> 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 3
## Operator Audit, Closed-Loop Canonicalization, and the Historical Obstruction Theorem
### Operator Audit, Closed-Loop Canonicalization, and the Historical Obstruction Theorem

**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 1–2  
**Status:** Phase 3 formalization + finite computational audit; not a completed proof of Legendre's conjecture

---

# 0. The Conclusions of This Phase First

Phase 3 obtains two types of result.

The first type is **negative results / corrections**:

> The Phase 2 approach of integrating $\varepsilon_{\mathrm{full}}$ alone into an infinite open-loop rollout is not equivalent to the original prime fitter. Although this simplified version remains monotone in finite tests and captures an approximately correct growth shape, it produces an overall scale shift of about $0.7904$ and therefore cannot serve as the canonical prime-fitting operator.

The second type is **exact positive results**:

> Once an integer candidate $q$ is placed in the $n$th continuous square cell and satisfies $q\equiv\pm1\pmod6$, if $q$ is not prime, its nontrivial divisibility certificate must come entirely from the ‘historical base space’ at scales $\le n$.

This gives primality within a square cell a fully local–global criterion.

---

# 1. Source Audit: The Original v2.0 Is a Closed-Loop Operator

Appendix A of the stored $\varepsilon_{\mathrm{full}}$ paper defines:

$$
\widehat A_{N+1}
=
A_N
\left(
\frac{N+1}{N}
\right)^{1+\varepsilon_{\mathrm{full}}(N)},
$$

and:

$$
\widehat p_{N+1}
=
(N+1)\widehat A_{N+1}
-
NA_N.
$$

In the original algorithm, the input $A_N$ is the actual cumulative mean of the currently known prime history:

$$
A_N
=
\frac1N
\sum_{k=1}^N p_k.
$$

Thus, the original v2.0 is a:

$$
\boxed{
\text{historical state}
\longrightarrow
\text{one-step fit}
}
$$

closed-loop operator, not an open-loop generator that never receives feedback after starting at $N=2$.

---

# 2. The Phase 2 Open-Loop Simplification

Phase 2 defined:

$$
A_2^{\mathrm{OL}}
=
\frac52,
$$

$$
A_{N+1}^{\mathrm{OL}}
=
A_N^{\mathrm{OL}}
\left(
\frac{N+1}{N}
\right)^{1+\varepsilon_{\mathrm{full}}(N)},
$$

$$
P_{N+1}^{\mathrm{OL}}
=
(N+1)A_{N+1}^{\mathrm{OL}}
-
NA_N^{\mathrm{OL}}.
$$

This operator is formally self-consistent, but it changes the original closed-loop adaptive operator into a pure integrator.

Phase 3 must check whether this modification preserves prime-fitting identity.

---

# 3. The Exact Condition for Open-Loop Monotonicity

Let:

$$
r_N
=
\left(
\frac{N+1}{N}
\right)^{1+\varepsilon_N}.
$$

Then:

$$
A_{N+1}=A_Nr_N.
$$

and:

$$
P_{N+1}
=
A_N
\left[
(N+1)r_N-N
\right].
$$

On the other hand:

$$
P_N
=
A_N
\left[
N-\frac{N-1}{r_{N-1}}
\right].
$$

Therefore:

$$
\boxed{
P_{N+1}-P_N
=
A_N
\left[
(N+1)r_N
-2N
+
\frac{N-1}{r_{N-1}}
\right].
}
$$

Since $A_N>0$, the necessary and sufficient condition for strict open-loop monotonicity is:

$$
\boxed{
(N+1)r_N
+
\frac{N-1}{r_{N-1}}
>
2N.
}
$$

This compresses the monotonicity question entirely into the relationship between $\varepsilon_N$ at adjacent scales.

---

# 4. Open-Loop Unboundedness

The large-scale branch is:

$$
\varepsilon_{\mathrm{asy}}(N)
=
\frac1L
+
\frac{1-\Lambda}{L^2}
+
\frac{\Lambda^2-\Lambda-1}{2L^3},
$$

where:

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

For sufficiently large $N$:

$$
\varepsilon_{\mathrm{asy}}(N)>0.
$$

Thus $1+\varepsilon_{\mathrm{asy}}(N)$ stays positive, and the multiplicative recurrence does not converge to a finite constant. Hence:

$$
A_N\to\infty,
$$

and consequently:

$$
P_N\to\infty.
$$

The main open-loop problem is therefore not unboundedness, but fidelity.

