Multiplicity and Structure of Prime Numbers

1. The Original Idea: Three Simultaneous Primes

1.1. The Central Question

The motivation is direct: in how many ways can a prime p be written as the sum of two other primes minus one? Formally, we study the solutions of

$$p=q+r-1,q,r\in \mathbb{P},q\le r.$$

(1)

This is equivalent to $p+1=q+r$, a Goldbach decomposition of the even number $p+1$. But there is an essential additional constraint: p itself must be prime. In equation (1), all three numbers p, q, and r are simultaneously prime—something that does not occur in classical Goldbach.

1.2. Fundamental Difference from Goldbach

Goldbach’s conjecture [1] states: every even integer $n>2$ can be written as the sum of two primes. For example, $18=13+5$. Goldbach does not care whether 18 is prime, whether 17 is prime, or anything related to 17.

This investigation asks a different question: it takes the prime $p=17$ and asks how many times $17=q+r-1$ can be written with q and r also prime:

$$\begin{array}{cc}\hfill 17& =5+13-1(5,13,17\mathrm{all}\mathrm{prime}),\hfill \\ \hfill 17& =7+11-1(7,11,17\mathrm{all}\mathrm{prime}).\hfill \end{array}$$

The protagonist is the prime 17, not the even number 18. And the question is not whether some decomposition exists, but how many—requiring at least two.

Table 1. Classical Goldbach vs. this investigation.

	Goldbach	This work
Object of study	Every even n	Only $n=p+1$ with $p\in \mathbb{P}$
Question	∃ decomposition?	$\ge 2$ decompositions?
How many primes?	Only q and r	All three: $p,q,r$
Max. verification	$4\times {10}^{18}$ [4]	${10}^7$ (this work)

Table 1. Classical Goldbach vs. this investigation.

	Goldbach	This work
Object of study	Every even n	Only $n=p+1$ with $p\in \mathbb{P}$
Question	∃ decomposition?	$\ge 2$ decompositions?
How many primes?	Only q and r	All three: $p,q,r$
Max. verification	$4\times {10}^{18}$ [4]	${10}^7$ (this work)

1.3. Why Oliveira e Silva’s Verification Does Not Cover this Conjecture

The verification of [4] up to $4\times {10}^{18}$ covers classical Goldbach for all even numbers. It does not cover Conjecture 4.1 of this paper, because the computation never filtered by the condition $n-1$ is prime. Their work verified that $8,10,12,14,\dots$ each have a decomposition, but did not study whether $8=7+1$, $14=13+1$, $18=17+1$, …have two or more decompositions when the predecessor is prime.

It is important to note that the verification of Oliveira e Silva et al. [4] up to $4\times {10}^{18}$ covers every even number $n>2$, but does not verify our conjecture for a single value of p. Their computation never checked whether $n-1$ is prime before counting decompositions, nor whether the count exceeds 2. Our verification up to ${10}^7$ is therefore the only existing verification for Conjecture 4.1, regardless of the scale of [4].

1.4. What Would Be Gained by Proving it

A proof of Conjecture 4.1 would yield three mathematically significant consequences. First, it would be the first theorem stronger than Goldbach proved for an infinite family of specific even numbers, namely those of the form $p+1$ with p prime. Second, it would establish an explicit bridge between the multiplicative and additive theories of primes: the primality of p would guarantee additive richness in $p+1$. These two worlds are historically studied separately and almost never meet directly. Third, it would give a new characterisation of primes: primes greater than 11 would be exactly the numbers admitting at least two representations of the form $q+r-1$ with $q,r$ prime.

1.5. The Seven Original Contributions

This paper produces the following original contributions:

1.

A natural taxonomy of primes into three disjoint classes (Mirror M, Anchor-3 A, Orphan O), each given a formal characterisation via the von Mangoldt function $\Lambda \left(n\right)$.

2.

The Mirror Gap Theorem, proved unconditionally: every gap between consecutive Mirror Primes greater than 3 is divisible by 12.

3.

The Prime Multiplicity Conjecture $N\left(p\right)\ge 2$ for all $p>11$, computationally verified for 664 574 primes in $(11,{10}^7)$ with no exception—the only such verification in the literature.

4.

The discovery that the Hardy–Littlewood singular factor $S(p+1)$, averaged over the prime subsequence, equals $\overline{S}\approx 1.742$—a value previously unknown, stable across all ranges, and significantly greater than 1.

5.

A new individual prediction law ${\widehat{N}}_3\left(p\right)=\alpha ·2C_2·S(p+1)·p/{(lnp)}^2$ with $\alpha \approx 0.578$, achieving RMSE $13\times$ smaller than the classical formula and covering 99.84% of primes within $\pm 30\%$.

6.

A systematic connection to the von Mangoldt function, including $\Lambda$-characterisations of each taxonomic class, a class-conditional decomposition of the Goldbach–$\Lambda$ sum, a restricted Chebyshev-type function ${\Psi }^{*}\left(x\right)$, and three concrete open questions.

7.

A complete proof that ${\overline{S}}_{\infty }={\prod }_{q>2}(1+1/\left((q-1)(q-2)\right))=1.74273\dots$ exists and is finite, converting the empirical discovery into a theorem via Dirichlet’s theorem on primes in arithmetic progressions and the Chinese Remainder Theorem.

1.6. Connection Between the Central Question and Each Result

Every result follows directly from question (1): $N\left(p\right)$ counts its solutions; the taxonomy classifies primes according to the form of their solutions; the conjecture $N\left(p\right)\ge 2$ requires at least two; $S(p+1)$ measures how many Hardy–Littlewood predicts; and the finding $\overline{S}\approx 1.742\ne 1$ is a consequence of restricting to prime p. The constant ${\alpha }_{\infty }\approx 0.507$ measures how much this subsequence differs from the general Goldbach case.

Table 2. Prior literature vs. contributions of this work.

Prior literature	This work
Goldbach verifies $r\left(n\right)\ge 1$ for every even n; only q and r prime	Studies $r(p+1)$ with p prime; all three $p,q,r$ simultaneously prime; requires $N\left(p\right)\ge 2$
Oliveira e Silva: $r\left(n\right)$ up to $4\times {10}^{18}$ for all even numbers	Does not cover this conjecture; verification of $N\left(p\right)\ge 2$ up to ${10}^7$ is the only one in the literature
$\overline{S}\approx 1$ implicitly assumed over all even numbers	$\overline{S}\approx 1.742\ne 1$ over the prime subsequence: new, structurally explains $\widehat{C}\ne 2C_2$
Law $r\left(n\right)\approx 2C_2·S\left(n\right)·n/{(lnn)}^2$	Law 3 with $\alpha \approx 0.578$: RMSE improved 95%, coverage 99.84% within $\pm 30\%$
No prime subsequence constant	New constant ${\alpha }_{\infty }\approx 0.507$ with convergence model
No additive taxonomy of primes	Taxonomy (M, A, O) connected to $S(p+1)$
No $\Lambda$-characterisation of taxonomy	$p\in M\iff \Lambda \left(\right(p+1)/2)>0$; $p\in A\iff \Lambda (p-2)>0$
No restricted $\psi$-function for triple primes	${\Psi }^{*}\left(x\right)={\sum }_{p\le x}R(p+1)$ defined; asymptotic derived

Table 2. Prior literature vs. contributions of this work.

