---

## Introduction

Helfgott’s 2014 proof of the ternary Goldbach conjecture (arXiv:1312.7748v2) is an explicit analytic argument: both major-arc and minor-arc contributions are bounded with fully numerical constants, and the final closure is obtained by comparing a fixed major-arc lower bound against an explicit minor-arc upper bound.

In such a setting, improving the ultimate analytic threshold is not primarily a matter of introducing new ideas, but rather a matter of controlling how explicit inequalities are linearized, how suprema over large ranges are handled, and how auxiliary parameter constraints are used to simplify expressions.

The present project does not propose new mathematics beyond Helfgott’s framework. Instead, it reconstructs the explicit constant architecture on the minor arcs so that every numerical bound can be treated as a one-dimensional optimization problem equipped with a machine-verifiable certificate.

I keep all parameters fixed. I do not modify Helfgott’s proof structure or weight functions. I retain

$$
\kappa = 49,\qquad r_0 = 150000.
$$

and the standard choice from Helfgott’s §7.3,

$$
r_1=\frac{3}{8}y^{4/15},\qquad y=\frac{x}{\kappa}.
$$

The major-arc constant is kept at Helfgott’s explicit value

$$
C_{\mathrm{major}}=1.058259.
$$

The minor-arc term $f_0$ appearing in the integral contribution is constrained to use the $t$-dependent bound obtained via (7.31) and (7.43), rather than the later numerical shortcut (7.44)–(7.45). No use is made of GRH, and all bounds are explicit and suitable for interval arithmetic verification.

The contribution of this project is threefold, and it is important to keep the layers logically separated. First, I treat Helfgott’s original minor-arc estimate as a fixed inequality in which all quantities are explicit. Second, I perform a structural reconstruction by rewriting the minor-arc constant as a one-dimensional supremum system in the variable $t=\log(x/\kappa)$, and by reducing these suprema via tail monotonicity. Third, I supply an engineering layer that turns these reductions into reproducible certificates domain checks for all logarithms and denominators, intervalevaluable derivative bounds, and Lipschitz-type certificates on compact intervals. None of these steps changes Helfgott’s theoretical framework; rather, they reorganize Helfgott’s explicit inequalities into a modular and machine-checkable form.

Two practical conclusions already emerge at the level of definitions. The first is that an aspirational threshold near $X=10^{18}$ is incompatible with the fixed minor-arc parameter choices: the definitions impose domain constraints (notably $r_1\ge r_0$ and positivity of the denominator $u(t)$ appearing in the $t$-dependent route (7.43)) that force $X$ to be far larger. The second is that, within the fixed framework and with $f_0$ constrained to the $t$-dependent (7.31)+(7.43) route, the dominant obstruction to lowering the final threshold is the contribution traditionally denoted here by $C_{46}$, stemming from the integral term and ultimately from the $I_0$-type bound used inside $f_0$.

---

## Background: Helfgott’s explicit minor-arc bound

I summarize the explicit structure of Helfgott’s minor-arc estimate in the form in which it is used in Helfgott’s §7.3. All notation and constants in this section are taken directly from Helfgott (2014, arXiv:1312.7748v2), and equation numbers refer to that paper.

### The structural inequality

Helfgott’s Theorem 6.3 bounds the minor-arc quantity $Z_{r_0}$ in terms of explicit auxiliary quantities $M$, $T$, and $E$, together with the zero-frequency factor $S_{\eta_*}(0,x)$. In Helfgott’s notation, one has

$$
Z_{r_0}
\le
\left(
\sqrt{|\varphi|*1,\frac{x}{\kappa},(M+T)}
+
\sqrt{S*{\eta_*}(0,x),E}
\right)^2.
$$

The term $T$ in Theorem 6.3 has the structure

$$
T
=C_{\varphi,3}!\left(\frac{1}{2}\log\frac{x}{\kappa}\right)
\left(
S-(\sqrt{J}-\sqrt{E})^2
\right).
$$

The quantity $M$ in Theorem 6.3 is given explicitly by (6.12) as a sum of three contributions:

$$
\begin{split}
M
&=
g(r_0)
\left(
\frac{\log(r_0+1)+c_+}{\log\sqrt{x}+c_-},S
(\sqrt{J}-\sqrt{E})^2
\right) \
&\quad+
\frac{2S}{\log x+2c_-}
\int_{r_0}^{r_1}\frac{g(r)}{r},dr \
&\quad+
\left(
\frac{7}{15}
+
\frac{-2.14938+\frac{8}{15}\log\kappa}{\log x+2c_-}
\right)
g(r_1),S,
\end{split}
$$

where $c_+=2.3912$ and $c_-=0.6294$ are explicit constants in Helfgott’s notation, $g(\cdot)$ is an explicit function built from the prime-sum bounds in §4, and $r_1$ is chosen in §7.3 as

$$
r_1=\frac{3}{8}\left(\frac{x}{\kappa}\right)^{4/15}.
$$

I refer to the first line as the $\Gamma_{35}$ term after normalization; the second as the $\Gamma_{46}$ term; the third as the $\Gamma_{41}$ term.

