Higher-Dimensional Extensions of the Transcendental Constants, Attractors Beyond π, e, φ, and Feigenbaum

This paper extends the transcendental-layer constants program by arguing that certain special constants
are not merely exceptional mathematical objects, but higher-order coherence resonance attractors
associated with distinct dimensional, summational, recursive, statistical, factorial, and alternating
structures of emergence.

The earlier transcendental-layer paper established π, e, φ, and Feigenbaum structure as primary invariants of closure, transformation, harmonic proportion, and threshold scaling.

The present paper develops the next stage: a wider family of special constants interpreted as resonance
markers of deeper coherence architecture.

In particular, the paper examines Euler-Mascheroni, Apéry’s constant, Conway’s constant, Khinchin’s
constant, Glaisher-Kinkelin, and Catalan’s constant as higher-order markers of asymptotic adjustment,
volumetric nesting, recursive cascade growth, statistical continued-fraction coherence, factorial
layering, and alternating phase structure.

The term “transcendental program” is used here in an ontological and structural sense, not as a claim that every constant discussed has been proven transcendental in the strict technical sense of number theory.

The central claim is that these constants should not be treated as a miscellaneous afterlife of famous
mathematical constants. Rather, they belong to a higher-order resonance family. Each marks a lawful
attractor within a distinct mode of coherence evolution. Some govern the boundary between discrete and
continuous summation. Some govern nested volumetric shelling. Some govern self-replicative growth
cascades. Some govern statistical order emerging from apparent irregularity. Some govern factorial
layering. Some govern alternating coherence structures. In this sense, the paper proposes a deeper
ontology of special constants within the transcendental-layer program.

The aim of the paper is not to claim that every special constant is automatically ontologically
fundamental. It is instead to show that a specific class of constants can be interpreted as coherence
resonance attractors arising in higher-dimensional or higher-order structures of emergence. This
broadens the transcendental layer from first-form invariants into a richer resonance architecture linking
mathematics, dimensionality, recursion, statistical order, and phase structure.

Keywords: transcendental program; special constants; coherence resonance attractors; Euler-Mascheroni
constant; Apéry’s constant; Conway’s constant; Khinchin’s constant; Glaisher-Kinkelin constant; Catalan’s
constant; constants ontology; coherence mathematics; higher-dimensional emergence