---

# 5. The Finite Machine-Audit Specification

This round reconstructs the implementation from the stored Appendix A:

- $\varepsilon_{\mathrm{table}}(2\ldots999)$: following the original sieve and finite-difference rules;
- $\varepsilon_{\mathrm{mid}}$: using the stored coefficients;
- Actual primes tested: the first $200000$;
- Largest actual prime:

$$
p_{200000}=2750159.
$$

Test:

1. The Phase 2 open loop;
2. The original v2.0 one-step closed-loop raw fit;
3. The original $6k\pm1$ snapping;
4. Square-cell fidelity;
5. Candidate primality.

Finite computation serves only falsification / diagnostics and is not promoted to a proof over an infinite domain.

---

# 6. Open-Loop Audit Results

Within:

$$
2\le N\le200000
$$

this finite test range:

$$
P_N^{\mathrm{OL}}
$$

are all strictly increasing.

The smallest observed increment is approximately:

$$
1.345866751.
$$

However, the position ratios are:

| $N$ | $P_N^{\mathrm{OL}}/p_N$ |
|---:|---:|
| $1000$ | 0.790063568 |
| $10000$ | 0.790435160 |
| $100000$ | 0.790450817 |
| $200000$ | 0.790429877 |

It can be seen that they quickly stabilize at a scale of approximately:

$$
\boxed{
0.7904
}
$$

this value.

Thus, it captures a similar growth shape but loses the correct integration constant / scale calibration.

Therefore:

$$
\boxed{
P^{\mathrm{OL}}
\text{ is not the canonical prime-fitting operator adopted subsequently in this paper.}
}
$$

---

# 7. Why the Open Loop Fails

$\varepsilon_{\mathrm{full}}(N)$ originally describes:

$$
\log A_N
$$

the finite-scale local slope of this quantity.

Integrating finite-scale slopes directly cell by cell is equivalent to assuming:

$$
\varepsilon_{\mathrm{finite\ difference}}
=
\varepsilon_{\mathrm{exact\ local\ generator}}.
$$

The two are generally unequal.

The open loop therefore produces:

$$
\text{shape recovery}
+
\text{normalization drift}.
$$

This is the fundamental problem with the Phase 2 simplification.

---

# 8. Audit of the Original One-Step Closed Loop

Return to the stored original algorithm:

$$
A_N
=
\frac1N\sum_{k=1}^Np_k,
$$

$$
\widehat A_{N+1}
=
A_N
\left(
\frac{N+1}{N}
\right)^{1+\varepsilon_{\mathrm{full}}(N)},
$$

$$
\widehat p_{N+1}
=
(N+1)\widehat A_{N+1}
-
NA_N.
$$

For the predictions corresponding to $1000\le N\le200000$:

| Metric | Result |
|---|---:|
| Raw prediction and actual $p_N$ lie in the same square cell | 91.0674% |
| Same square cell after $6k\pm1$ snapping | 91.0684% |
| Proportion with $\delta_\square(N)<1$ | 82.2428% |
| Snapped candidate itself is prime | 21.7913% |
| Snapped candidate is exactly $p_N$ | 0.5844% |
| Median raw absolute error | 153.512 |
| raw absolute error P99 | 673.290 |

Thus, the original closed-loop fitting is indeed quite accurate in position, but:

$$
\text{Very accurate position}
\not\Rightarrow
\text{The output itself is prime}.
$$

Merely:

$$
\text{geometric fit}
+
6k\pm1
$$

cannot carry a primality proof.

---

# 9. Phase 2 Square-Cell Fidelity Is Sufficient, but Not the Shortest Route

Phase 2 defines:

$$
\widehat P
\equiv_\square
P
$$

if:

$$
c_\square(\widehat P(N))
=
c_\square(p_N)
$$

holds for every $N$.

It remains a valid sufficient route.

However, it requires aligning the fitted sequence with the specific $N$th actual prime.

Legendre's conjecture actually needs only:

$$
\text{At least one integer provable to be prime in every square cell}.
$$

The next step therefore adopts a fixed-point certificate.

---

# 10. The $M6^*$ Exact Prime Fixed-Point Operator

Define:

$$
M6^*
=
\{m>1:
m\equiv\pm1\pmod6\}.
$$

For:

$$
q\in M6^*,
$$

define:

$$
\boxed{
T_{\mathrm{top}}(q)
=
\min
\{d\in M6^*:d\mid q\}.
}
$$

Then:

$$
\boxed{
T_{\mathrm{top}}(q)=q
\iff
q\in\mathbb P\cap M6^*.
}
$$

Thus $T_{\mathrm{top}}$ can serve as a noncircular prime-certificate operator.

---

# 11. The Core New Theorem of Phase 3: The Historical Obstruction Theorem

## Theorem 11.1

Let:

$$
n\ge2,
$$

and let the integer $q$ satisfy:

$$
n^2<q<(n+1)^2
$$

and:

$$
q\in M6^*.
$$

Then necessarily:

$$
\boxed{
T_{\mathrm{top}}(q)=q
}
$$

or:

$$
\boxed{
T_{\mathrm{top}}(q)\le n.
}
$$

## Proof

If $T_{\mathrm{top}}(q)=q$, the first case holds.