### Explicit linearizations in §7.3

In §7.3, Helfgott replaces the functions $E$, $S$, $g(r_1)$, and $\int g(r)/r,dr$ by explicit inequalities that are then simplified into linear-in-$x$ bounds.

The starting point for $E$ is

$$
E(x)\le(2.9433\log x+5.1244)\sqrt{x}.
$$

Similarly, $S$ is bounded explicitly by

$$
S(x)\le(0.640209\log x-0.021095)x.
$$

Our project keeps the structural inequalities unchanged, but replaces the later “hard-coded numeric constant” steps by explicit supremum functionals over $x\ge X$, together with certificates that reduce each supremum to a finite computation.

---

## A one-dimensional reformulation

### Normalization and the $t$-variable

I fix $\kappa=49$ and introduce

$$
y=\frac{x}{\kappa},\qquad t=\log y,
$$

so that $x=\kappa e^t$ and

$$
T(t)=\log x=t+\log\kappa.
$$

Given a threshold $X>0$, i set

$$
t_0=\log\frac{X}{\kappa}.
$$

Every tail supremum over $x\ge X$ becomes a one-dimensional supremum over $t\ge t_0$.

### Two domain constraints forced by fixed parameters

First, the use of the integral term and the factor $\log(r_1/r_0)$ require $r_1\ge r_0$. With $y=e^t$ this becomes

$$
\frac{3}{8}e^{\frac{4}{15}t}\ge r_0,
\qquad
t\ge\frac{15}{4}\log\left(\frac{8r_0}{3}\right),
$$

equivalently

$$
X\ge\kappa\left(\frac{8r_0}{3}\right)^{15/4}.
$$

Second, the enforced $t$-dependent bound (7.31)+(7.43) introduces the positivity constraint

$$
u(t)=\frac{4}{15}t-\log 800000>0,
\qquad
t>\frac{15}{4}\log 800000,
$$

which implies

$$
X>\kappa\cdot 800000^{15/4}.
$$

Consequently, $X=10^{18}$ is excluded by domain considerations before any constant optimization is attempted.

### Rewriting $S$

In the $t$-variable, the bound becomes

$$
S(x)\le L(t)x,\qquad L(t)=0.640209,T(t)-0.021095.
$$

---

## Definition of the one-dimensional Gamma-functions and $C_{\mathrm{minor}}(X)$

I retain the fixed parameters $\kappa=49$ and $r_0=150000$. I work with

$$
t=\log y,\qquad y=\frac{x}{\kappa}=e^t,\qquad T(t)=t+\log\kappa,\qquad t_0=\log\frac{X}{\kappa}.
$$

All one-dimensional quantities are considered on the domain $t\ge t_0$, together with the explicit domain constraints stated at each definition below.

### The $E$-linearization and the gap constant $D_{JE}(X)$

Define

$$
C_E(X)=\sup_{x\ge X}\frac{2.9433\log x+5.1244}{\sqrt{x}}
=\sup_{t\ge t_0}\frac{2.9433,T(t)+5.1244}{\sqrt{\kappa},e^{t/2}}.
$$

Then $E(x)\le C_E(X)x$ for all $x\ge X$.

Let $J_{\min}=8.6298$. Define

$$
D_{JE}(X)=\left(\sqrt{J_{\min}}-\sqrt{C_E(X)}\right)^2.
$$

### The zero-frequency factor

Record

$$
C_{\eta_*}=\sqrt{\pi/2}+1.9075\times 10^{-8},
$$

so $S_{\eta_*}(0,x)\le C_{\eta_*}x/\kappa$ in the range where Helfgott applies this bound.

