> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# Legendre Base-Space Program Phase 5
## Canonical Fit-Ordered Fiber, Fixed-Point Depth, and Low/High Obstruction Decomposition
### Canonical Fit-Ordered Fiber, Fixed-Point Depth, and Low/High Obstruction Decomposition

**Version:** v0.1  
**Date:** 2026-09-30  
**Author:** Neo.K × AI collaboration  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Research line:** Legendre Base-Space Prime-Fitting Program  
**Prerequisites:** Phase 1–4  
**Status:** Phase 5 formalization + finite machine audit; not a completed proof of Legendre's conjecture

---

# 0. Conclusions of This Phase

Phase 5 formally addresses ‘how exactly the prime-fitting operator selects candidates in each square cell’.

This round obtains three key conclusions:

1. The old V3 prime-position formula can canonically generate a fit center in every square cell, and asymptotically this center is only $O(\log n)$ from the left square boundary;
2. Merely selecting ‘the $6k\pm1$ nearest to the fit center’ is not a prime-valued section; the finite-test rank-1 hit rate is only about **17.1477%**;
3. The fixed-point depth under V3 ordering has almost the same distribution as simply scanning $M6^*$ from the left square boundary, so pure position fitting has shown no new obstruction-avoidance signal.

This program therefore upgrades the ‘prime-fitting base space’ from a single point to a:

$$
\boxed{
\text{fit-ordered local fiber}
}
$$

and adds a low-order historical-obstruction preconditioner that does not complete the entire primality test.

---

# 1. Source Audit

The existing research can be divided into three layers:

- **V3 / prime geometry**: provides an explicit prime-position approximation formula and a geometric positioning center;
- **$\varepsilon_{\mathrm{full}}$**: provides scale correction, but Phase 3 has confirmed that it cannot directly serve as an infinite open-loop generator;
- **TCGQT Prime Triplex Lab**: A is the prime residual, B generated composites, and C the $6k\pm1$ lanes; however, the lab uses a sieve / factorization to display the actual residual, so it is a verifier / visualization, not an independent generator.

Thus, Phase 5 adopts:

$$
\boxed{
\text{V3 as generator center}
+
T_{\mathrm{top}}\text{ as verifier}.
}
$$

---

# 2. The V3 Global Position Model

Let:

$$
L=\ln N,
\qquad
\Lambda=\ln\ln N.
$$

Define:

$$
\boxed{
V(N)
=
N\left[
L+\Lambda-1
+
\frac{\Lambda-2}{L}
-
\frac{\Lambda^2-6\Lambda+11}{2L^2}
+
\frac{\Lambda^3-9\Lambda^2+26\Lambda-24}{6L^3}
\right].
}
$$

It can be computed without knowing the $N$th actual prime.

---

# 3. The Square-Boundary Crossing Index

Define:

$$
\boxed{
\nu_3(n)
=
\min\{N:V(N)>n^2\}.
}
$$

and:

$$
\boxed{
x_3(n)
=
V(\nu_3(n)).
}
$$

This definition does not use $\pi(n^2)$ or the next actual prime.

---

# 4. Left-Boundary Localization of the Fit Center

By the minimality of $\nu_3(n)$:

$$
V(\nu_3(n)-1)
\le
n^2
<
V(\nu_3(n)).
$$

Thus:

$$
0
<
x_3(n)-n^2
\le
V(\nu_3(n))-V(\nu_3(n)-1).
$$

The asymptotic form of V3 is:

$$
V(N)
=
N\left(
\ln N+\ln\ln N-1
+
O\left(
\frac{\ln\ln N}{\ln N}
\right)
\right),
$$

Hence:

$$
V(N+1)-V(N)
=
O(\ln N).
$$

Also, from:

$$
V(\nu_3(n)-1)
=
\Theta(\nu_3(n)\ln\nu_3(n))
\le n^2
$$

we obtain:

$$
\ln\nu_3(n)=O(\ln n).
$$

Therefore:

$$
\boxed{
x_3(n)-n^2
=
O(\ln n).
}
$$

Thus, for sufficiently large $n$:

$$
n^2<x_3(n)<(n+1)^2.
$$

---

# 5. The Finite Center Audit

Within:

$$
10\le n\le20000
$$

this range, all V3 centers lie inside their square cells.

Finite offsets $x_3(n)-n^2$:

- Minimum: 0.000225
- Median: 8.783453
- P90: 16.188990
- Maximum: 19.764347

---

# 6. Canonical Fit-Ordered Local Fiber

Let:

$$
C_n^{(6)}
=
(n^2,(n+1)^2)\cap M6^*.
$$

Define the fitting score:

$$
\boxed{
S_n(q)=|q-x_3(n)|.
}
$$

Order by increasing score, breaking ties by increasing $q$, to obtain:

$$
\boxed{
J_n(1),J_n(2),\dots,J_n(m_n).
}
$$

This entire ordered fiber, not the single point $J_n(1)$, is the current canonical fit object.

