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 10¹0: 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 10¹0) 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 CAT ≈ −0. 51, and the separability finding are original to this paper.
Samir Hanna Safar (Sat,) studied this question.