A Golden-Ratio Ladder and a Delocalisation-Saturated Participation Bridge for the Hydrogen-Bond Network of Liquid Water

We organize the ultrafast hydrogen-bond timescales and far-infrared/terahertz band centres of liquid water on a single golden-ratio ladder, anchored to the 80 fs H-bond contraction time and indexed by a loop-free tetrahedral Bethe sequence. Both data sets fall inside the rung windows set by their shell index, placing them on one spacing from a single time anchor. A separate empirical energy anchor gives a calibrated map of the freezing and density maximum. A TIP4P/Ew analysis shows that participation is delocalisation-saturated rather than linear in shell size, and we fit a corresponding saturating law. The low-frequency intermolecular modes are delocalised over the network, with coherent shell-breathing extending over about one coordination shell, a result stable with temperature. The ladder placements are therefore an organizing construction with explicit calibrations. A statistical assessment (a scan over alternative logarithmic bases and a null-model randomisation) confirms that the bracketing does not statistically single out the golden ratio, so the physical conclusions rest on the measured participation law, not the ladder. Together these establish a calibrated bridge between hydrogen-bond-network topology and vibrational participation, with the ladder supplying the rung spacing and the molecular dynamics the participation law that populates it.