---

# 7. Failure of the Rank-1 Section

Let:

$$
F_3^{(1)}(n)=J_n(1).
$$

In finite experiments:

$$
\boxed{
\Pr[
T_{\mathrm{top}}(F_3^{(1)}(n))
=
F_3^{(1)}(n)
]
\approx
17.1477\%.
}
$$

Thus:

$$
\boxed{
J_n(1)
\text{ is not a global prime-valued section.}
}
$$

---

# 8. Fixed-Point Depth

Define:

$$
\boxed{
\tau_3(n)
=
\min
\left\{
r:
T_{\mathrm{top}}(J_n(r))
=
J_n(r)
\right\}.
}
$$

Legendre's conjecture is equivalent to:

$$
\boxed{
\tau_3(n)<\infty
\quad
\forall n.
}
$$

because $J_n$ enumerates all $M6^*$ candidates in the cell.

---

# 9. Baseline Fixed-Point Depth Audit

Finite range:

$$
10\le n\le20000.
$$

| Metric | $\tau_3$ |
|---|---:|
| rank-1 fixed-point rate | 17.1477% |
| median | 4 |
| P90 | 12 |
| P99 | 23 |
| P99.9 | 34 |
| max | 55 |

One of the worst cases is:

$$
n=14908,
\qquad
\tau_3(n)=55.
$$

Its fit center is:

$$
x_3(n)\approx222248468.330759,
$$

The first fixed point is:

$$
222248627.
$$

---

# 10. Signal–Carrier Audit: V3 vs Left Edge

Define another baseline that does not use V3 at all:

> Starting from $n^2+1$, order the $M6^*$ integers in the cell increasingly until the first fixed point.

Denote its depth by:

$$
\tau_L(n).
$$

Finite audit:

| Metric | V3 fit order | Left-edge order |
|---|---:|---:|
| rank-1 rate | 17.1477% | 17.0377% |
| median | 4 | 4 |
| P90 | 12 | 12 |
| P99 | 23 | 23 |
| P99.9 | 34 | 34 |
| max | 55 | 55 |

And the proportion for which:

$$
\tau_3(n)=\tau_L(n)
$$

holds is:

$$
52.4686\%.
$$

Mean rank difference:

$$
0.010905.
$$

There is therefore currently no evidence that V3's **position score itself** supplies additional obstruction avoidance.

---

# 11. Why?

Because:

$$
x_3(n)-n^2=O(\ln n).
$$

and adjacent $M6^*$ candidates are separated by only $2$ or $4$.

V3 fit ordering therefore still essentially performs local ordering near the left square boundary.

This proves that ‘position fitting’ and ‘factor-obstruction avoidance’ are two different kinds of information.

---

# 12. The Three-Layer Structure of Prime Fit

Thus, the most reasonable current Prime Fit should be written as:

$$
\boxed{
\text{Prime Fit}
=
\text{Position Fit}
+
\text{Low-Obstruction Avoidance}
+
\text{High-Obstruction Closure}.
}
$$

---

# 13. The Low-Order Obstruction Cutoff

Take a prespecified:

$$
z(n)
$$

such that:

$$
5\le z(n)<n
$$

holds for sufficiently large $n$.

Define:

$$
\mathfrak A_{\le z}(n)
=
\{a\in\mathfrak A:a\le z(n)\},
$$

and:

$$
\boxed{
W_z(n)
=
6\prod_{a\in\mathfrak A_{\le z}(n)}a.
}
$$

---

# 14. Preconditioned Fit Fiber

Define:

$$
\boxed{
C_n^{(z)}
=
\left\{
q\in(n^2,(n+1)^2):
\gcd(q,W_z(n))=1
\right\}.
}
$$

Then order by:

$$
|q-x_3(n)|
$$

to obtain:

$$
J_n^{(z)}(1),J_n^{(z)}(2),\dots
$$

This generator removes only low-order factors and does not complete a primality test.

---

# 15. Low / High Obstruction Decomposition

High-order atoms:

$$
\mathfrak A_{>z}(n)
=
\{a\in\mathfrak A:z(n)<a\le n\}.
$$

For $q\in C_n^{(z)}$, low-order obstructions have been removed by construction.

Thus, $q$ is prime if and only if:

$$
\boxed{
q\not\equiv0\pmod a
\quad
\forall a\in\mathfrak A_{>z}(n).
}
$$

---

# 16. Kernel Factorization

Let:

$$
Q_n
=
6\prod_{a\in\mathfrak A_n}a.
$$

Then:

$$
\boxed{
Q_n
=
W_z(n)R_z(n),
}
$$

where:

$$
R_z(n)
=
\prod_{a\in\mathfrak A_{>z}(n)}a
$$

and:

$$
\gcd(W_z(n),R_z(n))=1.
$$

Therefore:

$$
\gcd(q,Q_n)=1
$$

is equivalent to:

$$
\boxed{
\gcd(q,W_z(n))=1
\quad\land\quad
\gcd(q,R_z(n))=1.
}
$$

---

# 17. Preconditioned Fixed-Point Depth

Define:

$$
\boxed{
\tau_{3,z}(n)
=
\min
\left\{
r:
T_{\mathrm{top}}(J_n^{(z)}(r))
=
J_n^{(z)}(r)
\right\}.
}
$$

This round performs only an engineering experiment:

$$
\boxed{
z(n)
=
\left\lfloor
(\ln(n+1))^2
\right\rfloor.
}
$$

This is not a theorem, nor is optimality claimed.

