We take as sole axiom the logical instability of non-existence—the postulate that a state of nothing is not self-consistent, so existence is necessary—and trace its physical consequences. The axiom is metaphysical and pre-temporal; what follows from it is a chain of mathematical necessities. Because the totality of existence admits no external reference frame, its ground state must be self-consistent under its own description; together with the CPT theorem of quantum field theory on curved spacetime, this forces the ground state to be the CPT-invariant de Sitter vacuum dS4/CPT=RP4. From this real projective four-space we derive, as a connected sequence, the spacetime dimension n=4, the unique Cohesion Field action, the Clifford tori and the golden ratio φ as the self-similar stable modes, the Z3 ternary class structure, and the shell number N=33=3M with M=11. These fix the geometric origin of the IHC predictions developed in the companion papers: the fine-structure constant α⁻¹=137.036, the weak mixing angle sin²θW=3φ⁻¹/8, the coherence parameter βcoh=6cos(π/23), the vacuum suppression factor β=1344.5 (matching the observationally required value to 0.04%), an exactly constant dark-energy equation of state w=−1 derived from the conformal structure (Einstein-frame flatness), the dark-energy fraction ΩΛ=(5−√5)/4=0.69098 (from ρm/ρΛ=1/√5, derived via the fold involution and the normalisability of the vacuum mode at the identified pole of RP4), and the charged-lepton mass scaling m∝φᵏ. We distinguish throughout what is deductive from what is not: the geometric and structural content follows from the axiom by standard mathematics, while the quantitative normalisations connecting this structure to measured magnitudes—a single overall mass scale, and the unit coefficient of the self-observation correction e⁻ᵅ to the electron mass—are identified as the points where the metaphysical axiom must be bridged to physical numbers, and are flagged explicitly. For the e⁻ᵅ correction we derive the exponential form (by two independent routes), the electromagnetic coupling, and the selection of a timeless topological count over a dynamical magnitude, isolating the single residual identification to one sharply stated point.

This paper serves as the foundational companion to the IHC series.