The Synchronization Theorem: A Minimal-Product Characterization of Four Coprime Periods

This paper proves a genuine mathematical theorem that replaces a previously circular claim in the Emergence Canvas Model. The original construction asserted four "dynamic primitives" with periods that are prime, then observed that primes are pairwise coprime, and called this the "Synchronization Condition." This is circular: any four distinct primes are automatically pairwise coprime—coprimality does no selective work. What This Paper Does It restates the construction so that it is no longer circular and proves a genuine theorem in its place: Among all four-element sets of pairwise coprime integers ≥ 2, the product is minimized uniquely by {2, 3, 5, 7}, and primality is a consequence of the minimization, not an input to it. The proof is an exchange argument: any composite, prime power, or larger prime can be replaced with a smaller unused prime, strictly decreasing the product without violating pairwise coprimality. Repeating this exchange until no smaller unused prime remains yields the four smallest primes. The result is verified independently by exhaustive search over all 455,126 four-element subsets of {2, ..., 60}: the minimum product found is 210, achieved by exactly one subset, {2, 3, 5, 7}, and every element is prime. What This Theorem Does and Does Not Imply What follows: The theorem provides a non-circular replacement for a previously circular claim, closing a genuine logical gap. Later constructions in the model can be built from the same four numbers rather than introducing new unexplained constants at each step. This is a real, checkable economy—but it is an aesthetic and structural virtue of the model's bookkeeping, not an empirical confirmation. What does not follow: The theorem does not extend to gauge structure, spatial dimensionality, or particle creation. Three attempted routes to such an extension were tested and each was found to be either trivial, arbitrary, or generic rather than specific to this result: · Ranking the four primes and taking the first three reproduces "1, 2, 3" only because ranking any three items always yields ranks 1, 2, 3—true of any three-element subset of anything, and therefore explains nothing specific to this theorem.· Dropping one primitive to leave three requires an unmotivated choice of which primitive to discard.· Splitting the four primes by parity (2 is even, 3/5/7 are odd) gives a 1-versus-3 split, but this split holds for any set of primes that includes 2 and would hold regardless of the specific values {2, 3, 5, 7}. A correction identified: The paper identifies and corrects an arithmetic inconsistency in the existing eigenvalue derivation that depends on this theorem's output. The doublet diagonal entry stated as a = t_2 t_3 t_4 = 1/42 does not yield the claimed eigenvalues 1/6 and 1/42. The value that actually reproduces the claimed result is a = 2t_2 t_4 = 2/21—a different combination, never stated in the derivation as written. The final eigenvalues {1/6, 1/42, 1/210} are correct given a = 2/21; the stated justification for that entry is not. Why This Matters The paper draws a hard boundary around what this theorem implies. It correctly reproduces the numerical content used later in the model's eigenvalue construction (Pillar III), but only once an additional, independent postulate about gauge multiplicity is also supplied—and that additional postulate cannot itself be derived from the theorem. Gauge multiplicity and spatial dimensionality remain independent postulates. Treating any of the three failed routes above as a genuine derivation would be exactly the numerology failure this paper is trying to avoid. The theorem should be cited for exactly what it establishes—a specific, provable minimization result—and not stretched to underwrite the model's broader and empirically untested claims. It is a genuine mathematical result that corrects a logical gap, but it does not and cannot bear the weight of a physical theory. Keywords: synchronization theorem, coprime periods, minimal product, emergence canvas model, dynamic primitives, mathematical proof, exhaustive verification, eigenvalue inconsistency, gauge multiplicity, independent postulates, scope boundary