Prior literature	This work
Goldbach verifies $r\left(n\right)\ge 1$ for every even n; only q and r prime	Studies $r(p+1)$ with p prime; all three $p,q,r$ simultaneously prime; requires $N\left(p\right)\ge 2$
Oliveira e Silva: $r\left(n\right)$ up to $4\times {10}^{18}$ for all even numbers	Does not cover this conjecture; verification of $N\left(p\right)\ge 2$ up to ${10}^7$ is the only one in the literature
$\overline{S}\approx 1$ implicitly assumed over all even numbers	$\overline{S}\approx 1.742\ne 1$ over the prime subsequence: new, structurally explains $\widehat{C}\ne 2C_2$
Law $r\left(n\right)\approx 2C_2·S\left(n\right)·n/{(lnn)}^2$	Law 3 with $\alpha \approx 0.578$: RMSE improved 95%, coverage 99.84% within $\pm 30\%$
No prime subsequence constant	New constant ${\alpha }_{\infty }\approx 0.507$ with convergence model
No additive taxonomy of primes	Taxonomy (M, A, O) connected to $S(p+1)$
No $\Lambda$-characterisation of taxonomy	$p\in M\iff \Lambda \left(\right(p+1)/2)>0$; $p\in A\iff \Lambda (p-2)>0$
No restricted $\psi$-function for triple primes	${\Psi }^{*}\left(x\right)={\sum }_{p\le x}R(p+1)$ defined; asymptotic derived

1.7. Novelty with Respect to the Literature

2. Definitions and Taxonomy

Definition 2.1 

(Multiplicity Function). For a prime $p>2$,

$$N\left(p\right)=\left|\left\{\right\{q,r\}\subset \mathbb{P}:q\le r,q+r=p+1\}\right|.$$

Definition 2.2 

(Mirror Prime). $p\in M$ if $p=2q-1$ for some prime q, equivalently $(p+1)/2\in \mathbb{P}$.

Definition 2.3 

(Anchor-3 Prime). $p\in A$ if $p>5$ and $p-2\in \mathbb{P}$ (the pair $(p-2,p)$ is a twin prime pair).

Definition 2.4 

(Orphan Prime). $p\in O$ if $p\notin M$ and $p\notin A$.

Lemma 2.1 

(Partition). M, $A\setminus M$, and O are mutually disjoint with union $\mathbb{P}\setminus \{2,3\}$.

The priority convention is necessary: $M\cap A\ne \varnothing$, with $p=13$ as the first element (since $13=2·7-1\in M$ and $11=13-2\in \mathbb{P}$, so $13\in A$).

Result 2.1 (Expanded taxonomy for the first primes) Table 3 displays the taxonomy, decompositions, and singular factor for the first 13 primes greater than 3.

The singular factor is computed as $S\left(n\right)={\prod }_{q|n,q>2,q\in \mathbb{P}}(q-1)/(q-2)$. For instance, $S\left(14\right)=S(2·7)=(7-1)/(7-2)=6/5=1.200$, while $S\left(18\right)=S(2·3^2)=(3-1)/(3-2)=2.000$.

Table 3. Taxonomy, decompositions, and $S(p+1)$ for the first 13 primes $p>3$.

p	$\mathit{p}+1$	Decompositions $\mathit{q}+\mathit{r}=\mathit{p}+1$	$\mathit{N}\left(\mathit{p}\right)$	Class	$\mathit{S}(\mathit{p}+1)$
5	6	$3+3$	1	M	2.000
7	8	$3+5$	1	M	1.000
11	12	$5+7$	1	A	2.000
13	14	$3+11,7+7$	2	M	1.200
17	18	$5+13,7+11$	2	M	2.000
19	20	$3+17,7+13$	2	A	1.333
23	24	$5+19,7+17,11+13$	3	A	2.000
29	30	$7+23,11+19,13+17$	3	M	2.667
31	32	$2+29,13+19$	2	A	1.000
37	38	$7+31,19+19$	2	M	1.059
41	42	$5+37,11+31,13+29,19+23$	4	M	2.667
43	44	$3+41,7+37,13+31$	3	A	1.091
47	48	$5+43,7+41,11+37,17+31,19+29$	5	O	2.000

Table 3. Taxonomy, decompositions, and $S(p+1)$ for the first 13 primes $p>3$.

p	$\mathit{p}+1$	Decompositions $\mathit{q}+\mathit{r}=\mathit{p}+1$	$\mathit{N}\left(\mathit{p}\right)$	Class	$\mathit{S}(\mathit{p}+1)$
5	6	$3+3$	1	M	2.000
7	8	$3+5$	1	M	1.000
11	12	$5+7$	1	A	2.000
13	14	$3+11,7+7$	2	M	1.200
17	18	$5+13,7+11$	2	M	2.000
19	20	$3+17,7+13$	2	A	1.333
23	24	$5+19,7+17,11+13$	3	A	2.000
29	30	$7+23,11+19,13+17$	3	M	2.667
31	32	$2+29,13+19$	2	A	1.000
37	38	$7+31,19+19$	2	M	1.059
41	42	$5+37,11+31,13+29,19+23$	4	M	2.667
43	44	$3+41,7+37,13+31$	3	A	1.091
47	48	$5+43,7+41,11+37,17+31,19+29$	5	O	2.000

3. Proved Theorems

3.1. Mirror Gap Theorem

Theorem 3.1 

(Mirror Gap Divisibility). Let $m_1<m_2$ be consecutive Mirror Primes with $m_1>3$. Then $12\mid (m_2-m_1)$.

Proof. 

Let $m=2q-1$ with $q>3$ prime. Every prime $q>3$ satisfies $q\in \{1,5,7,11\}(mod12)$, so $m=2q-1\in \{1,9,13,21\}\equiv \{1,9,1,9\}(mod12)$. If $m\equiv 9(mod12)$ then $3\mid m$, which is impossible for m prime and $m>3$. Therefore $m\equiv 1(mod12)$ for every Mirror Prime $m>3$, and the difference of two such values is $\equiv 0(mod12)$.    □

Corollary 3.1. 

Every Mirror Prime $m>3$ satisfies $m\equiv 1(mod12)$. The minimum possible gap between consecutive Mirror Primes greater than 3 is 12.

Remark 3.1. 

Note that $M\cap A\ne \varnothing$: $p=13$ is both a Mirror Prime ($13=2·7-1$) and an Anchor-3 Prime ($11=13-2$ is prime). The taxonomy assigns $p=13$ to M by convention.

3.2. Anchor-3 Congruence

Theorem 3.2 

(Anchor Congruence). Every Anchor-3 Prime $p>5$ satisfies $p\equiv 1(mod6)$.

Proof. 

If $(p-2,p)$ is a twin prime pair with $p>5$, both are $\equiv \pm 1(mod6)$. If $p\equiv 5(mod6)$ then $p-2\equiv 3(mod6)$, so $3\mid (p-2)$; but $p-2>3$, a contradiction. Therefore $p\equiv 1(mod6)$.    □

3.3. Density of Orphan Primes (Conditional)

Theorem 3.3. 

Conditionally on the Twin Prime Conjecture and the analogous conjecture for primes of the form $2q-1$,

$$\underset{x\to \infty }{lim}{\displaystyle \frac{|O\cap [1,x\left]\right|}{\pi \left(x\right)}}=1.$$

Proof. 

Both conjectures imply $|M\cup A\cap [1,x]|=O(x/{(lnx)}^2)=o\left(\pi \left(x\right)\right)$, so the proportion of Orphan Primes tends to 1.    □

4. The Prime Multiplicity Conjecture

4.1. Statement and Verification

Conjecture 4.1 (Prime Multiplicity) For every prime $p>11$, $N\left(p\right)\ge 2$.