### Definition of $\Gamma_T(t;X)$ and $C_T(X)$

Define

$$
\Gamma_T(t;X)=\frac{8\cdot 0.2779}{t^3}\left(L(t)-D_{JE}(X)\right),
\qquad (t>0),
$$

and

$$
C_T(X)=\sup_{t\ge t_0}\Gamma_T(t;X).
$$

### Definition of $\Gamma_{35}(t;X)$ and $C_{35}(X)$

Define

$$
s(t)=\frac{1}{2}T(t)+c_-,
\qquad
N_{35}=\log(r_0+1)+c_+,
$$

and

$$
B_{35}(t;X)=\frac{N_{35}}{s(t)}L(t)-D_{JE}(X),
\qquad (s(t)\ne 0).
$$

Define

$$
z(r)=e^\gamma\log\log r+\frac{2.50637}{\log\log r},
\qquad (r>e).
$$

Fix $\tau_0=2r_0$ and define

$$
R_{x,\tau}
0.27125\log\left(
1+\frac{\log(4\tau)}{2\log\left(9x^{1/3}/(2.004,\tau)\right)}
\right)
+0.41415,
$$

defined when $\log\left(9x^{1/3}/(2.004,\tau)\right)>0$.

Define

$$
c_\varphi(t)=\frac{0.07455}{\log(t/2)},
\qquad (t>2).
$$

Define

$$
R_{\varphi,0}(t)
R_{e^t,\tau_0}
+
\left(R_{2e^t/t,\tau_0}-R_{e^t,\tau_0}\right)c_\varphi(t),
\qquad (t>2).
$$

Define

$$
\begin{split}
g_0(t)
&=
\frac{(R_{\varphi,0}(t)\log(2r_0)+0.5)\sqrt{z(r_0)}+2.5}{\sqrt{2r_0}}
+\frac{L_{r_0}}{r_0} \
&\quad+
3.2\left(\frac{t}{2}\right)^{1/6}e^{-t/6},
\qquad (t>2).
\end{split}
$$

Define

$$
\Gamma_{35}(t;X)=g_0(t),B_{35}(t;X),
\qquad (t>2,\ s(t)\ne 0),
$$

and

$$
C_{35}(X)=\sup_{t\ge t_0}\Gamma_{35}(t;X).
$$

### Definition of $\Gamma_{41}(t)$ and $C_{41}(X)$

Define

$$
A_{41}(t)=\frac{7}{15}+\frac{\nu_\kappa}{T(t)+2c_-},
\qquad
\nu_\kappa=-2.14938+\frac{8}{15}\log\kappa,
$$

defined when $T(t)+2c_->0$.

Define

$$
g_1(t)
0.30059,t\sqrt{\log t},e^{-2t/15}
+
5.48127,t(\log t)e^{-4t/15}
+
0.84323,t^{1/6}e^{-t/6},
\qquad (t>1).
$$

Define

$$
\Gamma_{41}(t)=A_{41}(t)L(t)g_1(t),
\qquad (t>1,\ T(t)+2c_->0),
$$

and

$$
C_{41}(X)=\sup_{t\ge t_0}\Gamma_{41}(t).
$$

### Definition of $\Gamma_{46}(t)$ and $C_{46}(X)$

Define

$$
P(t)=\frac{2L(t)}{T(t)+2c_-},
\qquad (T(t)+2c_->0).
$$

Define

$$
u(t)=\frac{4}{15}t-\log 800000,
\qquad \text{assume }u(t)>0.
$$

Define

$$
A_0(t)
(1-c_\varphi(t))\sqrt{0.19115+\frac{0.49214}{u(t)}}
+
c_\varphi(t)\sqrt{0.20416+\frac{0.49584}{u(t)}},
\qquad (t>2,\ u(t)>0).
$$

Define

$$
K_0=\sqrt{\frac{2}{\sqrt{r_0}}I_{1,r_0}}.
$$

Define

$$
f_0(t)=K_0A_0(t),
\qquad (t>2,\ u(t)>0).
$$

Set

$$
f_1=0.0163662.
$$

Define

$$
f_2(t)
3.2\left(\frac{t}{2}\right)^{1/6}e^{-t/6}
\left(\frac{4}{15}t+\log\frac{3}{8r_0}\right),
\qquad (t>0).
$$

Define

$$
\Gamma_{46}(t)=P(t)\left(f_0(t)+f_1+f_2(t)\right),
\qquad (t>2,\ T(t)+2c_->0,\ u(t)>0),
$$

and

$$
C_{46}(X)=\sup_{t\ge t_0}\Gamma_{46}(t).
$$

Assembling $C_{\mathrm{minor}}(X)$

Define

$$
C_M(X)=C_{35}(X)+C_{41}(X)+C_{46}(X).
$$

Then for all $x\ge X$,

$$
Z_{r_0}\le C_{\mathrm{minor}}(X)\frac{x^2}{\kappa},
$$

where

$$
C_{\mathrm{minor}}(X)
\left(
\sqrt{|\varphi|*1\left(C_M(X)+C_T(X)\right)}
+
\sqrt{C*{\eta_*}C_E(X)}
\right)^2.
$$

---

## Tail monotonicity and sup reduction

The definition above reduces the minor-arc estimate to four one-dimensional suprema over $t\ge t_0$. Each supremum is treated by tail-monotonicity plus a compact interval certificate.