Otherwise, $q$ has a proper divisor. Let:

$$
d=T_{\mathrm{top}}(q).
$$

Then $d$ is a nontrivial factor, so:

$$
d\le\sqrt q.
$$

From:

$$
q<(n+1)^2
$$

we obtain:

$$
\sqrt q<n+1.
$$

Hence:

$$
d<n+1.
$$

Since $d,n$ are integers:

$$
\boxed{
d\le n.
}
$$

This completes the proof.

---

# 12. Base-Space Meaning

This theorem states:

$$
\boxed{
\text{local composite failure at scale }n
\Longrightarrow
\text{past obstruction at scale }\le n.
}
$$

That is, a composite failure in the $n$th square cell does not require a ‘future scale’ for its explanation.

All its divisibility certificates already exist in the earlier global historical base space.

---

# 13. The Historical Obstruction Set

Define:

$$
\mathcal D_n
=
\{
d\in\mathbb Z:
5\le d\le n,\,
d\equiv\pm1\pmod6
\}.
$$

For:

$$
q\in(n^2,(n+1)^2)\cap M6^*,
$$

define:

$$
\boxed{
\mathcal O_n(q)
=
\{
d\in\mathcal D_n:
d\mid q
\}.
}
$$

By the historical obstruction theorem:

$$
\boxed{
q\text{ is prime}
\iff
\mathcal O_n(q)=\varnothing.
}
$$

---

# 14. The Historical Obstruction Product

Define:

$$
\boxed{
H_n
=
\prod_{\substack{5\le d\le n\\d\equiv\pm1\pmod6}}
d.
}
$$

The empty product is defined as:

$$
1.
$$

---

# 15. The Historical Coprimality Criterion

## Theorem 15.1

If:

$$
q\in(n^2,(n+1)^2)\cap M6^*,
$$

then:

$$
\boxed{
q\in\mathbb P
\iff
\gcd(q,H_n)=1.
}
$$

## Proof

If $q$ is prime, since $q>n$, it cannot be divisible by any $1<d\le n$. Thus:

$$
\gcd(q,H_n)=1.
$$

Conversely, if $q$ is composite, Theorem 11.1 gives:

$$
T_{\mathrm{top}}(q)\le n.
$$

and:

$$
T_{\mathrm{top}}(q)\in M6^*.
$$

Hence:

$$
T_{\mathrm{top}}(q)\mid H_n
$$

and:

$$
T_{\mathrm{top}}(q)\mid q.
$$

therefore:

$$
\gcd(q,H_n)>1.
$$

This completes the proof.

---

# 16. The Local Update Law of the Historical Product

$H_n$ has the recurrence:

$$
H_2=1,
$$

and:

$$
\boxed{
H_{n+1}
=
H_n
\cdot
\begin{cases}
n+1,
&
n+1\equiv\pm1\pmod6,
\\[4pt]
1,
&
\text{otherwise}.
\end{cases}
}
$$

Historical obstruction itself is therefore also:

$$
\text{global state}
\longrightarrow
\text{local update}.
$$

---

# 17. Canonical Square-Cell Projection

For each:

$$
n\ge2,
$$

first let the geometry / skewness / other fitting models produce a real-valued center:

$$
x_n.
$$

Then define a deterministic cell projection:

$$
J_n(x_n)
\in
A_n\cap M6^*.
$$

For example:

$$
J_n(x)
=
\operatorname*{argmin}_{m\in A_n\cap M6^*}
|m-x|,
$$

and fix a deterministic tie-break.

Define:

$$
\boxed{
F_\square(n)
=
J_n(x_n).
}
$$

In this way:

$$
\boxed{
F_\square(n)
\in
(n^2,(n+1)^2)\cap M6^*
}
$$

holds automatically by construction.

---

# 18. The Actual Prime-Fitting Operator Criterion

There is then no need to compare with the $N$th actual prime.

We only need to prove:

$$
\boxed{
T_{\mathrm{top}}
\circ
F_\square
=
F_\square.
}
$$

That is:

$$
\boxed{
T_{\mathrm{top}}(F_\square(n))
=
F_\square(n)
\quad
\forall n\ge2.
}
$$

This is the fixed-point version of ‘the prime-fitting operator is the prime-fitting operator’.