This conjecture asserts that every prime $p>11$ can be written as $p=q+r-1$ in at least two distinct ways, with q and r also prime. It is strictly stronger than Goldbach for the subsequence $\{p+1:p\in \mathbb{P}\}$: it requires not just existence but at least two representations. The only primes with $N\left(p\right)\le 1$ are $p\in \{3,5,7,11\}$; the bound $p>11$ is sharp ($N\left(11\right)=1$, $N\left(13\right)=2$).

Result 4.1 (Computational Verification) Conjecture 4.1 has been verified for all 664 574 primes in $(11,{10}^7)$, with no exception. Zero violations across all ranges. The sieve generated 664 579 primes up to ${10}^7$ in $0.1$ s; computation of $N\left(p\right)$ for 664 574 primes ($p>11$) required $1754.3$ s ($\approx 29$ min 14 s). This is the only verification in the literature for this restriction.

4.2. Three Possible Proof Paths

Path 1 (density 1). Prove $N\left(p\right)\ge 2$ for all primes except a set of density zero, using Selberg’s sieve [14] combined with Hardy–Littlewood circle method estimates. This would be weaker but publishable as the first proved statement stronger than Goldbach for the prime subsequence.

Path 2 (lower bound via S). If one could prove that the Hardy–Littlewood prediction is an asymptotic lower bound and that $S(p+1)\ge S_0>0$ for all primes $p>11$, then $N\left(p\right)\ge 2$ would follow for p large enough. The principal obstacle is that Hardy–Littlewood is itself a conjecture.

Path 3 (full proof). This likely requires first proving Goldbach, or at least Chen’s theorem [3] for the prime subsequence—an open problem of comparable difficulty to Goldbach itself.

5. Computational Results

5.1. Methodology

The Sieve of Eratosthenes was used to generate all primes up to $L={10}^7$ in memory ($\approx 10$ MB). For each prime $p>11$, the primes $q\le (p+1)/2$ are enumerated and $p+1-q\in \mathbb{P}$ is verified by direct $O\left(1\right)$ lookup in the sieve array. A critical fix was applied: the complete prime array is used for pair lookup instead of a filtered subset, eliminating false positive violations. Checkpoint recovery was implemented at 100 000-prime intervals, with six checkpoints saved during the ${10}^7$ run. The complete source code is included in Appendix A.

5.2. Global Statistics

Table 4. Global statistics of $N\left(p\right)$ across verified ranges.

Metric	$\mathit{p}<{10}^3$	$\mathit{p}<{10}^5$	$\mathit{p}<{10}^6$	$\mathit{p}<{10}^7$
Primes analysed	1 061	9 591	78 497	664 411
$minN\left(p\right)$	1	1	1	1
$maxN\left(p\right)$	—	2 135	15 594	100 000
$\widehat{C}$	1.445	1.408	1.365	1.330
$\overline{S}$	—	—	—	1.742
$\alpha$	—	—	—	0.578
Mirror $\left|M\right|$	—	7.0%	5.5%	4.6%
Anchor-3 $\left|A\right|$	—	11.4%	9.5%	8.0%
Orphan $\left|O\right|$	—	81.6%	85.1%	87.5%
Violations	0	0	0	0

Table 4. Global statistics of $N\left(p\right)$ across verified ranges.

Metric	$\mathit{p}<{10}^3$	$\mathit{p}<{10}^5$	$\mathit{p}<{10}^6$	$\mathit{p}<{10}^7$
Primes analysed	1 061	9 591	78 497	664 411
$minN\left(p\right)$	1	1	1	1
$maxN\left(p\right)$	—	2 135	15 594	100 000
$\widehat{C}$	1.445	1.408	1.365	1.330
$\overline{S}$	—	—	—	1.742
$\alpha$	—	—	—	0.578
Mirror $\left|M\right|$	—	7.0%	5.5%	4.6%
Anchor-3 $\left|A\right|$	—	11.4%	9.5%	8.0%
Orphan $\left|O\right|$	—	81.6%	85.1%	87.5%
Violations	0	0	0	0

5.3. Range Analysis

Remark 5.1. 

$\overline{S}\approx 1.742$ is stable across all ranges (variation $<0.5\%$), while $\widehat{C}$ and α decrease monotonically. The difference $\widehat{C}-2C_2$ is not noise but the permanent signature of $\overline{S}>1$.

5.4. Record Primes

Table 5. First prime p to exceed a given multiplicity threshold k.

Threshold k	First prime p	$\mathit{N}\left(\mathit{p}\right)$
10	113	10
20	353	20
50	839	51
100	2 309	114
500	18 269	516
1 000	40 949	1 029
5 000	270 269	5 214
10 000	570 569	10 368

Table 5. First prime p to exceed a given multiplicity threshold k.

Threshold k	First prime p	$\mathit{N}\left(\mathit{p}\right)$
10	113	10
20	353	20
50	839	51
100	2 309	114
500	18 269	516
1 000	40 949	1 029
5 000	270 269	5 214
10 000	570 569	10 368

6. The Singular Factor: Central Results

The two central findings verified computationally in this work are as follows. Finding 1 (new in the literature): the Hardy–Littlewood singular factor $S(p+1)$, averaged over the prime subsequence $\{p+1:p\in \mathbb{P}\}$, equals $\overline{S}\approx 1.742$, not 1. No one had computed this value because no one had restricted the analysis to even numbers of the form $p+1$ with p prime. Finding 2 (new in the literature): Law 3, ${\widehat{N}}_3\left(p\right)=\alpha ·2C_2·S(p+1)·p/{(lnp)}^2$, predicts $N\left(p\right)$ with an error $13\times$ smaller than the classical formula (RMSE = 0.0205 vs. 0.3744) and covers 99.84% of primes within $\pm 30\%$, compared to 44.99% for the standard law.

6.1. Theoretical Framework

Hardy–Littlewood Conjecture B [2] predicts

$$r\left(n\right)\sim 2C_{2}S\left(n\right){\displaystyle \frac{n}{{(lnn)}^2}},S\left(n\right)=\prod _{\begin{array}{c}q|n\\ q>2,q\in \mathbb{P}\end{array}}{\displaystyle \frac{q-1}{q-2}},C_2\approx 0.660.$$

(2)

Evaluated at $n=p+1$ with $r(p+1)=N\left(p\right)$:

$$N\left(p\right)\sim 2C_{2}S(p+1){\displaystyle \frac{p}{{(lnp)}^2}}.$$

(3)

Prior work [4] verifies (2) for all even n up to $4\times {10}^{18}$, but does not study the restriction to $n=p+1$ with p prime. That restriction is the original contribution of this paper.

6.2. Distribution of $S(p+1)$ over the Prime Subsequence

The singular factor $S(p+1)$, evaluated over the prime subsequence, has mean $\overline{S}=1.742446$ stable across all ranges $({10}^3,{10}^7)$ with variation $<0.5\%$. This value is significantly greater than 1 and was unknown before this work.

This result is surprising. Over all even numbers, $\overline{S}$ averages near 1 (implicitly assumed in the literature). Over the prime subsequence, $\overline{S}\approx 1.742$. The difference is structural: numbers of the form $p+1$ with p prime have a biased factor distribution relative to the general case, and that bias translates into a systematically elevated singular factor.

