Omnidimensional Hypercomplex Spectral System (OHSS-17)-Pre-Spacetime Foundational and the Minimal Relational Architecture Before Spacetime
Year: 2026
Type: preprint
Abstract: OHSS-17: From a Pre-Spacetime Algebraic Structure to Spectral Time, Causality, Quantum Readout, and Particle-Like Sectors -V1 OHSS-17 is a long-term mathematical research program investigating a radical possibility: space, time, light, fields, particles, and quantum measurement may not be independent fundamental ingredients, but different readouts of a deeper algebraic structure that precedes spacetime. Rather than beginning with a pre-existing manifold, metric, distance, velocity, particle, or physical field, the project begins with a finite system of basis states, transition relations, and a real nonassociative multiplication law. It then asks, step by step and under explicit mathematical verification gates: Can an internal clock, historical directionality, causal structure, Lorentzian signature, null propagation, quantum observables, and particle-like representation sectors emerge from the consistency and symmetry of this structure itself? The project develops a real 17-dimensional algebra with basis e₀, e₁, e₂, …, e₁₆. These basis elements are organized into three principal sectors: Cₐₙ꜀ = span{e₀, e₁₆}, the central or anchor sector; Fₐᵣₘ = span{e₁, e₂, e₃}, the three-arm sector; Rᵣᵢₘ = span{e₄, …, e₁₅}, a twelve-position rim or address cycle. The element e₀ is the algebraic identity, while e₁₆ is a central anchor satisfying e₁₆² = e₀. The central sector admits the exact splitting P₊ = ½(e₀ + e₁₆), P₋ = ½(e₀ − e₁₆). An effective quotient is then defined by Q₁₆ = V₁₇ / ℝP₋, in which the images of e₀ and e₁₆ become a single effective unit: q(e₀) = q(e₁₆) = 1Q. This quotient does not replace or erase the raw algebra. It provides a typed effective carrier on which certain transformations become well-defined even when they are not automorphisms of one fixed raw multiplication table. A Frozen and Auditable Raw Algebra The project is based on a frozen multiplication table containing 17² = 289 ordered products of basis elements. The table is not altered to force later physical interpretations. OHSS-17 strictly distinguishes between: raw multiplication; left-multiplication operators and their operator envelope; the associator generated by nonassociativity; quotients and projections; spectral representations; metric exports; quantum readouts; and conditional physical bridges. One of the central raw-sector conditions is Fₐᵣₘ Rᵣᵢₘ = 0, Rᵣᵢₘ Fₐᵣₘ = 0. Thus, there is no direct raw binary arm–rim multiplication channel. However, a vanishing binary product does not imply that every higher relation vanishes. Because the algebra is nonassociative, the associator can generate a nonzero trilinear bridge without changing the raw multiplication table. The project also fixes left-nested evaluation and separately examines the Jacobi and Malcev behavior of the associated bracket, the left-multiplication envelope, and the derivation algebra. The framework is therefore intended as a reproducible algebraic construction, not as a collection of numerical coincidences or informal physical analogies. The Spectral Clock: A Returning Cycle and a Non-Returning History The primitive internal motion lives on the twelve-position rim. A one-address clock step is represented by Uclk, with Uclk¹² = I. The project makes a necessary distinction between two operations: the address microtick Uclk, and the completed status pulse Uclk³. A microtick advances the address by one position. A status pulse collects three microticks and completes the shortest transverse closure of the root-sector event cycle. The primitive oriented event pairs form a three-step cycle: (a₁, a₂) → (a₂, a₃) → (a₃, a₁). The reverse clock orientation produces the oppositely oriented cycle. Each address microtick is also connected, through an incidence map, to one primitive root edge. The clock therefore coordinates an oriented event record and an elementary displacement record without turning either object into a new raw product. The root-edge displacement associated with the clock separates into: a periodic transverse component; and a persistent uniform component. After three microticks, the transverse component cancels: P⊥ Iₑₚ (I + Uclk + Uclk²) = 0. The uniform displacement does not cancel. The first three cumulative window energies are E₁ = 6, E₂ = 6, E₃ = 0. One or two ticks retain a transverse defect. Three ticks produce the first complete transverse closure. After twelve microticks, the address clock returns to its original periodic configuration, while the accumulated history, event count, central record, and uniform displacement continue to grow. OHSS-17 therefore distinguishes between the return of the clock pointer and the return of physical or causal history. The clock is periodic, but its lifted history is not. This provides a mathematical mechanism by which a locally periodic structure can support a globally accumulating