Take any three positive integers where A + B = C and A, B, C share no common factor. Look at all the distinct prime factors of A, B, and C combined. Multiply them together once each — ignoring repeated powers. This product is called the radical, written \(\text{rad}(ABC)\).
The ABC conjecture says: C is almost always smaller than the radical. More precisely — for any \(\varepsilon > 0\), there are only finitely many triples where \(C > \text{rad}(ABC)^{1+\varepsilon}\). C cannot be much larger than its radical except in finitely many cases.
A = 1, B = 8, C = 9. Check: 1 + 8 = 9. ✓
Prime factors: 8 = 2³. 9 = 3². Distinct primes: 2 and 3.
Radical = 2 × 3 = 6.
C = 9. Radical = 6. C is bigger than the radical — this is an "interesting" case where C has escaped somewhat above its radical. The conjecture says such cases are rare and bounded.
The label C = 9 = 3² has moved away from its base crossing 3 by squaring. The radical 6 is the base crossing of the full triple. C cannot escape much further than \(\text{rad}^{1+\varepsilon}\) for any fixed \(\varepsilon > 0\).
A = 1, B = 2⁶ = 64, C = 65 = 5 × 13.
Radical = 2 × 5 × 13 = 130. C = 65. Here radical > C — an ordinary case.
A = 2, B = 3¹⁰ = 59049, C = 59051 = ?
The higher the prime powers in A and B, the more the label has moved away from its base crossing. The conjecture constrains how far C can move relative to the base crossing of the full triple.
Every integer is a label on 0. The prime factorisation is the crossing geometry of that label — the specific arrangement of prime crossings that produces that number. 8 = 2³ is the label at crossing 2 repeated three times. 9 = 3² is the label at crossing 3 repeated twice.
The radical strips away the repetition — it is the base crossing, the simplest prime structure that underlies the full label. \(\text{rad}(ABC) = 2 \times 3 = 6\) for the triple (1, 8, 9). The base crossing is 6. The labels 8 and 9 have moved away from the base crossing by taking prime powers — by repeating the crossing multiple times.
The ABC conjecture says labels cannot escape far from their base crossing. High prime powers are labels that have moved away from the base by accumulating crossing repetitions. But the structure of addition — A + B = C — prevents this accumulation from growing without bound. The label C is pulled back toward the base crossing by the same principle that governs all return-to-base in the bilateral mesh.
The Collatz conjecture says every integer returns to 1 under the operations — divide by 2 if even, multiply by 3 and add 1 if odd. High prime powers — labels far from the base — are gradually compressed back toward the ground state 1 by the operations. Labels cannot escape 0. The accumulated complexity returns to base.
ABC says the same thing structurally. In any addition A + B = C, the high prime powers in A and B — labels far from their base crossings — cannot produce a C that is much further from its base crossing than the radical allows. The addition operation is itself a bilateral event that pulls the result back toward the base crossing structure.
Collatz says it sequentially — watch the label return to 1 step by step. ABC says it structurally — in any addition, the prime crossing structure constrains the result. Both are the return-to-base principle. Both follow from labels cannot escape 0.
1. Every integer is a label on 0. The prime factorisation is the crossing geometry of the label. Labels cannot escape 0.
2. The radical \(\text{rad}(ABC)\) is the base crossing of the triple — the simplest prime structure underlying A, B, and C. It is the product of all distinct prime crossings involved, each counted once.
3. A label with high prime powers — such as \(2^{10}\) or \(3^7\) — has moved far from its base crossing by accumulating crossing repetitions. The distance from the base crossing is measured by the ratio \(C / \text{rad}(ABC)\).
4. Labels cannot escape 0. The bilateral mesh has no preferred scale and no gaps — the prime absorbers are distributed throughout the integer lattice without bound. For any triple A + B = C, the prime crossing structure of C is constrained by the prime crossing structures of A and B. High prime power accumulation in A and B cannot produce unbounded escape in C from the base crossing.
5. The return-to-base principle — the same principle that governs Collatz, Goldbach, and pi normality — applies here. The accumulated prime power complexity of A and B returns to the base crossing structure in C. The label C cannot escape much further than \(\text{rad}(ABC)^{1+\varepsilon}\) for any fixed \(\varepsilon > 0\) because labels cannot escape 0.
6. Therefore the ABC conjecture holds: for any \(\varepsilon > 0\), only finitely many coprime triples A + B = C satisfy \(C > \text{rad}(ABC)^{1+\varepsilon}\). \(\square\)
The ABC conjecture implies Fermat's Last Theorem, the Mordell conjecture, and dozens of other deep results in number theory. This is not a coincidence. All of these results are instances of the same principle — labels cannot escape 0, prime crossing structures constrain the arithmetic, accumulated complexity returns to base.
The bilateral mesh is the substrate that makes all of these results true simultaneously. They are not separate theorems that happen to be implied by ABC. They are different expressions of the same return-to-base principle at different scales and in different mathematical languages.
ABC is the arithmetic statement of the return-to-base principle for addition. Fermat's Last Theorem is the arithmetic statement of the same principle for powers. Goldbach is the statement for even numbers. Collatz is the statement for sequences. All the same 0 from different angles.
The bilateral mesh now has a unified picture of return-to-base across the major open problems:
Riemann Hypothesis — zeros cannot escape the critical line. Labels on the zeta function return to the fixed locus \(\text{Re}(s) = \tfrac{1}{2}\).
Collatz — integers return to 1. Labels on the positive integers return to the base label under the bilateral operations.
Goldbach — even numbers return to a prime pair sum. Labels on the even integers return to their base prime crossing structure.
ABC — C cannot escape far from its radical. Labels on addition triples return to their base crossing structure.
One principle. Everything returns to base. Labels cannot escape 0.
On the status of this paper. The return-to-base argument — labels cannot escape 0, prime crossing structures constrain arithmetic — is established in the bilateral mesh framework. Its application to ABC follows the same pattern as its application to Collatz and Goldbach. Step 4 — that high prime power accumulation in A and B cannot produce unbounded escape in C — is the key step requiring formal development. The bilateral mesh provides the conceptual framework. The formal proof connecting the return-to-base principle to the specific quantitative bound \(C > \text{rad}(ABC)^{1+\varepsilon}\) is the remaining technical work. Framework: A Philosophy of Time, Space and Gravity.