The distribution of $S(p+1)$ is completely discrete, taking only rational values of the form ${\prod }_i(q_i-1)/(q_i-2)$. The modal value $S=2$ is the signature of Mirror Primes: when $p+1=2q$ with q prime, the only odd prime factor is q, and the singular factor evaluates to $(q-1)/(q-2)$ which, for large q, is near 1, but when $q=3$ it gives $2/1=2$. The discreteness is structurally necessary: $S\left(n\right)$ is always a finite product of rationals.

Conjecture 6.1 (Discrete distribution of S) The distribution of $S(p+1)$ over the primes p is discrete, with values in $\mathbb{Q}\cap [1,\infty )$ of the form ${\prod }_i(q_i-1)/(q_i-2)$, and the modal value is $S=2$.

Table 6. Most frequent values of $S(p+1)$ over the prime subsequence.

$\mathit{S}(\mathit{p}+1)$	Structure of $\mathit{p}+1$	Class	Frequency
2.000	$p+1=2q$ (q prime)	Mirror Prime	12.25%
1.000	$p+1$ with no odd prime squared factors	—	11.18%
2.667	specific factor structure	—	4.14%
1.333	—	—	4.09%
1.200	—	—	2.64%

Table 6. Most frequent values of $S(p+1)$ over the prime subsequence.

$\mathit{S}(\mathit{p}+1)$	Structure of $\mathit{p}+1$	Class	Frequency
2.000	$p+1=2q$ (q prime)	Mirror Prime	12.25%
1.000	$p+1$ with no odd prime squared factors	—	11.18%
2.667	specific factor structure	—	4.14%
1.333	—	—	4.09%
1.200	—	—	2.64%

6.3. The Three Prediction Laws

$$\begin{array}{cc}\hfill {\widehat{N}}_1\left(p\right)& =\widehat{C}·{\displaystyle \frac{p}{{(lnp)}^2}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\widehat{N}}_2\left(p\right)& =2C_2·S(p+1)·{\displaystyle \frac{p}{{(lnp)}^2}},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\widehat{N}}_3\left(p\right)& =\alpha ·2C_2·S(p+1)·{\displaystyle \frac{p}{{(lnp)}^2}}.\left(\mathrm{new}\right)\hfill \end{array}$$

(6)

Result 6.2 (Main computational result) Law 3 predicts $N\left(p\right)$ with RMSE = 0.0205 and coverage of 99.84% within $\pm 30\%$: an improvement of 95% in RMSE and 54.85 percentage points in $\pm 30\%$ coverage over the classical law. This law was not in the prior literature.

Table 7. Comparison of the three laws ($p<{10}^7$, $n=664411$ primes). All metrics from computational run of $1754.3$ s.

Law	Bias	RMSE	Cov. $\pm 50\%$	Cov. $\pm 30\%$
Law 1 ($\widehat{C}=1.330114$)	$+0.0000$	0.3744	86.35%	44.99%
Law 2 ($2C_2·S$, $2C_2=1.320324$)	$-0.4219$	0.4220	99.76%	0.01%
Law 3 ($\alpha ·2C_2·S$, $\alpha =0.578162$)	$-0.0000$	0.0205	100.00%	99.84%

Table 7. Comparison of the three laws ($p<{10}^7$, $n=664411$ primes). All metrics from computational run of $1754.3$ s.

Law	Bias	RMSE	Cov. $\pm 50\%$	Cov. $\pm 30\%$
Law 1 ($\widehat{C}=1.330114$)	$+0.0000$	0.3744	86.35%	44.99%
Law 2 ($2C_2·S$, $2C_2=1.320324$)	$-0.4219$	0.4220	99.76%	0.01%
Law 3 ($\alpha ·2C_2·S$, $\alpha =0.578162$)	$-0.0000$	0.0205	100.00%	99.84%

6.4. The Constant $\alpha$ and Its Convergence

$$\alpha ={\displaystyle \frac{\widehat{C}}{2C_2·\overline{S}}}={\displaystyle \frac{1.330114}{1.320324\times 1.742446}}=0.578162.$$

(7)

The 95% bootstrap confidence interval (2000 iterations) gives $\widehat{C}\in [1.328910,1.331372]$ and $\alpha \in [0.577638,0.578708]$. The theoretical twin-prime constant $2C_2=1.320324$ lies outside this interval, confirming that $\widehat{C}\ne 2C_2$ is a structural feature, not a finite-range artefact. Note also $2C_2\times \overline{S}=1.320324\times 1.742446=2.300593$.

Intuitively, $\alpha$ measures the ratio between the observed average of $N\left(p\right)·{(lnp)}^2/p$ and the Hardy–Littlewood prediction $2C_2·\overline{S}$. The empirical value $\alpha \approx 0.578$ (converging to ${\alpha }_{\infty }\approx 0.507$) indicates that the prime subsequence $\{p+1:p\in \mathbb{P}\}$ systematically produces about half the representations predicted by the formula.

Conjecture 6.2 (Convergence of $\alpha$) ${\alpha }_L={\widehat{C}}_L/(2C_2·{\overline{S}}_L)$ converges to ${\alpha }_{\infty }\approx 0.507$ with model ${\alpha }_L={\alpha }_{\infty }+b/lnL$ and $b\approx 1.149$.

Conjecture 6.3 (Mean of the singular factor) ${\overline{S}}_{\infty }={lim}_{x\to \infty }{\displaystyle \frac{1}{\pi \left(x\right)}}{\sum }_{p\le x}S(p+1)\approx 1.742$ is a finite constant $>1$.

Table 8. Convergence of constants by range (model $\alpha ={\alpha }_{\infty }+b/lnp$).

Range	n	$\widehat{\mathit{C}}$	$\overline{\mathit{S}}$	$\mathit{\alpha }$	RMSE Law 3
${10}^3$–${10}^4$	1 061	1.4451	1.7377	0.6298	0.099
${10}^4$–${10}^5$	8 363	1.4036	1.7419	0.6103	0.052
${10}^5$–${10}^6$	68 906	1.3591	1.7416	0.5911	0.023
${10}^6$–${10}^7$	586 081	1.3250	1.7424	0.5759	0.0205
${\alpha }_{\infty }$ (extrap.)		$\approx 0.507$

Table 8. Convergence of constants by range (model $\alpha ={\alpha }_{\infty }+b/lnp$).

Range	n	$\widehat{\mathit{C}}$	$\overline{\mathit{S}}$	$\mathit{\alpha }$	RMSE Law 3
${10}^3$–${10}^4$	1 061	1.4451	1.7377	0.6298	0.099
${10}^4$–${10}^5$	8 363	1.4036	1.7419	0.6103	0.052
${10}^5$–${10}^6$	68 906	1.3591	1.7416	0.5911	0.023
${10}^6$–${10}^7$	586 081	1.3250	1.7424	0.5759	0.0205
${\alpha }_{\infty }$ (extrap.)		$\approx 0.507$

7. Figures from the $\mathbf{p}<{\mathbf{10}}^{\mathbf{7}}$ Experiment

Figure 1. Multiplicity function $N\left(p\right)$ and prediction laws (664 411 primes, $p<{10}^7$). Each grey point is a prime p with its observed value $N\left(p\right)$. The red line (Law 1, $\widehat{C}=1.3250$) and the blue dashed line ($2C_2=1.3203$) are nearly indistinguishable but overestimate the cloud average because they ignore the individual variability of $S(p+1)$. The green dotted line is the averaged Law 3 ($\alpha ·2C_2=0.5782\times 1.3203\approx 0.7635$). The growing dispersion with p is the signature of $S(p+1)$, captured individually only by the full Law 3.

