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This work develops a theoretical framework for understanding fractional cores and their implications in higher dimensional spaces.

June 4, 2026Open Access

M36b Fractional Cores - Rooms by Nesting Depth, the Two-Dimensional TetraCore, the Three-Dimensional PentaCore, and the Multiverse Direction

Key Points

This work develops a theoretical framework for understanding fractional cores and their implications in higher dimensional spaces.
Introduced a coordinate principle for higher operational structures involving Rooms and Corridors.
Developed mathematical formulations to explore the properties and ranks of various Cores.
Compared TAdd, TMult, and TApow in the context of rank classifications.
The TetraCore is established as a two-dimensional structure with specific coordinate points and ranks.
The new PentaCore is revealed as a three-dimensional framework, influencing both integer and hyper-dimensional structures.
Corridor interpolation allows for continuous connections between half-rank gravitational shadows.

Abstract

M36b develops the theory of Fractional Cores by introducing a coordinate principle for higher operational structures: a Room is a nesting depth, while a Corridor is the internal combiner transported through that nesting. For any conjugate pair (f, f^-1) and combiner C, the Room-n operation is Core₍, ₂ (a, b) = fⁿ (C (f^-n (a), f^-n (b) ) ). At n = 0 the scaffold cancels, so Room 0 is exactly the combiner. This turns Room 0 into a structural identity rather than a lower-rank shadow. The main result is that the TetraCore is two-dimensional. Its coordinate is TCr, k (a, b) = tetBʳ (SCk (slogBʳ (a), slogBʳ (b) ) ), where r is the TetraCore Room and k is the Symmetric-Core Corridor. The associated HyperCore rank is R = (k+1) (1-r) + 4r. Thus TC0, k = SCk, while the half-Rooms r = 1/2 form the TetraZeug ladder: TC1/2, 0 = TetraZeug at R = 2. 5, TC1/2, 1 = MultitetraZeug at R = 3, TC1/2, 2 = ApowtetraZeug at R = 3. 5. M36b also demystifies the Lucky Zone: multiplication appears as TC0, 1 = SC1 = Mult, a direct Room-0 Corridor identity. It then studies the open problem of the R=4 Triad member, comparing TAdd, TMult, and TApow, with the Apow analogy favouring TApow while leaving the final identification open. The monograph further introduces Corridor interpolation. For s in 0, 1, TCr, k+s interpolates between adjacent Corridors, with rank R = (k+s+1) (1-r) + 4r. At r = 1/2 this gives a continuous passage between half-rank gravitational shadows, for example from R = 2. 5 to R = 3. Finally, M36b lifts the same coordinate logic to the PentaCore. The PentaCore is three-dimensional, PCx, y, z, with PC Room x, TC Corridor y, and SC Corridor z. Its rank formula is R = ( (z+1) (1-y) +4y) (1-x) + 5x. The pure Addition direction PCx, 0, 0 gives R = 1 + 4x, which passes through the known integer and half-integer HyperCore ranks from R = 1 to R = 5, including the Inter-Super-Log Etage (ISLE) at R = 4. 5. In general, the Cₙ-Core has dimension n-2, so higher Cores become higher-dimensional holographic lifts of lower-Core content. Conjecturally, TetraCore has gravitational connections to GUT at R=2. 5 Heun Hyperzeug with its TetraZeug. In the Multiverse setting, also conjecturally, PentaCore with its dyadic fractional Rooms has influence on both integer Etages (and Triads) as well as each hyperzeug, including the new one: ISLE at R=4. 5.

M36b Fractional Cores - Rooms by Nesting Depth, the Two-Dimensional TetraCore, the Three-Dimensional PentaCore, and the Multiverse Direction

Key Points

Abstract

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