First Results on the Purple Program: The Continuity Dichotomy at the Zero Pole, and the Joint Law of the Two Determinations

This paper settles the first two problems of the research program posed in Purple Numbers: Foundations (DOI 10.5281/zenodo.21168196). Problem 5 (continuity at the Zero pole) is solved by a dichotomy theorem: the determination weight T — being locally constant — extends uniquely to a continuous function on the profinite integers, taking the value 0 at the pole itself (the pole is undetermined: perfectly balanced) and the extreme values ±ln 15015 at ±1 (beside the full spectrum, determination is maximal); whereas the reach ρ, the dig κ, and the magnitude log n admit no continuous extension, being unbounded on every profinite neighbourhood. Residue functionals cross to the pole; magnitude functionals do not: the framework's Zero/Infinity division is a topological theorem. Problem 3 (joint determination) is resolved empirically over all 455 million primes below 10^10: the archimedean determination A(p) and the finite determination T(p) are correlated, but the correlation obeys the clean two-term law corr(A,T) ≈ −0.51/ln x, vanishing at exactly the rate the independence postulate (Axiom 6 of the Foundations) requires; the archimedean reading alone leaves the direction of the reach essentially undecided (within ±2% of even, against the finite reading's 8%–92% range); and conditionally on T the residual effect of A is below one percent. The measured coupling constant −0.51 joins the two constants found earlier in the program, giving three independent instances of the same second-order structure: a limit constant plus a 1/ln x correction. The classical neighbours of these results — the Lemke Oliver–Soundararajan biases and standard profinite topology — are attributed throughout.Notes field: Third paper of the Purple Mathematics series. Solves Problem 5 (continuity dichotomy theorem at the Zero pole, with elementary proof) and empirically resolves Problem 3 (joint law of archimedean and finite determination over 455 million primes to 10^10) of the research program stated in Purple Numbers: Foundations. Classical ingredients (profinite topology; Chinese Remainder Theorem; the Lemke Oliver–Soundararajan consecutive-prime biases, arXiv:1603.03720) are attributed in the text; the dichotomy statement, the extension values at the pole, the measured coupling constant C_AT ≈ −0.51, and the separability finding are original to this paper.