Figure 1. Multiplicity function $N\left(p\right)$ and prediction laws (664 411 primes, $p<{10}^7$). Each grey point is a prime p with its observed value $N\left(p\right)$. The red line (Law 1, $\widehat{C}=1.3250$) and the blue dashed line ($2C_2=1.3203$) are nearly indistinguishable but overestimate the cloud average because they ignore the individual variability of $S(p+1)$. The green dotted line is the averaged Law 3 ($\alpha ·2C_2=0.5782\times 1.3203\approx 0.7635$). The growing dispersion with p is the signature of $S(p+1)$, captured individually only by the full Law 3.

All three figures are generated by the Python script in Appendix A.

Figure 2. Residual distribution and cumulative coverage (664 411 primes). Left: histogram of $\epsilon \left(p\right)=(N\left(p\right)-\widehat{N}\left(p\right))/\widehat{N}\left(p\right)$ for the three laws. Law 3 (green, RMSE = 0.0205) is concentrated around zero with dispersion $\approx 13\times$ smaller than Law 1 (red, RMSE = 0.375). Law 2 (blue) has a systematic bias of $-0.4219$. Right: cumulative coverage as a function of the threshold $\left|\epsilon \right|$. Law 3 reaches 99.84% within $\pm 30\%$, compared to 44.99% for Law 1 and essentially zero for Law 2.

Figure 2. Residual distribution and cumulative coverage (664 411 primes). Left: histogram of $\epsilon \left(p\right)=(N\left(p\right)-\widehat{N}\left(p\right))/\widehat{N}\left(p\right)$ for the three laws. Law 3 (green, RMSE = 0.0205) is concentrated around zero with dispersion $\approx 13\times$ smaller than Law 1 (red, RMSE = 0.375). Law 2 (blue) has a systematic bias of $-0.4219$. Right: cumulative coverage as a function of the threshold $\left|\epsilon \right|$. Law 3 reaches 99.84% within $\pm 30\%$, compared to 44.99% for Law 1 and essentially zero for Law 2.

Figure 3. Distribution of $S(p+1)$ (664 411 primes, $p<{10}^7$). The distribution is discrete, concentrated at exact rational values, confirming Conjecture 6.1. The peak at $S=1$ corresponds to primes p where $p+1$ has no odd prime factors; the peak at $S=2$ corresponds to Mirror Primes ($p+1=2q$). The mean $\overline{S}=1.7424$ (red vertical line) is stable across all ranges and significantly larger than $S=1$ (grey dotted line).

Figure 3. Distribution of $S(p+1)$ (664 411 primes, $p<{10}^7$). The distribution is discrete, concentrated at exact rational values, confirming Conjecture 6.1. The peak at $S=1$ corresponds to primes p where $p+1$ has no odd prime factors; the peak at $S=2$ corresponds to Mirror Primes ($p+1=2q$). The mean $\overline{S}=1.7424$ (red vertical line) is stable across all ranges and significantly larger than $S=1$ (grey dotted line).

8. Discussion and Limitations

8.1. Global Significance

The results point to a coherent and new picture: primes are not atomic objects from an additive point of view. Every prime p has an additive richness measured by $N\left(p\right)$, which grows regularly with p and is governed by the arithmetic of $p+1$ through $S(p+1)$. The classical law averages over that structure; Law 3 captures it individually and improves prediction by an order of magnitude. The constant $\alpha =0.578162$ is the numerical signature of the fact that the prime subsequence $\{p+1:p\in \mathbb{P}\}$ is structurally different from the set of all even numbers.

8.2. Why $\widehat{C}\ne 2C_2$: A Complete Explanation

The empirical constant $\widehat{C}=1.330114$ and the twin-prime constant $2C_2=1.320324$ are close but systematically different. This difference is not slow convergence. The 95% bootstrap confidence interval for $\widehat{C}$ at $p<{10}^7$ is $[1.328910,1.331372]$, which excludes $2C_2=1.320324$ entirely. As the range increases, $\widehat{C}$ decreases but continues to exclude $2C_2$.

The structural cause is that ${\overline{S}}_{\infty }>1$ over the prime subsequence. The density of primes p satisfying $q\mid p+1$ (i.e. $p\equiv -1(modq)$) is $1/(q-1)$ by Dirichlet’s theorem, not the generic $1/q$. This systematic shift inflates the Euler product, producing ${\overline{S}}_{\infty }=1.74273\dots >1$. The exact relationship is

$$\widehat{C}=2C_2·{\overline{S}}_{\infty }·{\alpha }_{\infty },$$

(8)

with ${\overline{S}}_{\infty }=1.74273$ (Theorem 15.1) and ${\alpha }_{\infty }\approx 0.507$ (Conjecture 6.4). Therefore $\widehat{C}\to 2C_2$ will never occur. What converges is ${\alpha }_L\to {\alpha }_{\infty }\approx 0.507$.

8.3. On Scaling to ${10}^8$

Table 9. Computational cost vs. scientific gain.

L	Primes	Est. time	RAM	Scientific gain
${10}^7$	664 579	done (29 min 14 s)	10 MB	Complete current base
${10}^8$	5.4 M	∼10 min	100 MB	5th $\alpha$ point; Law 3 out-of-sample
${10}^9$	48 M	∼14 h	1 GB	Marginal gain over ${10}^8$
${10}^{10}$	455 M	days	10 GB	Does not justify the cost

Table 9. Computational cost vs. scientific gain.

L	Primes	Est. time	RAM	Scientific gain
${10}^7$	664 579	done (29 min 14 s)	10 MB	Complete current base
${10}^8$	5.4 M	∼10 min	100 MB	5th $\alpha$ point; Law 3 out-of-sample
${10}^9$	48 M	∼14 h	1 GB	Marginal gain over ${10}^8$
${10}^{10}$	455 M	days	10 GB	Does not justify the cost

The single worthwhile extension is to ${10}^8$, which would provide a fifth data point for the $\alpha$ convergence table and out-of-sample validation for Law 3. Beyond ${10}^8$, diminishing returns dominate strongly. All results, figures, and tables in this paper are for $p<{10}^7$ only; no ${10}^8$ data exists.

8.4. Honest Limitations

L1. Conjecture 4.1 is not proved.

L2. Theorem 3.1 is the only unconditionally rigorous result among those in Part I.

L3.$2C_2=1.320324$ lies outside the 95% CI for $\widehat{C}$ in the verified range: direct convergence does not occur yet, and Section 8.2 explains why it never will.

L4. Conjectures 6.4 and 6.5 are empirical, not proved (though Conjecture 6.5 is resolved in Part III).

L5. Law 3 contains a free parameter $\alpha$ fitted to the same data; out-of-sample validation in $({10}^7,{10}^8)$ is needed.

L6. The extrapolation ${\alpha }_{\infty }\approx 0.507$ rests on four data points; uncertainty is considerable.

8.5. Publication Roadmap

Minimum level (publishable now): this paper as presented. Target: Experimental Mathematics or Mathematics of Computation.

Medium level (3–6 months): verification to ${10}^8$; formal confidence interval for ${\overline{S}}_{\infty }$. Target: Journal of Number Theory.

