The KM Axiom System for Dynamic Prime Generation** *Two-Clock Legality, Semi-Conservative Limits, and Arithmetic-Geometric Bridges

We formulate the KM axiom system for dynamic prime generation. The system uses two clocks: an additive magnitude horizon and a multiplicative dependency depth. Integers and primes may be generated as latent objects at one depth, while activation as a factor witness is controlled by the current horizon. Missing a prime not exceeding the square root leaves its square uncovered; activating a prime above the square root is anticipatory. We prove that the prime witnesses not exceeding \(\sqrt X\) form the unique minimal library that certifies every composite up to \(X\). Finite KM stages are partial arithmetic structures. Their directed limit is the standard natural-number semiring, and every finite computation agrees with ordinary arithmetic. The global illegality predicates and bridge axioms are not definitionally conservative. We call this combination *semi-conservative*: finite arithmetic and the direct limit are fixed, while global statements involving activation, spectral pressure, and absolute illegal chains may have additional strength. The global KM axioms prohibit infinite essential chains of anticipatory factor calls and require every bridge-generated illegal call to propagate to the next expansion layer. Explicit bridge axioms connect off-critical zeta zeros and failures of the twin-prime, Goldbach, and Legendre statements to essential illegal seeds. Within KM, these bridges yield the corresponding KM theorems by contradiction. Over the bridge-free KM base, however, each bridge is equivalent to its associated classical conclusion. Thus the bridge mechanism provides an exact decomposition of logical strength, rather than an unconditional proof of an open conjecture. We introduce signed spectral debt \(d\psi-dx\), derive its centered logarithmic-derivative transform, and establish an exact carry-dependency flow decomposition. The framework is covariant under exponential partition bases and admits a universal direct-limit characterization, finite-type two-parameter persistence invariants, and extensions to prime ideals and Dirichlet \(L\)-functions. Radical-height and elliptic-curve channels provide KM extensions of the abc and Birch–Swinnerton-Dyer conjectures. The proposed square-root proof of abc is shown to reverse a crucial inequality; abc is instead formulated through bounded radical debt and the Frey–Szpiro transduction. Curve-indexed spectral charge yields KM–BSD rank and leading-term theorems relative to explicit bridges. Their strengths are again equivalent to the corresponding classical statements over the bridge-free base. Finally, we isolate the abstract factorization boundary: well-founded divisibility guarantees atomic decompositions, whereas uniqueness additionally requires every atom to be prime. Unique factorization is therefore an algebraic base theorem and is not equivalent to RH or any prime-distribution estimate. ## Keywords dynamic prime generation; additive completion; multiplicative monoids; KM axioms; temporal legality; anticipatory factors; absolute illegal chains; semi-conservative extension; direct limits; signed spectral debt; Riemann hypothesis; generalized Riemann hypothesis; twin-prime conjecture; Goldbach conjecture; Legendre conjecture; abc conjecture; modified Szpiro conjecture; Frey curves; Birch–Swinnerton-Dyer conjecture; elliptic curves; arithmetic-geometric bridges; factorial monoids; partition covariance.