# 02 — CPL: $70\%/80\%/90\%/99\%$ Target Ladder

## 1. Basic Definitions

$$
P_q:\quad
\liminf_{T\to\infty}
\frac{N_0^s(T,2T)}{N(T,2T)}\ge q.
$$

What is being studied is a proportion certificate, not an RH progress bar.

---

## 2. Target Table

| Node | Status | Strongest Current Basis |
|---|---|---|
| $P_{2/3}$ | Unconditionally proved | Claude Theorem B |
| $P_{67.25}$ | Unconditionally proved | Claude Theorem D |
| $P_{68.185}$ | certificate ceiling statement | bandwidth-one configuration-wise class; not an "achieved proportion" |
| $P_{70}$ | Unproved | same-route rough support $\approx1.04$; conditional 4th-moment route can reach $13/18$ |
| $P_{80}$ | Unproved | same-route rough support $\approx1.26$ |
| $P_{90}$ | Unproved | same-route rough support $\approx1.70$ |
| $P_{99}$ | Unproved | paper does not provide support threshold; extrapolation is prohibited |
| $P_{100}^{density}$ | conditional mechanism endpoint | full PCC or sufficiently rich higher moments can yield density $1$, which still does not equal RH |

---

## 3. Horizontal-multiplicity Common Proportion Coordinates

The related Goldston–Suriajaya / GLSS route provides a comparable constant $C$: if for the corresponding horizontal / symmetric-diagonal count we obtain

$$
\mathcal H(T)\le(C+o(1))N(T),
\qquad 1\le C<2,
$$

then the simple-critical proportion can be lower-bounded by at least:

$$
2-C.
$$

Therefore, the CPL targets can be converted to:

| Target | Required $C$ |
|---|---:|
| $70\%$ | $C\le1.30$ |
| $80\%$ | $C\le1.20$ |
| $90\%$ | $C\le1.10$ |
| $99\%$ | $C\le1.01$ |
| density $100\%$ | $C\to1$ |

These coordinates are convenient for comparing different certificates without mistaking them for the same proof method.

---

## 4. The True Research Plane

Use at least two-dimensional coordinates:

$$
(\sigma,k),
$$

where:

- $\sigma$: controllable pair-correlation Fourier support / bandwidth;
- $k$: the order of the spectral moment that can be reliably obtained.

View the proportion as:

$$
q=q(\sigma,k,\mathcal I),
$$

where $\mathcal I$ represents additional arithmetic correlation information.

### Axis A: Support expansion

$$
1\to1.04\to1.26\to1.70\to\cdots
$$

The main QCI is off-diagonal prime correlations.

### Axis B: Moment expansion

$$
2\to4\to6\to\cdots
$$

The main QCI is higher additive correlations / Hardy--Littlewood-strength asymptotics.

### Axis C: Certificate enrichment

Without increasing support / moments, add zero-side block geometry, additional invariants, or stronger configuration constraints.

This axis is currently the most unknown, and it may also be where we later connect with topology/QCI research.

---

## 5. $P_{70}$: The First Real Breakthrough Point

### Route A — Support

To cross from the current unconditional:

$$
\sigma=1
$$

to approximately:

$$
\sigma\approx1.04.
$$

The number only increases by $0.04$, but the proof regime undergoes a qualitative change, because it begins to require currently unknown off-diagonal prime-pair information.

### Route B — Fourth Moment

If the $HL^*(4,\lambda)$ condition from the paper is obtained:

$$
P\ge\frac{13}{18}\approx72.22\%.
$$

Therefore, $P_{70}$ is not pure fantasy; there is a clear conditional bridge.

### Route C — Broader bandwidth-one certificate

The $68.185\%$ ceiling given in Remark 1.1 indicates: even if not restricted to the window optimisation of §7.1, some broader bandwidth-one configuration-wise class still gets stuck before $70\%$.

So if Route C is to succeed, one must first clarify the **exact set of assumptions** of this ceiling, and find information or invariants that do not belong to it.

---

## 6. $P_{80}$ and $P_{90}$

The paper only states the support values as roughly:

$$
\sigma_{80}\approx1.26,
$$

$$
\sigma_{90}\approx1.70.
$$

In the future, we cannot just cite these two numbers; we need to reconstruct which extremal law they come from, establishing a recalculable:

$$
q\mapsto\sigma_{\min}(q).
$$

---

## 7. $P_{99}$

The strictest research discipline at present:

$$
\boxed{\text{Do not guess }\sigma_{99}.}
$$

It is known that we can define:

$$
P_{99}:q=0.99,
$$

which in $C$-coordinates is equivalent to:

$$
C\le1.01.
$$

The true first question should be:

$$
\boxed{
\inf\{\mathcal I:\ \mathcal I\text{ is sufficient to deduce }P_{99}\}
}
$$

rather than doing curve fitting from $1.04, 1.26, 1.70$.

---

## 8. QCI Interpretation

CPL is a clean stress test for the Quantitative Closure Interface:

$$
\text{spectral structure}
\to
\text{prime-side representation}
\to
\text{support/moments}
\to
\text{uniform asymptotics}
\to
\text{proportion certificate}.
$$

Every time $q$ is increased, one should record:

1. Which information domain is insufficient;
2. Whether what is lacking is support, moment, configuration constraint, or arithmetic input;
3. Whether the new information is unconditionally valid;
4. Whether the error is uniform;
5. What the sharpness domain of the certificate is.