Twin-Prime Starts, Additive Block Coverage, and a Finite Automaton in Goldbach Space

1. Introduction

The binary Goldbach conjecture asserts that every even integer greater than 2 is a sum of two primes. It remains open. The twin-prime conjecture, asserting infinitely many pairs $(p,p+2)$ of primes, also remains open. Classical context comes from the circle method and the Hardy–Littlewood heuristic viewpoint, while Chen’s theorem and subsequent sieve developments provide strong partial results in nearby additive prime problems.

This note does not attempt to prove Goldbach, and it does not claim any theorem beyond the tested finite range. Instead, it records an empirical phenomenon: when one restricts attention to twin-prime starts, their additive sums induce a block structure in Goldbach space, and the resulting coverage pattern compresses into a small symbolic automaton.

The two central themes are:

(1)

the structural relevance of twin-prime starts in the tested Goldbach range;

(2)

the finite automaton that emerges from the presence pattern generated by that structure.

A third theme, secondary but useful, is computational: the same structure suggests local filters and centered search strategies for finding Goldbach representations. In the present note, however, the main focus is descriptive and structural.

2. Twin-Prime Starts and Additive Blocks

Definition 1

(Twin-prime start). An integer $a\ge 3$ is called atwin-prime startif both a and $a+2$ are prime.

Thus the first starts are

$$3,5,11,17,29,41,59,71,\dots$$

Definition 2

(Start-sum set). Let

$$S=\{a+c:a,c\mathrm{are}\mathrm{twin}\text{-}\mathrm{prime}\mathrm{starts}\}.$$

In the computations below, S is truncated to the verified finite range.

Definition 3

(Twin-start block). For twin-prime starts $a,c$, define the associated block

$$B(a,c)=\{a+c,a+c+2,a+c+4\}.$$

Equivalently, if $s=a+c\in S$, then the block generated by s is

$$\{s,s+2,s+4\}.$$

We define the block union

$$\mathcal{U}=\bigcup _{s\in S}\{s,s+2,s+4\}.$$

An even integer n is called covered if $n\in \mathcal{U}$, and uncovered otherwise.

Observation 1.

By definition,

$$n\in \mathcal{U}⟺n\in S\mathrm{or}n-2\in S\mathrm{or}n-4\in S.$$

So coverage is determined by local membership of S around n.

3. Presence Classes in Goldbach Space

The block system above can be compared with actual Goldbach representations.

For an even integer n, one may classify its Goldbach decompositions $n=p+q$ according to whether p and q belong to some twin-prime pair.

We use the following classes:

• $GG$: both summands are twin-members;
• $GI$: exactly one summand is a twin-member;
• $II$: neither summand is a twin-member.

This yields a natural three-bit presence code for each even integer n:

$$({\mathbf{1}}_{GG}\left(n\right),{\mathbf{1}}_{GI}\left(n\right),{\mathbf{1}}_{II}\left(n\right)).$$

In the tested range, the following six macrostates occur:

$$\begin{array}{ccc}P& =& 100,\hfill \\ B& =& 110,\hfill \\ T& =& 111,\hfill \\ F& =& 011,\hfill \\ E& =& 010,\hfill \\ R& =& 101.\hfill \end{array}$$

Their qualitative meanings are:

State	Interpretation
T	total state: $GG,GI,II$ all present
B	boundary state: $GG$ and $GI$ present, but not $II$
F	defect state: $GI$ and $II$ present, but not $GG$
E	exceptional defect: only $GI$ present
P	pure twin state: only $GG$ present
R	rare mixed state: $GG$ and $II$ present, but not $GI$

4. The Finite Automaton

Ordering even integers increasingly,

$$6,8,10,12,\dots ,$$

we read the corresponding macrostate sequence as a symbolic dynamical word. This induces a finite automaton by recording allowed transitions between consecutive states.

4.1. Empirical State Counts up to ${10}^6$

In the tested range up to ${10}^6$, the state counts are:

State	B	E	F	P	R	T
Count	56	2	31	14	1	499894

Thus the automaton is overwhelmingly dominated by the total state T.

4.2. Dominant Transitions

The most important observed transitions are:

$$T\to T,T\to B,B\to T,T\to F,F\to F,F\to T.$$

Empirically, the symbolic system has:

• a dominant stable phase T;
• a small boundary layer B;
• a localized defect state F;
• a unique exceptional degeneration through E.

4.3. Local Defect Words

The typical defect clusters appear in the local words

$$TTFFF,TFFFT,FFFTT,$$

depending on the alignment of the observation window.

The initial exceptional cluster appears in the words

$$TBFEE,BFEET,FEETT.$$

This suggests that the automaton is not merely a summary of counts: it has a rigid local grammar.

4.4. Empirical Stabilization

One of the strongest observations in the tested range is the following:

Observation 2.

Up to ${10}^6$, every non-total state is confined to a finite low region. In fact, after 4208, every tested even integer lies in the total state $T=111$.

This statement is purely empirical, but exact in the verified range.

5. Observed Defects

The uncovered even integers are exactly the following 33 values:

$$\begin{array}{cc}& 94,96,98,\hfill \\ & 400,402,404,\hfill \\ & 514,516,518,\hfill \\ & 784,786,788,\hfill \\ & 904,906,908,\hfill \\ & 1114,1116,1118,\hfill \\ & 1144,1146,1148,\hfill \\ & 1264,1266,1268,\hfill \\ & 1354,1356,1358,\hfill \\ & 3244,3246,3248,\hfill \\ & 4204,4206,4208.\hfill \end{array}$$

So the defect set consists of exactly 11 consecutive triples.

