Epistemic note
This project constitutes a deductive structure based on constraints and invariants,
not a philosophical hypothesis nor a physical theory.
Its contents are not evaluable in isolation:
each individual page presupposes the full conceptual structure of the site.
Any judgment formulated without examination of the complete context
is necessarily partial and non-conclusive.
The distinction between structural deduction and interpretation
is an integral part of the model.
This section is intended for readers who seek a more technical perspective. The goal is not to provide a definitive derivation, but to show how spherical geometry can impose structural constraints on the number of possible periods, when elements are distributed coherently on the surface of a hemisphere.
The final value (approximately 28 periods) does not arise from a single closed formula, but from the interaction between:
We consider a hemisphere of radius R representing the
hemisphere of ordinary matter.
Chemical elements (or, more generally, stable configurations)
are represented as points distributed on its surface in a:
The area of a hemisphere is:
Area_hemisphere = 2 · π · R²
We now introduce A_elem, which does not represent a physical constant,
but a minimal conceptual area associated with a distinguishable
configuration (atomic or extended state).
The maximum number of elements that can be placed on the hemisphere is therefore:
N_hem ≈ Area_hemisphere / A_elem = (2 · π · R²) / A_elem
Once a physical or informational criterion for A_elem is fixed,
the number of available “slots” on the hemisphere becomes finite
and estimable.
To avoid preferential clustering and local correlations, a quasi-uniform distribution of elements is assumed. A natural approach is to use a constant azimuthal rotation based on an irrational angle.
Let n denote the element index:
φₙ = n · φ₀ (mod 2π) θₙ = function(n / N_hem) · (π/2)
where φ₀ is an irrational angle (for example related to the golden ratio),
chosen to minimize periodic repetitions and local correlations,
analogously to classical problems of uniform distribution
on curved surfaces.
The hemisphere is divided into spherical bands, each associated
with a period of the extended periodic table.
Each band occupies an angular interval
[θ_k, θ_{k+1}].
The area of a band is:
Area_band_k ≈ 2 · π · R² · (cos θ_k − cos θ_{k+1})
Assuming an approximately constant surface density, the number
of elements in period k is proportional to the band area:
Elements_k ∝ (cos θ_k − cos θ_{k+1})
This directly links:
Summing the contributions of all bands until the hemisphere is completely saturated:
Σ_k Elements_k ≈ N_hem
Since the function cos θ varies rapidly near the equator
(θ = π/2), the bands become progressively more “dense”
in terms of configurations, requiring increasingly extended periods.
When the entire hemisphere is filled, the maximum number of bands compatible with a coherent and non-degenerate distribution is on the order of:
k_max ≈ 28
This value does not represent an absolute ontological limit, but the limit of the classical atomic organization regime imposed by the geometry of the hemisphere.
If periods beyond the seventh exist, they cannot manifest as chemical elements in the classical sense. The very notion of an “element” presupposes stability, well-defined time, and atomic describability — conditions that cease to hold beyond a certain physical regime.
In this model, such periods are not sought in ordinary space nor under standard experimental conditions, but are compatible only with extreme regimes, near the collapse of classical physical description. In this sense, regions close to black holes are not places that “contain” new elements, but structural signals of the limit beyond which matter changes regime.
Black holes therefore do not represent a destination or a repository, but a process: a local approach toward the center of the Sphere of Everything, where surface representation loses meaning and only the fundamental rules of existence remain valid.
If one hemisphere represents ordinary matter, the opposite hemisphere can be interpreted as hosting:
Within this framework, the complementary hemisphere may also host a comparable number of periods, leading to a total on the order of 56 conceptual periods across the entire Sphere of Everything.
This analysis proposes a working structure: spherical geometry is not a graphical artifact, but a constraint that guides the extension of the concept of the periodic table beyond the plane.
The key idea is that geometry itself suggests where one regime of matter ends and where it becomes necessary to introduce a new one.
A further consequence of the model concerns the relationship between existence and temporal duration. This topic is developed on the page Existence without duration.
For a more detailed mathematical formalization of the model, see the Mathematical appendix.