The Finite–Infinite Boundary

The Finite–Infinite Boundary (FIB) is an epistemic and structural lens for why infinite aperiodic sequences, the primes above all, escape every bounded-memory representation, and every finite representation exhibits cycles that belong to the representation. This integrated edition makes the lens exact in three settings and honest in a fourth. In the profinite setting, Stone duality identifies sieve-expressible sets with clopen subsets of ℤ̂ and with unary regular languages; the closure of the primes in ℤ̂ is the unit group, a Haar-null set; the singular series is a regularised ratio of Haar measures; and the memory cost of the residue is exactly Θ(log m) in unary and open in binary. In the spectral setting, the explicit formula is lossless and finite truncation is the boundary, with log-periodic artifacts at frequencies just above the cut, dual to the periodic artifacts of a finite sieve. In the geometric setting, the critical line is a meridian of the Riemann sphere in the recentred chart w = s − ½, the functional equation is the half-turn about the polar axis, and the two natural circle charts behave oppositely: the Cayley chart is faithful and crowds every zero toward the point at infinity, while the exponential chart is periodic and, by Hlawka’s theorem, smears the zeros uniformly around the circle, except at the periods 2π/log pᵏ where Landau’s formula makes the primes audible with bias −Λ(pᵏ). The Hardy Z-function realises the phase budget θ(T)/π as hard-wall sign flips, at least five-twelfths of the time provably and always in the first 414 zeros computed here; S(T) is the phase debt. On the modular surface the scattering determinant is φ(s) = ξ(2s − 1)/ξ(2s), the discrete spectrum is unconditionally on the Selberg line and the continuous spectrum carries the Riemann zeros as resonances on Re s = ¼ iff RH: the aperiodic residue sits at the cusp. The BTZ black hole, a one-generator quotient, has periodic Matsubara resonances and is therefore the periodic shadow of this picture; the cusp is the geometry the programme’s BTZ–Selberg correspondence was missing. The Route to White is run with a null and an out-of-sample replication; the Route to Black is inverted into a duality test. Two measured results now carry two independent licences. A Cusp Count Theorem shows that the scattering term of the modular Weyl law equals the Riemann zero count to height 2T up to O(log T), so the discrete spectrum pays for the continuous one in the currency of zeros; the Eisenstein series carries both the arithmetic involution s ↦ 1 − s and the modular involution τ ↦ −1/τ, and the latter is the Hawking–Page transition of three-dimensional gravity at the elliptic point τ = i, with the Farey tail of Ford circles as its partition function. What the cosmological links prove is structural identity, not RH; the edition says so. Applications to spectral computing, medicine and cosmology are stated with labels.