---

# 19. The Equivalent Historical Coprimality Version

By Theorem 15.1:

$$
T_{\mathrm{top}}(F_\square(n))
=
F_\square(n)
$$

is equivalent to:

$$
\boxed{
\gcd
\left(
F_\square(n),
H_n
\right)
=
1.
}
$$

The main proof obligation therefore becomes:

$$
\boxed{
\forall n\ge2,
\qquad
\gcd
\left(
F_\square(n),
H_n
\right)
=
1.
}
$$

---

# 20. The Main Sufficient Theorem of Phase 3

## Theorem 20.1

If there exists:

$$
F_\square:
\mathbb Z_{\ge2}
\to
\mathbb Z
$$

such that every $n\ge2$ satisfies:

$$
F_\square(n)
\in
(n^2,(n+1)^2)\cap M6^*
$$

and:

$$
\gcd
\left(
F_\square(n),
H_n
\right)=1,
$$

Legendre's conjecture holds.

## Proof

By Theorem 15.1, $F_\square(n)$ is prime.

Also:

$$
n^2
<
F_\square(n)
<
(n+1)^2.
$$

Thus, for every $n\ge2$, the square cell contains a prime.

When $n=1$:

$$
2,3\in(1,4).
$$

Therefore:

$$
\forall n\ge1,
\qquad
\exists p\in\mathbb P:
n^2<p<(n+1)^2.
$$

This completes the proof.

---

# 21. Fixed-Point Residual

Define:

$$
\boxed{
R_\square(n)
=
F_\square(n)
-
T_{\mathrm{top}}(F_\square(n)).
}
$$

Since:

$$
T_{\mathrm{top}}(q)\le q,
$$

we have:

$$
R_\square(n)\ge0.
$$

and:

$$
\boxed{
R_\square(n)=0
\iff
F_\square(n)\text{ is prime}.
}
$$

The entire program can therefore be compressed to:

$$
\boxed{
R_\square(n)\equiv0.
}
$$

---

# 22. Local–Global Reformulation

The global historical state:

$$
H_n
$$

accumulates multiplicative obstructions at all scales $\le n$.

The local fitted output:

$$
F_\square(n)
$$

lies in:

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

The two actually meet at:

$$
\boxed{
\gcd(F_\square(n),H_n).
}
$$

The problem is therefore rewritten as:

$$
\boxed{
\text{global historical obstruction}
\cap
\text{local fitted candidate}.
}
$$

---

# 23. The Core Difficulty Has Not Disappeared

We must explicitly retain that:

$$
\gcd(F_\square(n),H_n)=1
$$

for every $n$ remains a nontrivial global proof obligation.

This paper has not ‘defined away’ Legendre's conjecture.

It only transforms:

$$
\forall n\,\exists p
$$

into:

$$
\forall n:
\quad
\gcd(F_\square(n),H_n)=1.
$$

This form is worth studying because:

1. $F_\square$ can be constructed from the existing fitting theory;
2. $H_n$ has a simple local update law;
3. Composite failure is necessarily witnessed at an earlier scale;
4. The problem becomes explicit operator–divisibility compatibility.

---

# 24. The Entry Point to Phase 4: The Prime-Fit Obstruction Field

The next phase directly studies:

$$
F_\square(n)\bmod d
$$

for:

$$
d\in\mathcal D_n.
$$

Define:

$$
\boxed{
\mathfrak R(n,d)
=
F_\square(n)\bmod d.
}
$$

Its domain is:

$$
\boxed{
\mathcal O
=
\{(n,d):
n\ge2,\,
d\in M6^*,\,
5\le d\le n\}.
}
$$

From:

$$
\gcd(F_\square(n),H_n)=1
$$

we see that the final target is equivalent to:

$$
\boxed{
\mathfrak R(n,d)\neq0
\quad
\forall(n,d)\in\mathcal O.
}
$$

If this nonzero invariant can be deduced from the global structure of the fitting operator, Legendre's conjecture closes.

---

# 25. Phase 3 Status

```text
Legendre Base-Space Program
Phase 3 — Operator Audit and Historical Obstruction

epsilon_full source audit:                 COMPLETE
Phase-2 open-loop rollout:                 FALSIFIED AS CANONICAL FITTER
Open-loop monotonic criterion:             DERIVED
Open-loop finite monotonicity:             PASS to N=200000
Open-loop normalization fidelity:          FAIL
Original closed-loop operator:             RESTORED
Square-cell finite fidelity audit:         COMPLETE
6k±1 alone as prime certificate:           FAIL
T_top exact prime certificate:             ADOPTED
Past-factor obstruction theorem:           PROVED
Historical obstruction set:                DEFINED
Historical product H_n:                    DEFINED
Local coprimality prime criterion:          PROVED
Fixed-point fitting criterion:             PROVED EQUIVALENT
Legendre sufficient theorem:               PROVED
Remaining global obligation:               gcd(F_square(n), H_n)=1
Legendre proof status:                     OPEN
Next phase:                                OBSTRUCTION RESIDUE FIELD
```