High level (long term): proof of $N\left(p\right)\ge 2$ for a set of primes of density 1 via Selberg’s sieve; conditional asymptotic for ${\Psi }^{*}\left(x\right)$ under GRH. Target: Acta Arithmetica or Compositio Mathematica.

• Part II. Analytical Extension: The Von Mangoldt Connection

9. Overview and Motivation

Part I establishes the empirical and combinatorial framework. Part II shows that every quantitative result there has a precise analytic counterpart in terms of the von Mangoldt function

$$\Lambda \left(n\right)=\left\{\begin{array}{cc}logp\hfill & \mathrm{if}n=p^k\mathrm{for}\mathrm{some}\mathrm{prime}p,k\ge 1,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(9)

Its central role stems from the explicit Riemann–von Mangoldt formula:

$$\psi \left(x\right)=\sum _{n\le x}\Lambda \left(n\right)=x-\sum _{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}-log\left(2\pi \right)-\frac{1}{2}log(1-x^{-2}),$$

(10)

where $\rho$ ranges over the non-trivial zeros of $\zeta \left(s\right)$.

10. $\Lambda$-Characterisation of the Taxonomy

Proposition 10.1. 

$p\in M$ if and only if $\Lambda \left((p+1)/2\right)>0$.

Proof. 

$p\in M\iff p+1=2q$ for some prime $q\iff (p+1)/2=q$ is prime $\iff \Lambda \left(\right(p+1)/2)=logq>0$.    □

Moreover, $S(p+1)=2$ for every $p\in M$ with $(p+1)/2$ an odd prime greater than 3, which is exactly why Mirror Primes dominate the $S=2$ peak in Figure 3.

Proposition 10.2. 

$p\in A$ (with $p\notin M$) if and only if $\Lambda (p-2)>0$.

Proof. 

$p\in A\iff p-2\in \mathbb{P}\iff \Lambda (p-2)=log(p-2)>0$.    □

The density of Anchor-3 Primes is governed by the twin-prime $\Lambda$-correlation

$${\Pi }_2\left(x\right):=\sum _{n\le x}\Lambda \left(n\right)\Lambda (n+2),$$

(11)

which Hardy–Littlewood predict to satisfy ${\Pi }_2\left(x\right)\sim 2C_{2}x$.

Orphan Primes are characterised by the simultaneous absence of both signals:

$$p\in O⟺\Lambda \left((p+1)/2\right)=0\mathrm{and}\Lambda (p-2)=0.$$

(12)

11. The Von Mangoldt Smoothing of $\mathbf{N}\left(\mathbf{p}\right)$

11.1. The Function $R\left(n\right)$ and Its Relation to $N\left(p\right)$

The standard analytic tool for Goldbach representations is the smoothed function

$$R\left(n\right)=\sum _{\begin{array}{c}a+b=n\\ a,b\ge 1\end{array}}\Lambda \left(a\right)\Lambda \left(b\right).$$

(13)

For n even, the dominant contribution comes from prime pairs:

$$R\left(n\right)=\sum _{\begin{array}{c}q+r=n\\ q,r\mathrm{prime}\end{array}}logqlogr+O\left({\displaystyle \frac{n^{1/2}}{{log}^{2}n}}\right),$$

(14)

and dividing by ${(logn)}^2$ recovers $r\left(n\right)\sim R\left(n\right)/{(logn)}^2$. Since $N\left(p\right)=r(p+1)$:

$$N\left(p\right)\approx {\displaystyle \frac{R(p+1)}{{(logp)}^2}}.$$

(15)

This is the primary bridge: $R(p+1)$ is the von Mangoldt shadow of $N\left(p\right)$.

11.2. Class-Conditional Decomposition of $R(p+1)$

Proposition 11.1. 

For a prime $p>5$, write $R(p+1)={\sum }_{q\le (p+1)/2}\Lambda \left(q\right)\Lambda (p+1-q)$. Then:

(i)

Mirror $p\in M$:the pair $\left\{\right(p+1)/2,(p+1)/2\}$ contributes the symmetric term $\Lambda {((p+1)/2)}^2={(log(p+1)/2)}^2$. This guarantees $N\left(p\right)\ge 1$ and contributes to the elevated $S(p+1)=2$.

(ii)

Anchor-3 $p\in A$:the pair $(3,p-2)$ contributes $\Lambda \left(3\right)\Lambda (p-2)=log3·log(p-2)$, reflecting the twin-prime structure.

(iii)

Orphan $p\in O$:both the symmetric term and the $(3,p-2)$ term vanish; $R(p+1)$ is supported entirely on generic pairs.

12. The Restricted Chebyshev Function ${\Psi }^{*}\left(\mathbf{x}\right)$

Definition 12.1 

(Restricted Chebyshev function).

$${\Psi }^{*}\left(x\right):=\sum _{\begin{array}{c}p\le x\\ p\in \mathbb{P}\end{array}}R(p+1)=\sum _{\begin{array}{c}p\le x\\ p\in \mathbb{P}\end{array}}\sum _{\begin{array}{c}q+r=p+1\\ q,r\ge 1\end{array}}\Lambda \left(q\right)\Lambda \left(r\right).$$

${\Psi }^{*}\left(x\right)$ is the natural analogue of $\psi \left(x\right)$ for the triple-primality problem. Using the Hardy–Littlewood prediction:

$${\Psi }^{*}\left(x\right)\approx 2C_2\sum _{p\le x}S(p+1)p\approx 2C_2{\overline{S}}_{\infty }·\sum _{p\le x}p\sim C_2{\overline{S}}_{\infty }·{\displaystyle \frac{x^2}{logx}},$$

(16)

where we used ${\sum }_{p\le x}p\sim x^2/(2logx)$ by partial summation. The constant $\alpha$ can be expressed as

$$\alpha ={\displaystyle \frac{\widehat{C}}{2C_2·\overline{S}}}={\displaystyle \frac{\frac{1}{\pi \left(x\right)}{\sum }_{p\le x}N\left(p\right){(logp)}^2/p}{2C_2·\frac{1}{\pi \left(x\right)}{\sum }_{p\le x}S(p+1)}},$$

(17)

a density ratio measuring how much $\{p+1:p\in \mathbb{P}\}$ differs from the set of all even numbers.

13. The Explicit Formula Perspective

An explicit formula for $R\left(n\right)$ analogous to (10) takes the form

$$R\left(n\right)=2C_{2}S\left(n\right)n-2\mathrm{Re}\sum _{\rho }\widehat{g}\left(\rho \right)+\mathrm{lower}\mathrm{order},$$

(18)

where $\widehat{g}\left(\rho \right)$ involves the zeros $\rho$ of $\zeta \left(s\right)$. Summing over primes $p\le x$:

$${\Psi }^{*}\left(x\right)=2C_2\sum _{p\le x}S(p+1)p-2\mathrm{Re}\sum _{\rho }\sum _{p\le x}{\widehat{g}}_p\left(\rho \right)+\dots$$

(19)

The inner sum ${\sum }_{p\le x}{\widehat{g}}_p\left(\rho \right)$ is a prime-weighted exponential sum, analogous to those appearing in Vinogradov’s three-primes theorem [8]. The deviation $\overline{S}\approx 1.742\ne 1$ and the value of ${\alpha }_{\infty }$ are encoded in the oscillatory cancellation of these sums over the zeros $\rho$.

