FN05: The Arithmetic Realisation of Persistent Interfaces

Foundational Note 01 introduces an undivided whole together with the primitive operations of bifurcation and reconstruction. Every admissible bifurcation generates a persistent shared interface, while reconstruction recovers the original structure through that interface.The present note constructs one arithmetic–topological realisation of this operational scheme. The arithmetic whole is not identified with one privileged integer. For every real parameter s > 1, the positive integers are equipped with the zeta-normalised probability mass μs(n) = n−sζ(s) . The resulting probability space Ws = (N, μs) is the zeta-normalised arithmetic whole. The Euler product expresses the same normalisation as a product of independent prime-valuation channels. The multiplicative branch structure is encoded by Aval = Q×>0∼=⨁︂p∈PZ. Prime numbers therefore label primitive coordinate directions of the multiplicative valuation group. Positive rational numbers, including fractions, require no prior metric or geometric container: denominators are represented by negative prime valuations. The ordinary logarithm is one additive readout of the valuation lattice: log q =Σ︂pvp(q) log p. Distinct prime logarithms are linearly independent over Q. Moreover, for any two distinct primes p and q, the subgroup Z log p + Zlog q is dense in R. Hence logarithmic group completion converts discrete valuation data into a countable but arbitrarily fine additive scale readout. This logarithmic densification is not the creation of the real continuum: the image remains countable. The compact continuous phase interface instead arises from Pontryagin duality: ˆ︃Aval ∼=Π︂p∈PU(1). For a finite prime subsystem S, AS∼= Z|S| =⇒ ˆ︂AS∼= T|S|. The canonical evaluation map and Pontryagin biduality provide reconstruction: Aval ∼= ˆ︃ ˆ︃Aval. The zeta-normalised measure and the character interface are joined by the Euler–Pontryagin transform ˆ︂μs(z) =Σ︂n≥1μs(n)χz(n) =Π︂p1 − p−s1 − zpp−s , s > 1. Thus the Euler factors encode both multiplicative constitution and phase readout. The note also distinguishes two mathematically different uses of the zeta function. For s > 1, division by ζ(s) is a genuine positive probability normalisation. By contrast, 1 + 2 + 3 + · · · diverges in the ordinary sense. Its zeta-regularised value Regζ∞Σ︂n=1n = ζ(−1) = −1/12 is an analytic-continuation or finite-part residue, not the ordinary sum of the natural numbers. Equivalently, Σ︂n≥1ne−εn = 1ε2 −1/12 + O(ε2) as ε → 0+. The term −1/12 may therefore be studied as a regularised additive residue after subtraction of the divergent bulk, but it is not a probability, a total size of the whole, or an ordinary additive reconstruction value. A further strict arithmetic result is gcd{p2 − 1 : p > 3 prime} = 24. Accordingly, w24(p) = (p2 − 1)/24 defines an integer weight on every prime channel above 3, and W24(qr) = W24(q) +W24(r) defines an additive prime-square interface ledger. This arithmetic occurrence of 24 is not identified with the structural group order 24 of FBT04A, with the regularised residue −1/12 = −2/24, with a dimension, or with a physical capacity. Finally, multiplicative constitution and additive coupling are separated. Under an explicitly stated Goldbach hypothesis, an even integer N = p + q with p, q > 3 carries the load L24(p, q) =( N2 + (p − q)2 − 4)/48 . For fixed N, the most balanced admissible prime pair minimises this quadratic load. The Goldbach branch is conditional and does not enter the unconditional Pontryagin reconstruction theorem. No identification is made between the logarithm and energy squared, entropy, spacetime area, or quantum fluctuations. The stable conclusion is mathematical: multiplicative normalisation, prime-valuation bifurcation, logarithmic scale readout, compact character interfaces, additive regularisation, and bidual reconstruction form distinct but compatible layers of one arithmetic realisation of FN01.