Remark 1.

These 11 triples are precisely the appearances of the defect state F, together with the initial exceptional pattern involving E.

6. The Additive Law Behind the Defect State

We now explain the arithmetic origin of the defects in terms of the set S.

6.1. Consecutive 12-Gaps in S

In the tested range up to ${10}^6$, the consecutive gaps of length 12 in S are exactly:

$$\begin{array}{cc}& (88,100),\hfill \\ & (394,406),\hfill \\ & (508,520),\hfill \\ & (778,790),\hfill \\ & (898,910),\hfill \\ & (1108,1120),\hfill \\ & (1138,1150),\hfill \\ & (1258,1270),\hfill \\ & (1348,1360),\hfill \\ & (3238,3250),\hfill \\ & (4198,4210).\hfill \end{array}$$

Each such gap produces the interior triple

$$\{s+6,s+8,s+10\}.$$

Thus these gaps generate exactly:

$$\begin{array}{cc}& (88,100)\mapsto \{94,96,98\},\hfill \\ & (394,406)\mapsto \{400,402,404\},\hfill \\ & \dots \hfill \\ & (4198,4210)\mapsto \{4204,4206,4208\}.\hfill \end{array}$$

6.2. Why the Gap Produces Exactly Three Defects

If s and $s+12$ are consecutive elements of S, then there is no start sum in

$$s+2,s+4,s+6,s+8,s+10.$$

The block generated by s covers

$$s,s+2,s+4,$$

while the next block, generated by $s+12$, covers

$$s+12,s+14,s+16.$$

Hence the three even integers

$$s+6,s+8,s+10$$

lie strictly between the two blocks and are uncovered.

This explains why each observed defect occurs as a triple.

6.3. Exact Finite-Range Biconditional

The following statement is purely empirical, but exact in the tested range.

Proposition 1

(Empirical defect law up to ${10}^6$). In the tested range up to ${10}^6$, the following are equivalent:

(i) 

$(s,s+12)$ is a consecutive gap in the start-sum set S;

(ii) 

$\{s+6,s+8,s+10\}$ is an uncovered triple for the block union $\mathcal{U}$.

Moreover, this correspondence is exact and exhaustive: there are exactly 11 such consecutive 12-gaps in S, and they produce exactly the 11 observed defect triples.

Finite-range verification. 

This was checked exhaustively in the tested range up to ${10}^6$. The list of all observed uncovered triples coincides exactly with the list obtained from consecutive 12-gaps in S via the map

$$(s,s+12)⟼\{s+6,s+8,s+10\}.$$

No extra uncovered triples occur, and no extra consecutive 12-gaps in S occur in the range. □

7. From Additive Law to Automaton

The previous section explains the defect state F arithmetically.

The automaton and the additive law are therefore two descriptions of the same finite-range phenomenon:

• the automaton gives the symbolic global picture;
• the 12-gap law gives the local arithmetic cause of the defect state.

This can be summarized as follows:

Twin-prime starts induce an additive block system in Goldbach space; the presence pattern of that system compresses into a finite automaton; and the observed defect state is generated exactly, in the tested range, by consecutive 12-gaps in the twin-start sum set.

8. Computational Remarks

Although the main aim of this note is structural, the same picture suggests some algorithmic observations.

8.1. Centered Search Versus Naive Search

A naive Goldbach search usually finds a representation quickly by testing small primes first. By contrast, a centered search tests pairs

$${\displaystyle \frac{n}{2}}-d,{\displaystyle \frac{n}{2}}+d$$

with increasing d.

Empirically, these strategies serve different purposes:

• the naive search is better for finding some representation quickly;
• the centered search is better for finding more central representations and for exposing the geometry of the automaton.

8.2. Local Congruence Filtering

A local congruence filter around $n/2$ can remove many candidate offsets before primality checks. In the experiments, this reduced the number of actual primality tests without changing the centered representation found.

These computational remarks are secondary here; they are included only to indicate that the structural picture may also support local search heuristics.

9. Scope and Limitations

This note is strictly experimental. We explicitly do not claim:

• a proof of Goldbach’s conjecture;
• a proof of the infinitude of twin primes;
• any asymptotic theorem;
• any validity beyond the tested finite range.

In particular, words such as law, stability, automaton, and defect are used here in a finite-range empirical sense.

The contribution of the note is therefore limited and precise:

to isolate and document an exact empirical structure, verified in finite range, linking twin-prime starts, additive block coverage, a finite symbolic automaton, and a local 12-gap mechanism for all observed defects.

10. Concluding Remarks

The experimental picture that emerges is surprisingly rigid.

(1)

Twin-prime starts act as an additive memory in the tested Goldbach range.

(2)

Their start sums generate a block system with overwhelmingly dominant coverage.

(3)

The resulting presence pattern compresses into a small automaton with a dominant total state T.

(4)

All observed non-total behavior is confined to a finite low region.

(5)

All observed defect triples are explained exactly by the consecutive 12-gaps of the start-sum set.

Whether this phenomenon reflects a deeper infinite structure is left completely open. The present note makes no claim in that direction. It records only that, in the verified range up to ${10}^6$, the phenomenon is exact.