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

Abstract

The O12–O19 sequence establishes that the physically relevant observable on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ is the canonical pair-level quantity $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$, with capacity exponent $\delta_{\mathrm{pair}} \approx 7.44$ consistent with the phenomenological target $\delta_{\mathrm{pair}} \in [7.4, 10.6]$. 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 $\sigma_{\mathrm{BI}}$ within the effective cascade window $[n_0, n_1]$ defined by the pre-saturation regime. Under two structural hypotheses — that $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$ is a monotone function of an effective fibre-level amplitude [H1], and that this amplitude inherits the Born–Infeld saturation bound $A_n^{\max} = c_{\mathrm{BI}}/\sqrt{\lambda_n}$ [H2] — the persistence condition is equivalent to $\delta_{\mathrm{pair}} \in [\delta_{\min}, \delta_{\max}]$, where the endpoints are determined by $\sigma_{\mathrm{BI}}$, $n_0$, and $n_1$. The lower bound $\delta_{\min} \approx 7.4$ is shown to be structurally consistent with the minimal fibre-level exponent $2\,\delta_{c,\min}$ inherited from O16–O18; the upper bound $\delta_{\max} \approx 10.6$ retains a phenomenological component, imported via the O7 structural relation from the lepton mass window, until $\sigma_{\mathrm{BI}}$ is derived from first principles. The intrinsic derivation of $\sigma_{\mathrm{BI}}$ from the Born–Infeld constraint $|\partial_t \chi_v| \leq c_{\mathrm{BI}}$ 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