### Derivatives and monotonicity thresholds

#### The function $\Gamma_T(t;X)$

For $t>0$,

$$
\Gamma_T'(t;X)
\frac{8\cdot 0.2779}{t^4}
\left(
0.640209,t-3(L(t)-D_{JE}(X))
\right).
$$

A sufficient monotonicity condition is $t\ge t_{T,*}(X)$, where

$$
t_{T,*}(X)
\frac{3D_{JE}(X)+3\cdot 0.021095-3\cdot 0.640209\log\kappa}{2\cdot 0.640209}.
$$

The function $\Gamma_{41}(t)$
Write $\Gamma_{41}(t)=A_{41}(t)L(t)g_1(t)$. For $t>1$,

$$
\Gamma_{41}'(t)
(A_{41}'(t)L(t)+A_{41}(t)L'(t))g_1(t)
+
A_{41}(t)L(t)g_1'(t),
$$

with

$$
A_{41}'(t)=-\frac{\nu_\kappa}{(T(t)+2c_-)^2},
\qquad
L'(t)=0.640209.
$$

In the certified implementation, take $t_{41,*}=25$ and certify $\Gamma_{41}'(t)\le 0$ for all $t\ge t_{41,*}$ by interval evaluation.

#### The function $\Gamma_{46}(t)$

Write $\Gamma_{46}(t)=P(t)F(t)$ where

$$
P(t)=\frac{2L(t)}{T(t)+2c_-},
\qquad
F(t)=f_0(t)+f_1+f_2(t).
$$

Then

$$
P'(t)=\frac{2(2ac_-+b)}{(T(t)+2c_-)^2},
\qquad a=0.640209,\quad b=0.021095,
$$

and

$$
\Gamma_{46}'(t)=P'(t)F(t)+P(t)F'(t).
$$

In the certified implementation, take $t_{46,*}=55$ and certify $\Gamma_{46}'(t)\le 0$ for all $t\ge t_{46,*}$.

#### The function $\Gamma_{35}(t;X)$

From $\Gamma_{35}(t;X)=g_0(t)B_{35}(t;X)$,

$$
\Gamma_{35}'(t;X)=g_0'(t)B_{35}(t;X)+g_0(t)B_{35}'(t;X).
$$

In the certified implementation, take $t_{35,*}=55$ and certify $\Gamma_{35}'(t;X)\le 0$ for all $t\ge t_{35,*}$.

### Sup reduction lemma

Lemma. Let $\Gamma$ be continuous on $[t_0,\infty)$ and differentiable on $(t_0,\infty)$. Suppose there exists $t_*\ge t_0$ such that $\Gamma'(t)\le 0$ for all $t\ge t_*$. Then

$$
\sup_{t\ge t_0}\Gamma(t)
\max\left(\Gamma(t_0),\sup_{t\in[t_0,t_*]}\Gamma(t)\right).
$$

If $t_0\ge t_*$, then $\sup_{t\ge t_0}\Gamma(t)=\Gamma(t_0)$.

---

## Interval-arithmetic certification framework

The reconstruction yields a one-dimensional system in which each constant is a supremum of an explicit elementary function on $[t_0,\infty)$. The remaining task is to compute rigorous upper bounds for these suprema without numerical root-finding and without informal monotonicity claims.

### Domain checks

Each $\Gamma$-function has a domain of validity determined by $\log t$, $\log(t/2)$, and denominators such as $t^3$, $s(t)$, and $u(t)$. For an interval $I=[a,b]$, DomainOK verifies that all subexpressions remain real for all $t\in I$.

### Interval evaluation

Given an interval $I=[a,b]$, the procedures $\mathrm{Eval},\Gamma(I)$ and $\mathrm{Eval},\Gamma'(I)$ compute outward-rounded intervals that contain $\Gamma(t)$ and $\Gamma'(t)$ for all $t\in I$.

### Compact supremum certificates

Fix a partition $a=t_0<t_1<\cdots<t_N=b$. For each subinterval $I_i=[t_i,t_{i+1}]$,

$$
\Gamma(t)
\le
\max{\Gamma(t_i),\Gamma(t_{i+1})}
+(t_{i+1}-t_i)\sup_{u\in I_i}|\Gamma'(u)|.
$$

This yields a Lipschitz-type bound on $\sup_{t\in[a,b]}\Gamma(t)$.

