Structural Projection and Invariant Ratios From Activated Trajectories to Relational Signatures

This work introduces a pre-physical structural framework for projecting activated relational histories into dimensionless relational signatures while preserving all structural invariants fixed by a minimal axiomatic set.The construction deliberately avoids introducing metric spacetime, physical dynamics, quantum fields, inflationary mechanisms, or phenomenological parameters. Instead, it operates entirely at a structural level, addressing the question of which relational signatures are admissible once activation has occurred.A discrete structural support sequence is fixed by layered slicing and a no-double-counting rule. Under reversibility and minimal coherence stabilization enforced by Structural Coherence Units (SCUs), every activated trajectory admits a canonical triple partition. This selects the ratio 5 : 27 : 68, not as a numerically exact prediction or empirical fit, but as a canonical representative of an admissible structural class invariant under projection, coarse-graining, and structural equivalence.Projection via truncated simplex inheritance leads to discrete angular organization and robust structural constraints, including suppression of low-order angular modes, absence of primordial tensor structure, expansion–collapse duality, and a structural preference for effective three-dimensionality.The framework does not claim direct physical observables. Any connection to cosmological data or phenomenology requires additional structure and belongs to a later stage of the theoretical program. Optional appendices discuss how non-ideal structural configurations may generate persistent large-scale signatures without invoking physical causality.The work is intended as a foundational contribution to structural and relational approaches to cosmology and theoretical physics.