Omnidimensional Hypercomplex Spectral System (OHSS-17)-Pre-Spacetime Foundational and the Minimal Relational Architecture Before Spacetime

This work develops a pre-spacetime relational foundation for the OHSS-17 program. It begins from a primitive layer that contains no assumed spacetime, metric, physical units, fields, mass, energy, curvature, particles, or observers. Instead, the construction starts from three primitive distinguishable carriers, generated relational states, direct-readability and mediated-closure conditions, bounded relational reach, and a left-nested closure path.

The manuscript introduces a minimal primitive architecture in which three carriers generate three relational roots, each carrying a four-state closure cycle. This produces twelve primitive relational states without adding them externally. A key structural principle is the separation between generative origin and operational coupling: primitive carriers generate relational offspring, but do not directly couple back to them after generation. This principle later appears as a structural separation between carrier-sector and relational-sector images.

The work then presents a conditional minimality and representation theorem for the first effective algebraic realization of the primitive relational architecture. Under explicit requirements of faithfulness, linear faithfulness, readability preservation, non-degenerate signed return, no carrier-offspring coupling, bounded relational reach, and primitive left-nested closure, the first minimal effective realization has the sector form

C ⊕ F ⊕ R,

with two central-condition images, three carrier images, and twelve relational-state images. This gives a 17-basis effective structure as a conditional minimal realization of the stated primitive architecture, not as an unrestricted uniqueness claim.

The central sector represents direct readability and mediated closure. The carrier sector represents the three primitive carriers. The relational sector represents the twelve generated relational states. The effective rules include central readability, mediated closure return, non-central self-return, separation between carrier and relational sectors, bounded relational admissibility, and a left-nested product evaluation rule.

A dedicated section also introduces the spectral Lorentzian arm form. This is not presented as physical spacetime or as a direct derivation of physical special relativity. Rather, it is a spectral signed structure arising from the effective arm sector. The arm self-return rules lead to a spectral signature pattern (+---)_S, producing the spectral form

q_S(dX_S) = χ_S² dτ_S² − (dξ¹)² − (dξ²)² − (dξ³)².

Here χ_S, dτ_S, and dξᵃ are spectral quantities, not physical constants or physical spacetime coordinates. The physical Lorentzian interval appears only after a separate projection and calibration step, where χ_S dτ_S is mapped to c dt and dξᵃ is mapped to dxᵃ. Thus the work carefully distinguishes between a spectral Lorentzian structure and the physical spacetime interval.

Overall, the manuscript proposes a rigorous relational route from a non-geometric primitive layer to a first effective 17-sector algebraic realization, while explicitly separating primitive structure, effective algebra, spectral form, and later physical projection. The results are presented as conditional mathematical constructions and as a framework for further development, not as completed empirical physics.