pass J can become a paper draft once BP-1 and BP-3 exist in full theorem-proof form

BP-1 and BP-3 are the fastest “breakthrough-like” closures because they are internally controlled: they depend mostly on your chosen admissible class and on explicit inequalities, not on external astrophysical modelling.


## Status

- BP-1: initial theorems proved in `breakthrough/BREAKTHROUGH_PASS_1.md` (density sandwich, coverage profile lower bound, TV defect bound)
- BP-3: initial dynamics degradation bounds proved in `breakthrough/BREAKTHROUGH_PASS_1.md` under bounded shear and ∂λω=0

- Breakthrough Pass 2: forcing-robust BP-3 anti-collapse bounds and BP-2 coupled-subtorus equidistribution in the linear model (see `breakthrough/BREAKTHROUGH_PASS_2.md`).

- Breakthrough Pass 3: bounded-distortion uniformity and Fourier-mode defect diagnostic (see `breakthrough/BREAKTHROUGH_PASS_3.md`).

- Breakthrough Pass 4: true convergence mechanism via phase diffusion and exponential Fourier damping (see `breakthrough/BREAKTHROUGH_PASS_4.md`).

- Breakthrough Pass 5: unifies transport + diffusion into SADM and defines Stability Signature (SSG) with explicit decay conditions (see `breakthrough/BREAKTHROUGH_PASS_5.md`).

- Breakthrough Pass 6: diffusion-based convergence on coupled subtori under SOL, plus coupled stability signature (see `breakthrough/BREAKTHROUGH_PASS_6.md`).

- Breakthrough Pass 7: patch anti-collapse proofs by adding MBC (branch certification) so lower mass bounds are fully rigorous under checkable hypotheses (see `breakthrough/BREAKTHROUGH_PASS_7.md`).
