Projective Persistence and the Physical Sub-Spectrum: A Dynamical Selection Criterion for the Capacity Exponent

The O12–O19 sequence establishes that the physically relevant observable on Heis₃(Z/qZ) is the canonical pair-level quantity σₚₐᵢᵣᶜᵃⁿ(n), with capacity exponent δₚₐᵢᵣ ≈ 7.44 consistent with the phenomenological target δₚₐᵢᵣ ∈ [7.4, 10.6]. However, the mechanism selecting $[7.4, 10.6]$ as the physical sub-spectrum has not yet been identified. The present paper proposes a projective persistence criterion: a fibre , q-c\ is physically admissible if and only if the canonical observable reaches a Born–Infeld saturation threshold σBI within the effective cascade window [n₀, n₁] defined by the pre-saturation regime. Under two structural hypotheses — that σₚₐᵢᵣᶜᵃⁿ(n) is a monotone function of an effective fibre-level amplitude [H1], and that this amplitude inherits the Born–Infeld saturation bound Aₙᵐᵃˣ = cBI/√λₙ [H2] — the persistence condition is equivalent to δₚₐᵢᵣ ∈ [δₘᵢₙ, δₘₐₓ], where the endpoints are determined by σBI, n₀, and n₁. The lower bound δₘᵢₙ ≈ 7.4 is shown to be structurally consistent with the minimal fibre-level exponent 2\,δc,min inherited from O16–O18; the upper bound δₘₐₓ ≈ 10.6 retains a phenomenological component, imported via the O7 structural relation from the lepton mass window, until σBI is derived from first principles. The intrinsic derivation of σBI from the Born–Infeld constraint |∂ₜ χᵥ| ≤ cBI is identified as the central open problem of the programme. Keywords. spectral admissibility, non-injective projection, projective persistence, Born–Infeld saturation, Weil representation, canonical pair observable, capacity exponent, physical sub-spectrum