The Dyadic Tritone Seam Mirror Invariance (Z2) and Time Loop Symmetry with a Bilax-Return Validation

RSM presents a unified formulation of the Resonance-Synchronization Model (RSM) that combines three tightly linked strands of analysis: Fluid-dynamic description of breaking shore-bound gravity waves, centred on a pump-augmented melt-down kernel and a coarse-graining operator for statistical “lumps”. Recursive bifurcation as a renormalisation flow that generates the observed foam-ladder spectrum m∈2Z∪7Z. Bilax-Return validation, a rigorous statistical falsifier that tests the gauge-even/odd covenant I(ιx)=I(x), P(ιx)=−P(x) on windowed data. The resulting manuscript is a single, copy-ready LaTeX file that can be submitted as a concise RSM proposal. All equations, definitions and terminology are taken verbatim from the supplied PDFs (doi 10.5281/zenodo.17877364, doi) and from the author’s own notes; no external sources are introduced. When a gravity wave approaches an even shoreline slope it “surfs on itself’’: parcels that formerly belonged to the wave become indistinguishable from those belonging to the sea. In the Resonance-Synchronization Model (RSM) this phenomenon is expressed through three mutually consistent ideas: Permutation-invariant labelling – the distinction between “surfer’’ and “surfed’’ is a pure gauge choice. Coherence pockets – localized intervals where a coherence functional Θ spikes, erasing the crest–trough label (the 0D-adjacency event). Curvature-ladder selection – the steepness κ of a curvature wall determines the admissible foam-ring counts m(κ)=m0(κκ0)1/3,m∈2Z∪7Z.(1) The present note gathers the essential mathematics, the physical interpretation, and a concrete validation pipeline into a single, self-contained document. We treat the dyadic–tritone hinge as a concrete model of a two-sheet lift with a sheet-exchange involution. The same covenant is then imposed on a breaking free-surface wave: parcel relabeling and crest/trough labeling are treated as gauge freedoms, while publishable diagnostics are required to be invariant under sheet exchange. The surf-zone “self-surfing” picture is formalized using (i) coarse-graining operators (“lumps”) that avoid choosing a privileged “surfer,” (ii) a refinement semigroup that generates recursive foam bifurcations, and (iii) a pump-augmented meltdown kernel in which a coherence functional can drive an effective mass through zero, opening a transient adjacency pocket. The validation spine is a single falsifier: the Bilax Return, requiring closure of gauge-even diagnostics and sign flip of a single gauge-odd parity marker across the paired windows.