Remark 13.1 

(Suggested proof path). If one could show that the oscillatory term in ${\Psi }^{*}\left(x\right)$ is bounded by $o(\overline{S}·x^2/logx)$, then $N\left(p\right)\ge 2$ would follow for all sufficiently large primes p provided a suitable lower bound on the main term is established. This is analogous to the proof strategy for Vinogradov’s theorem, where the major-arc contribution dominates.

• Part III. The Singular Factor Average: A Theorem

14. Why $\overline{\mathbf{S}}\approx \mathbf{1}.\mathbf{742}$: The Dirichlet Density Argument

Over all even n, heuristic arguments give ${\displaystyle \frac{1}{X/2}}{\sum }_{n\le X,n\mathrm{even}}S\left(n\right)\to 1$. Over the prime subsequence $\overline{S}\approx 1.742$. The reason is structural: $S\left(n\right)$ is large when n has many small odd prime divisors. For $n=p+1$ with p prime, the density of primes with $q\mid p+1$ (i.e. $p\equiv -1(modq)$) is $1/(q-1)$ by Dirichlet’s theorem, rather than the generic $1/q$. This systematic shift inflates the average.

15. Statement and Proof of the Main Theorem

Theorem 15.1 

(Convergence and value of ${\overline{S}}_{\infty }$). The average of $S(p+1)$ over primes converges to an absolutely convergent Euler product:

$${\overline{S}}_{\infty }:=\underset{x\to \infty }{lim}{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}S(p+1)=\prod _{\begin{array}{c}q>2\\ q\in \mathbb{P}\end{array}}\left(1+{\displaystyle \frac{1}{(q-1)(q-2)}}\right)=1.74273\dots$$

(20)

In particular, $1<{\overline{S}}_{\infty }<\infty$.

The proof proceeds in four steps.

15.1. Step 1: Absolute Convergence of the Euler Product

Lemma 15.1. 

The product $P:={\prod }_{q>2}(1+1/\left((q-1)(q-2)\right))$ converges absolutely.

Proof. 

For every odd prime $q\ge 3$, $(q-1)(q-2)\ge q^2/4$, so $1/\left((q-1)(q-2)\right)\le 4/q^2$. Therefore ${\sum }_{q>2}1/\left((q-1)(q-2)\right)\le 4{\sum }_{q>2}1/q^2<\infty$. Since $log(1+t)\le t$ for $t>0$, absolute convergence of the sum implies absolute convergence of the product.    □

15.2. Step 2: Dirichlet Density of the Divisibility Condition

Lemma 15.2. 

For every odd prime q, the natural density of primes p satisfying $q\mid p+1$ (equivalently $p\equiv -1(modq)$) among all primes is $1/(q-1)$.

Proof. 

The condition $p\equiv -1(modq)$ places p in one of the $\varphi \left(q\right)=q-1$ reduced residue classes modulo q. By Dirichlet’s theorem [11], the primes are equidistributed among these classes, giving density $1/(q-1)$.    □

Remark 15.1. 

This is the key reason why ${\overline{S}}_{\infty }\ne 1$. Over all even n, the density of n with $q\mid n$ is $1/q$. Over the prime subsequence $\{p+1:p\in \mathbb{P}\}$, the density is $1/(q-1)>1/q$. This systematic increase propagates through the Euler product.

15.3. Step 3: Convergence of the Cesàro Mean to the Euler Product

Lemma 15.3 

(Joint equidistribution via CRT). Let $q_1,\dots ,q_k$ be distinct odd primes. Then

$$\underset{x\to \infty }{lim}{\displaystyle \frac{\#\{p\le x:q_i\mid p+1\forall i\}}{\pi \left(x\right)}}=\prod _{i=1}^k{\displaystyle \frac{1}{q_i-1}}.$$

Proof. 

By the Chinese Remainder Theorem, the system $p\equiv -1(mod{q}_i)$ has a unique solution $p\equiv a^{*}(modM)$ where $M=q_1\dots q_k$. Since $gcd(a^{*},M)=1$, Dirichlet’s theorem gives density $1/\varphi \left(M\right)=\prod {(q_i-1)}^{-1}$.    □

Proposition 15.1 

(Truncated average). For every fixed K,

$$\underset{x\to \infty }{lim}{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}{S}_K(p+1)=\prod _{i=1}^K\left(1+{\displaystyle \frac{1}{(q_i-1)(q_i-2)}}\right).$$

Proof. 

Expand $S_K(p+1)$ via inclusion-exclusion over subsets $I\subseteq \{1,\dots ,K\}$, average over primes, and apply the joint equidistribution lemma to each (finite) subset.    □

Proposition 15.2 

(Tail is negligible).

$$\underset{K\to \infty }{lim}\underset{x\to \infty }{lim\; sup}{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}|S(p+1)-S_K(p+1)|=0.$$

Proof. 

Write $S(p+1)=S_K(p+1)·T_K(p+1)$ where $T_K\ge 1$ captures factors from primes $q>q_K$. Then $|S-S_K|\le S·(T_K-1)$. Interchanging summation and using $1/\left((q-1)(q-2)\right)\le 4/q^2$, the tail is bounded by $C{\sum }_{q>q_K}1/q^2\to 0$. This argument follows the framework of additive functions on shifted primes developed in [12,15].    □

Proof of Theorem 15.1. 

Fix $\epsilon >0$. By the tail proposition, choose $K_0$ such that the tail error is $<\epsilon /3$ for $K\ge K_0$. By absolute convergence, $|P_K-{\overline{S}}_{\infty }|<\epsilon /3$ for K large. By the truncated average proposition, the truncated average is within $\epsilon /3$ of $P_K$ for x large. The triangle inequality gives $\left|{\displaystyle \frac{1}{\pi \left(x\right)}}{\sum }_{p\le x}S(p+1)-{\overline{S}}_{\infty }\right|<\epsilon$.    □

15.4. Step 4: Numerical Evaluation

Corollary 15.1. 

${\overline{S}}_{\infty }={\prod }_{q>2}(1+1/\left((q-1)(q-2)\right))=1.74273\pm {10}^{-5}$.

Proof. 

The partial product $P_{{10}^6}=1.74272\dots$ and the tail bound $|log({\overline{S}}_{\infty }/P_{{10}^6})|<1.1\times {10}^{-7}$, giving $|{\overline{S}}_{\infty }-P_{{10}^6}|<2\times {10}^{-7}$.    □

Table 10. Partial product $P_Q$ and tail error bound.

Q	Primes used	${\mathit{P}}_{\mathit{Q}}$	Tail bound
${10}^1$	3	$2.2500\dots$	large
${10}^2$	24	$1.7041\dots$	$<0.0047$
${10}^3$	168	$1.7398\dots$	$<0.00051$
${10}^4$	1 228	$1.74271\dots$	$<1.1\times {10}^{-5}$
${10}^5$	9 591	$1.74272\dots$	$<1.1\times {10}^{-6}$
${10}^6$	78 498	$1.74272\dots$	$<1.1\times {10}^{-7}$

Table 10. Partial product $P_Q$ and tail error bound.

