# Legendre Base-Space Program — Errata for Phase 12–14
## Literature-boundary correction issued in Phase 15

**日期：** 2026-10-02  
**狀態：** CORRECTION / supersedes the affected frontier claims in Phase 12–14

---

## E1. Incorrect prior statement

Phase 12–14 used variants of the statement:

> the unconditional “every-square” almost-prime frontier is $P_3$, and $P_2$ between every sufficiently large pair of consecutive squares is not known.

This is **incorrect**.

---

## E2. Correct literature boundary

Classical short-interval results are much stronger for ordinary $P_2$ numbers.

Iwaniec–Laborde (1981) prove that every sufficiently large interval of length:

$$
x^{0.45}
$$

contains a $P_2$ number.

Wu (2010) improves the all-$x$ exponent to:

$$
\boxed{
\frac{101}{232}
\approx
0.435345.
}
$$

Therefore every sufficiently large consecutive-square interval:

$$
(n^2,(n+1)^2)
$$

already contains a $P_2$, because:

$$
(n^2)^{101/232}
=
n^{101/116}
=
o(n),
$$

while the square-cell width is:

$$
2n+1.
$$

---

## E3. What remains valid in Phase 12–14

The exact algebraic statements remain valid:

- cube-root RootRough survivor $\Rightarrow$ prime or semiprime;
- least-factor semiprime row decomposition;
- prime-cofactor switching;
- exact $P_2$ boundary identity;
- constant ledger;
- square-anchor Möbius phase;
- symmetric-center decomposition.

What changes is their **novelty / frontier interpretation**.

---

## E4. Correct positioning of the cube-root boundary

Our set:

$$
\mathcal S_3(n)
=
\{q\in(n^2,(n+1)^2):P^-(q)>X^{1/3}\}
$$

is not the set of all $P_2$ numbers.

A generic $P_2$ may be:

$$
q=pr
$$

with:

$$
p\ll X^{1/3}.
$$

So proving:

$$
\mathcal S_3(n)>0
$$

is a **rough-$P_2$ / prime-or-balanced-semiprime statement**, stronger and structurally different from the classical existence of an arbitrary $P_2$.

---

## E5. Campbell 2026 interpretation

Campbell’s every-square $P_3$ theorem is an **explicit all-$n$ theorem** with an explicit computational / sieve split.

It should not be described as the strongest asymptotic multiplicative-order result in generic short intervals, because classical all-sufficiently-large-$x$ $P_2$ short-interval theorems are stronger in multiplicative order and interval exponent.

---

## E6. Correct open target

The present program’s genuine target remains:

$$
\boxed{
\text{prime occupancy / parity closure}
}
$$

inside every square cell.

The cube-root rough framework is retained because it creates the exact two-state boundary:

$$
\boxed{
\text{prime}\sqcup\text{semiprime},
}
$$

not because ordinary $P_2$ existence is open.