The O12--O19 sequence establishes that the physically relevant observable on Heis₃ (Z/qZ) is the canonical pair-level quantity ₀₈ₑ^can (n), with capacity exponent ₀₈ₑ 7. 44 consistent with the phenomenological target ₀₈ₑ 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 \c, q-c\ is physically admissible if and only if the canonical observable reaches a Born--Infeld saturation threshold ₁₈ within the effective cascade window n₀, n₁ defined by the pre-saturation regime. Under two structural hypotheses --- that ₀₈ₑ^can (n) is a monotone function of an effective fibre-level amplitude~H1, and that this amplitude inherits the Born--Infeld saturation bound Aₙ^ = c_/ₙ~H2 --- the persistence condition is equivalent to ₀₈ₑ _, _, where the endpoints are determined by ₁₈, n₀, and n₁. The lower bound _ 7. 4 is shown to be structurally consistent with the minimal fibre-level exponent 2\, ₂, inherited from O16--O18; the upper bound _ 10. 6 retains a phenomenological component, imported via the O7 structural relation from the lepton mass window, until ₁₈ is derived from first principles. The intrinsic derivation of ₁₈ from the Born--Infeld constraint |ₜ ᵥ| c_ is identified as the central open problem of the programme.
Jérôme Beau (Mon,) studied this question.