---

# 18. Preconditioned Finite Audit

Within:

$$
10\le n\le20000
$$

the results are:

| Metric | Original V3 fiber | $z=(\ln(n+1))^2$ |
|---|---:|---:|
| rank-1 fixed-point rate | 17.1477% | 45.9357% |
| median | 4 | 2 |
| P90 | 12 | 4 |
| P99 | 23 | 8 |
| P99.9 | 34 | 12 |
| max | 55 | 17 |

One of the worst preconditioned cases is:

$$
n=19079,
\qquad
\tau_{3,z}(n)=17,
$$

Its cutoff is:

$$
z(n)=97.
$$

The source of finite improvement is clear: it comes from explicitly removing low-order composite stripes, not from V3 itself acquiring primality.

---

# 19. The Generator / Verifier Boundary

## The Generator May Use

- $n,n^2$;
- V3 / position formulas;
- A fixed cutoff $z(n)$;
- Residue exclusion for $a\le z(n)$.

## The Generator Must Not Use

- Actual primality;
- The next actual prime;
- A $\pi(n^2)$ oracle;
- A complete scan over all $a\le n$;
- The complete $T_{\mathrm{top}}$ result.

## The Verifier May Use

$$
T_{\mathrm{top}}
$$

or the complete obstruction field.

---

# 20. Fit-Depth Envelope

What is genuinely worth proving is the existence of:

$$
\boxed{
\tau_{3,z}(n)\le R(n)
}
$$

and that the first $R(n)$ candidates must still lie inside the square cell.

Since adjacent $M6^*$ candidates are at most $4$ apart, candidate depth $R$ has physical width $O(R)$.

Also:

$$
x_3(n)-n^2=O(\ln n).
$$

Thus, if we can prove:

$$
\boxed{
\tau_{3,z}(n)=o(n),
}
$$

the fixed point must still lie, for sufficiently large $n$, in:

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

Hence:

$$
\boxed{
\text{sublinear fit-depth}
\Longrightarrow
\text{Legendre}.
}
$$

This global sublinear depth has not currently been proved.

---

# 21. Canonical Prime-Fitting Base Space v1

The cleanest current prime-fitting base space is:

$$
\boxed{
\mathcal B_{\mathrm{PF}}^{(z)}
=
\bigsqcup_{n\ge2}
\left(
C_n^{(z)},
S_n
\right),
}
$$

where:

$$
S_n(q)=|q-x_3(n)|.
$$

Each local fiber simultaneously carries:

- Square-cell geometry;
- The V3 fit center;
- The $M6^*$ lane restriction;
- Low-obstruction preconditioning;
- The high-obstruction residue field;
- A fixed-point verifier.

---

# 22. The Most Important Negative Result of This Round

Without adding obstruction structure, V3 fit ordering is almost homogeneous with the simplest left-edge M6 ordering.

Thus:

$$
\boxed{
\text{Prime-Fitting Base Space}
\neq
\text{Position-Fitting Base Space}.
}
$$

The genuinely additional information must be:

$$
\boxed{
\text{obstruction geometry}.
}
$$

---

# 23. The Entry Point to Phase 6

The next phase directly studies high-order obstructions:

$$
\boxed{
z(n)<a\le n.
}
$$

The question becomes:

> After the generator has excluded low-order factors, can high-order atoms still completely cover the initial segment of the preconditioned fit fiber?

We should study:

1. The maximum number of hits from each high-order atom on the initial segment of the fit fiber;
2. CRT intersections of different high-order atoms;
3. Why the union bound for high-order obstructions fails / when it can be corrected;
4. Whether the square anchor can provide high-order non-alignment;
5. Whether $\tau_{3,z}(n)=o(n)$ can be proved.

---

# 24. Phase 5 Status

```text
Legendre Base-Space Program
Phase 5 — Canonical Fit-Ordered Fiber

V3 global fit center:                      DEFINED
Square crossing index from V3:             DEFINED
V3 center left-edge localization:          DERIVED
Finite center-in-cell audit:               PASS
Rank-1 prime section:                      FALSIFIED
Fit-ordered local fiber:                   DEFINED
Fixed-point depth tau_3:                   DEFINED
V3 vs left-edge signal/carrier audit:      NO DISTINCT ADVANTAGE FOUND
Low-obstruction cutoff z(n):               DEFINED PARAMETRICALLY
Preconditioned fit fiber:                  DEFINED
Low/high obstruction factorization:        PROVED
Experimental log^2 cutoff audit:           IMPROVES DEPTH FINITELY
Generator/verifier boundary:               FIXED
Sublinear fit-depth sufficient route:      IDENTIFIED
Global sublinear depth theorem:             OPEN
Legendre proof status:                     OPEN
Next phase:                                HIGH-OBSTRUCTION CLOSURE
```