arrow of history without assuming an external linear time coordinate at the fundamental level. Transported Raw Fibres and the Effective Quotient A one-step address transformation is not treated as an automorphism of one permanently fixed raw multiplication frame. Instead, the project defines a transported family of raw products: μₙ(x, y) = Tₙ μ(T₋ₙx, T₋ₙy). The one-step map then acts as an isomorphism between successive raw fibres: T₁ μₙ(x, y) = μₙ₊₁(T₁x, T₁y). The transported products satisfy a two-fibre periodicity: μₙ₊₂ = μₙ. Thus, the twelve-address history retains twelve distinct event positions while alternating between two raw multiplication frames. At the quotient level, the clock defect lies in the removed P₋ direction. Consequently, the induced transformation becomes an effective automorphism on Q₁₆. The raw algebra and the quotient therefore play different roles: the raw fibres preserve the full nonassociative history; the quotient supports a unified effective clock action; and the central effective unit 1Q remains fixed. The λ Ladder and Spectral Transport The project introduces an internal ladder of real spectral weights: λₖ = φ^(k/16), for k = 0, 1, …, 16, where φ is the golden ratio. The ladder satisfies λ₀ = 1, λ₈ = √φ, λ₁₆ = φ, and λₖ λ₁₆₋ₖ = φ. These quantities do not modify raw multiplication. They are not primitive probabilities, lengths, energies, or interaction strengths. They are typed spectral transport and readout coefficients. The project shows that λ-weighted transport can preserve the event-cycle structure when operators and incidence maps are transported consistently. It does not create additional time, change the number of clock ticks, or turn the λ weights into raw physical units. The λ ladder therefore acts as a structured spectral deformation of representation coordinates, not as an alteration of the underlying algebraic law. Window Closure and Internal Damping OHSS-17 constructs internal window and damping operators on the rim. The interaction window detects the completion of the shortest three-tick event block. The weighted spectral damping operator is treated separately. It is not promoted to a fundamental clock or universal event generator. The static damping operator does not generally commute with the clock or with the completed three-tick pulse. It is therefore classified as a transported response or coarse-graining mechanism associated with the clock fibres, not as the primitive scheduler of internal events. A clock-covariant family may instead be defined by transporting the damping operator along the clock orbit. This distinction prevents a response operator from being mistaken for the generator of time itself. From the Central Sector and the Arms to Lorentzian Signature Lorentzian geometry is not assumed inside the raw multiplication table. It appears only after a typed metric readout is exported from the central and arm sectors. The central direction contributes one positive component, while the three arms contribute three negative components, producing the signature (1, 3). For a vector v = t e₀ + x₁e₁ + x₂e₂ + x₃e₃, the exported quadratic form is g(v, v) = t² − x₁² − x₂² − x₃². This gives the standard causal classification. If g(v, v) > 0, the vector is timelike. If g(v, v) < 0, the vector is spacelike. If g(v, v) = 0 with v ≠ 0, the vector is null. A null vector is not the zero vector. Its metric norm vanishes because the temporal and spatial contributions cancel exactly, while the vector itself retains a nonzero direction and transport content. The null vectors form the causal cone separating timelike and spacelike directions, with future and past branches determined by the sign of the temporal component. Light is therefore not inserted into the raw algebra as a predefined particle or velocity. It first appears as the causal boundary between temporal and spatial directions after the Lorentzian metric readout has been constructed. Internal Propagation and the Speed Coefficient In the transverse traceless perturbation sector, the project obtains a propagation law of the form ω² = N₀² |d|². For unit internal proper lapse, N₀ = 1, this becomes ω² = |d|². The internal causal propagation coefficient is therefore cint = 1. Within the declared internal normalization, this means one address microtick ↔ one primitive root edge of unit norm. Three concepts that were initially independent then become linked: the clock tick provides an internal unit of ordering; the root edge provides an internal unit of displacement; the null cone supplies the causal boundary relating them. The completed status pulse Uclk³ contains three primitive root-edge steps. Its net coarse-grained displacement has norm √3, but this net displacement is not used to redefine the primitive causal speed. The relation cint = 1 is an internal dimensionless normalization. It is not, by itself, a derivation of the measured SI value of the speed
Author: SOLTAN, MAGED
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169676390