Practical Note III: Fixed Inner Quantum-Horizon Echo Branches in O4b Binary Black-Hole Data

This technical note reports an O4b diagnostic test of fixed inner quantum-horizon echo-delay branches in public binary black-hole gravitational-wave data. The tested delay is not fitted to the strain data. In the source frame, the physical branches are $\Delta t_{\rm src}(M;\alpha)=\alpha,\frac{r_S(M)}{c}\ln\frac{r_S(M)}{\ell_P}$, with $\alpha \in {2/3,4/3}$. The observer-frame delay is obtained by the standard cosmological time-dilation factor, $\Delta t_{\rm det}=(1+z)\Delta t_{\rm src}$. The physical content tested here is the fixed logarithmic mass scaling and the two discrete branch coefficients, not the standard factor $(1+z)$. Unlike Practical Note I, which used a blind random-30 O4a sample, the present Note III uses an LVK Table-8 highlight-30 diagnostic subset from the GWTC-5.0/O4b release. The subset is not selected by echo outcome. It is selected from events highlighted by LVK-side search information and detector-significance reporting. The analysis is therefore not presented as a blind population validation, but as a high-confidence O4b diagnostic test on externally motivated events. The analysis keeps the H1/L1-only baseline used in the O4a note. Virgo data are part of O4b, but they are not included in the primary endpoint here in order to preserve methodological continuity with the O4a H1/L1 analysis. A full H1/L1/V1 coherent echo search is left to specialist pipelines and future robustness tests. The pipeline uses public GWOSC strain data, 32-second event-centred windows, component-mass branches computed from $m_1$ and $m_2$, and 5000 null trials. A branch is counted as clean scale-compatible when it satisfies the fixed criterion $n_{\rm detectors}\geq 2$, sign coherence, and $0.85 \leq \lambda_{\rm obs/pred} \leq 1.15$, where $\lambda_{\rm obs/pred}=\Delta t_{\rm obs}/\Delta t_{\rm pred}$. For the O4b Table-8 highlight-30 H1/L1-only run, 29 events were usable in the final analysis-success subset. The physical branch set $\alpha={2/3,4/3}$ produced clean scale compatibility in 21/29 events, with 35 clean rows and an event-level Fisher probability $p_{\rm Fisher}=1.27882\times 10^{-8}$, corresponding to approximately $5.57\sigma$ one-sided Gaussian-equivalent significance before any additional global trials accounting. The same event subset, strain inputs, null calibration, and clean criterion were applied to non-physical control coefficients $\alpha \in {0.45,0.55,0.80,1.00,1.60}$. The physical branch set outperformed all tested control branches in clean-event count, clean-row count, and Fisher-combined event-level probability. The strongest control branch by Fisher probability was $\alpha=0.55$, with 16 clean events, 18 clean rows, and $p_{\rm Fisher}=6.17229\times 10^{-8}$. Control branches are not expected to vanish, because a finite-width echo/correlation pattern may produce partial responses at nearby non-physical coefficients. The discriminating question is whether the physical branch pair ${2/3,4/3}$ organises the event-level delayed-correlation structure better than the controls. In this O4b Table-8 highlight diagnostic test, it does. This note should be read as a reproducible technical report and as a prompt for independent replication. The central claim is deliberately limited: in an externally motivated O4b high-confidence subset, the fixed physical branch pair $\alpha={2/3,4/3}$ produces the dominant clean scale-compatible component-mass echo-candidate structure among the branch sets tested here.