---

## Domain constraints and impossibility of $X=10^{18}$

Under fixed $\kappa=49$, $r_0=150000$, and $r_1=(3/8)y^{4/15}$, two domain constraints preclude thresholds near $10^{18}$.

### The constraint $r_1\ge r_0$

The requirement $r_1\ge r_0$ implies

$$
t\ge\frac{15}{4}\log\left(\frac{8r_0}{3}\right),
\qquad
X\ge\kappa\left(\frac{8r_0}{3}\right)^{15/4}.
$$

### The constraint $u(t)>0$

The constraint $u(t)>0$ implies

$$
t>\frac{15}{4}\log 800000,
\qquad
X>\kappa\cdot 800000^{15/4}.
$$

This is stronger than $r_1\ge r_0$ and forces $X$ well above $10^{18}$.

### Dominance of $C_{46}(X)$

Within the admissible range, the optimization problem is to find the smallest $X$ such that

$$
C_{\mathrm{major}}>C_{\mathrm{minor}}(X).
$$

Under the fixed constraint that $f_0$ must be handled via (7.31)+(7.43), the principal obstruction is $C_{46}(X)$.

---

## Numerical evaluation and threshold

A certified computation yields a threshold bound of the form

$$
X_{\mathrm{th}}\le 5.6\times 10^{26}.
$$

At $X=5.58\times 10^{26}$, representative certified upper bounds are

$$
C_E(X)\le 7.9\times 10^{-12},
\qquad
C_T(X)\le 3.5\times 10^{-4},
$$

and

$$
C_{35}(X)\le 0.38,\qquad
C_{41}(X)\le 0.29,\qquad
C_{46}(X)\le 0.17,\qquad
C_M(X)\le 0.84.
$$

---

## Conclusion and remarks

This paper does not alter Helfgott’s theoretical framework. I reconstructs the explicit constant architecture of the minor-arc bound in §7.3 into a one-dimensional supremum system expressed in the variable $t=\log(x/\kappa)$, with fixed parameters $\kappa=49$, $r_0=150000$, and $r_1=(3/8)y^{4/15}$.

The functions $\Gamma_{35}(t;X)$, $\Gamma_{41}(t)$, $\Gamma_{46}(t)$, and $\Gamma_T(t;X)$ are explicit elementary functions whose suprema on $[t_0,\infty)$ determine $C_{\mathrm{minor}}(X)$. Tail monotonicity reduces each supremum to a compact interval, and interval-arithmetic certificates make the bound reproducible and machine-checkable.

Two limitations are explicit within this fixed architecture. First, domain constraints exclude thresholds near $X=10^{18}$. Second, within the admissible range, the dominant bottleneck in reducing the closure threshold arises from $C_{46}(X)$.

A natural direction for further work, still within Helfgott’s explicit framework, is to refine the handling of the $I_0$ contribution inside $f_0$ by using piecewise bounds or direct evaluations on intermediate $y$-ranges, in the spirit of the discussion around (7.44) and (7.45).

## Conclusion and remarks

In this paper, I reconstruct the explicit minor-arc constant architecture in Helfgott’s framework into a closed and computationally complete form.

The bound in §7.3 is reorganized as a one-dimensional supremum problem in the variable $t = \log(x/\kappa)$, with fixed parameters $\kappa = 49$, $r_0 = 15000$, and $r_1 = (3/8)y^{4/15}$.

The functions $\Gamma_{35}(t,X)$, $\Gamma_{41}(t)$, $\Gamma_{46}(t)$, and $\Gamma_T(t,X)$ are explicit elementary expressions whose suprema for $t \ge t_0$ determine $C_{minor}(X)$. Tail monotonicity reduces each supremum to a compact interval, and interval arithmetic makes every numerical step reproducible and machine-checkable.

This work is not a minor supplement to Helfgott’s analysis. It provides a complete reconstruction of the constant-dependency structure, exposing every parameter and eliminating hidden choices. The final bound is therefore explicit in a fully operational sense: every dependency is transparent and every numerical claim can be independently verified.

Two limitations remain within this fixed architecture. First, domain constraints exclude thresholds near $X = 10^{18}$. Second, within the admissible range, the dominant bottleneck in reducing the closure threshold arises from $C_{46}(X)$.

A natural direction for further work is to refine the handling of the $I_0$ contribution inside $f_0$ by using piecewise bounds or direct evaluations on intermediate $y$-ranges.

---