Kaleidoscopic Harmonic Analysis on F₁: From Weyl Reflections and the Combinatorial Weyl Hamiltonian to Quantum Chaos, Cardy Entropy, and the Riemann Hypothesis

We establish a comprehensive, self-contained theoretical framework that redefines arithmetic sequences, prime number distribution, and multiplicative number theory as macroscopic physical manifestations of a combinatorial vacuum over the field with one element (F₁). By exploiting the formal algebraic inversion between unconstrained integer partitions $p(n)$ and the Weyl reflection kernel generated by generalized pentagonal numbers, we formalize the Discrete Partition--Weyl Transform (DPWT). This discrete harmonic analysis proves that any arbitrary numerical function can be uniquely decomposed into a linear combination of translated partition volumes. Within this architecture, prime numbers emerge not as stochastic anomalies, but as irreducible topological solitons carrying unit winding charge, yielding an exact, closed-form algebraic formula for the m-th prime without division or trial reduction. Extending this spectral sieve to multiplicative number theory, we decompose the M{\"o}bius, Mertens, and Liouville functions into partition spectra, resolving the breakdown of P{\"o}lya's Conjecture as a deterministic phase-resonance beating of linear pseudo-periods. Furthermore, we derive the unique, self-adjoint Combinatorial Weyl Hamiltonian (HB) via the Hermitian symmetrization of Toeplitz inversion matrices, establishing its tripartite dynamical structure: HB = Hₖᵢₙ + V_C₁ + HTd. We prove that the alternating parity weights of the Weyl kernel act as a topological gauge flux across the F₁ lattice, driving the eigenvalue spectrum into the Gaussian Unitary Ensemble (GUE) universality class of quantum chaos without requiring an external magnetic field. While bare partition kinetics induce sub-exponential volume divergence, we demonstrate that the incorporation of the Todd--Bernoulli boundary confinement operator HTd preserves strict operator self-adjointness (HB = HB^), confining all quantized resonance eigenvalues strictly to the real axis. Finally, by establishing an isomorphism between Gutzwiller's semiclassical trace formula for HB and the Guinand--Weil explicit formula, we map the asymptotic topological entropy of the partition vacuum to the Cardy formula of a $c=1$ Conformal Field Theory (CFT), framing the Riemann Hypothesis as a holographic entropy bound of quantum gravity on F₁.