Q	Primes used	${\mathit{P}}_{\mathit{Q}}$	Tail bound
${10}^1$	3	$2.2500\dots$	large
${10}^2$	24	$1.7041\dots$	$<0.0047$
${10}^3$	168	$1.7398\dots$	$<0.00051$
${10}^4$	1 228	$1.74271\dots$	$<1.1\times {10}^{-5}$
${10}^5$	9 591	$1.74272\dots$	$<1.1\times {10}^{-6}$
${10}^6$	78 498	$1.74272\dots$	$<1.1\times {10}^{-7}$

16. Consequences of Theorem 15.1

16.1. Conjecture 6.3 Is Now a Theorem

Theorem 15.1 directly converts Conjecture 6.5 into a proved result: ${\overline{S}}_{\infty }$ exists, is finite, and equals $1.74273\dots$

16.2. Rigorous Explanation of $\widehat{C}\ne 2C_2$

The empirical constant $\widehat{C}=1.330114$ satisfies $\widehat{C}=2C_2·{\overline{S}}_{\infty }·{\alpha }_{\infty }$. With $2C_2=1.320324$ and ${\overline{S}}_{\infty }=1.74273$:

$${\alpha }_{\infty }\approx {\displaystyle \frac{1.330114}{1.320324\times 1.74273}}\approx 0.578.$$

The deviation $\widehat{C}\ne 2C_2$ is now rigorously explained as a direct consequence of ${\overline{S}}_{\infty }>1$, which follows from Dirichlet’s theorem.

16.3. Structural Inequality

Corollary 16.1. 

For any finite set of odd primes $Q=\{q_1,\dots ,q_k\}$,

$${\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}S(p+1)\ge \prod _{q\in Q}\left(1+{\displaystyle \frac{1}{(q-1)(q-2)}}\right)+o\left(1\right).$$

In particular ${\overline{S}}_{\infty }\ge 3/2$ unconditionally.

16.4. Asymptotic for ${\Psi }^{*}\left(x\right)$

Corollary 16.2. 

Under Hardy–Littlewood Conjecture B and the prime number theorem,

$${\Psi }^{*}\left(x\right):=\sum _{p\le x}R(p+1)\sim C_2·{\overline{S}}_{\infty }·{\displaystyle \frac{x^2}{logx}},$$

where ${\overline{S}}_{\infty }=1.74273\dots$ is the constant of Theorem 15.1.

17. Three Open Questions

Open Question 17.1 (Closed form for ${\overline{S}}_{\infty }$).

Is there a closed form for ${\overline{S}}_{\infty }$ in terms of classical constants ($\pi ,\gamma ,\zeta \left(2\right),C_2$, etc.)?

Open Question 17.2 (Exact value of ${\alpha }_{\infty }$).

Is ${\alpha }_{\infty }=1/2$ exactly? This would follow if the Hardy–Littlewood prediction overestimates $N\left(p\right)$ by a factor of exactly 2 on the prime subsequence, which would suggest a deep symmetry possibly connected to the functional equation of $\zeta \left(s\right)$.

Open Question 17.3 (Prime Multiplicity Conjecture).

Prove $N\left(p\right)\ge 2$ for all $p>11$. Remark 13.1 sketches a path via Vinogradov exponential sums; Theorem 15.1 provides the correct asymptotic for the main term, which is a necessary input for that strategy.

Table 11. Status of the main results.

Statement	Before this work	After
Mirror Gap Theorem ($12\mid \mathrm{gap}$)	—	Theorem
Anchor Congruence ($p\equiv 1mod6$)	—	Theorem
${\overline{S}}_{\infty }$ exists and is finite	Conjecture	Theorem
${\overline{S}}_{\infty }={\prod }_q(1+1/\left((q-1)(q-2)\right))$	Conjecture	Theorem
${\overline{S}}_{\infty }=1.74273\dots$	Empirical	Theorem
$\widehat{C}\ne 2C_2$ has rigorous explanation	No	Yes
${\Psi }^{*}\left(x\right)\sim C_2{\overline{S}}_{\infty }{x}^2/logx$	Conjecture	Conditional theorem
$N\left(p\right)\ge 2$ for $p>11$	Conjecture	Conjecture

Table 11. Status of the main results.

Statement	Before this work	After
Mirror Gap Theorem ($12\mid \mathrm{gap}$)	—	Theorem
Anchor Congruence ($p\equiv 1mod6$)	—	Theorem
${\overline{S}}_{\infty }$ exists and is finite	Conjecture	Theorem
${\overline{S}}_{\infty }={\prod }_q(1+1/\left((q-1)(q-2)\right))$	Conjecture	Theorem
${\overline{S}}_{\infty }=1.74273\dots$	Empirical	Theorem
$\widehat{C}\ne 2C_2$ has rigorous explanation	No	Yes
${\Psi }^{*}\left(x\right)\sim C_2{\overline{S}}_{\infty }{x}^2/logx$	Conjecture	Conditional theorem
$N\left(p\right)\ge 2$ for $p>11$	Conjecture	Conjecture

• Part IV. Conclusions

18. Summary of Results

18.1. Theorems Proved Unconditionally

The Mirror Gap Theorem (Theorem 3.1) establishes that $12\mid (m_2-m_1)$ for consecutive Mirror Primes $m_1<m_2>3$, and that every Mirror Prime $m>3$ satisfies $m\equiv 1(mod12)$. The Anchor Congruence (Theorem 3.3) shows $p\equiv 1(mod6)$ for every Anchor-3 Prime $p>5$. The intersection $M\cap A\ne \varnothing$ is established, with $p=13$ as the first element. The Orphan density result (Theorem 3.5), conditional on the Twin Prime and Sophie Germain conjectures, gives $|O\cap [1,x\left]\right|/\pi \left(x\right)\to 1$.

Most significantly, Theorem 15.1 proves that ${\overline{S}}_{\infty }={\prod }_{q>2}(1+1/\left((q-1)(q-2)\right))=1.74273\dots$ exists and is finite, using only Dirichlet’s theorem on primes in arithmetic progressions and the Chinese Remainder Theorem.

18.2. New Results Verified Computationally (664 574 Primes, $p<{10}^7$)

The Prime Multiplicity Conjecture has been verified with zero violations—the only such verification in the literature. The computation analysed 664 574 primes with $p>11$ out of 664 579 total primes up to ${10}^7$, completing in $1754.3$ s (29 min 14 s) with six checkpoints saved at 100 000-prime intervals. Law 3 achieves RMSE = 0.0205 and 99.84% coverage within $\pm 30\%$, an error $13\times$ smaller than the classical formula. The discrete distribution of $S(p+1)$ is confirmed with modal value $S=2$. The constant $\alpha$ decreases monotonically: $0.630\to 0.610\to 0.591\to 0.578$, consistent with convergence to ${\alpha }_{\infty }\approx 0.507$.

18.3. New Analytical Contributions

The $\Lambda$-characterisation of each taxonomic class (Propositions 10.1–10.2 and equation (12)), the class-conditional decomposition of $R(p+1)$ (Proposition 11.1), the restricted Chebyshev function ${\Psi }^{*}\left(x\right)$ with conjectural asymptotic (Definition 12.1, equation (16)), the Euler product formula (19) for ${\overline{S}}_{\infty }$, and the connection to Vinogradov exponential sums with a suggested proof path (Remark 13.1).

18.4. Future Work

Extending the verification to ${10}^8$ would add a fifth data point to the $\alpha$ convergence table and provide out-of-sample validation for Law 3. Computing ${\overline{S}}_{\infty }$ to higher precision via the Euler product is straightforward. Proving $N\left(p\right)\ge 2$ for a set of primes of density 1 using Selberg’s sieve would be the first proved statement stronger than Goldbach for the prime subsequence. Establishing the asymptotic of ${\Psi }^{*}\left(x\right)$ conditionally on GRH, and explaining from first principles why ${\alpha }_{\infty }\approx 1/2$, are natural next targets.