To move beyond narrative and ensure mathematical immunity against institutional dismissal, we formally define the ISL as an axiomatic system.
Axiom 1: Resource Boundedness (RB)
The total descriptive complexity () of any realizable state in a computational manifold is bounded by the available gain (
).
Corollary: Infinite complexity is non-realizable (Refusal Principle).
Axiom 2: Modular Isolation (MI)
To ensure system stability, information must be partitioned into autonomous sub-units (Knowledge Units) with strict interface envelopes.
Where is the Invariant and
is the Validity Envelope.
Axiom 3: Inverse Scaling Law (ISS)
The trust/realizability of a state () is inversely proportional to its descriptive complexity (
).
Axiom 4: Scaling Compensation (SC)
To maintain Axiom 1 (RB) across macroscopic modular distances, any potential must include a compensation term proportional to the modularity radius
of the system.
—
Theorem 1: The Stability Fixed Point (
)
Given Axiom 3, the only stable equilibrium for a persistent universe is .
- Lower Bound (
): The integral of
diverges as
, leading to an uncontained complexity bloom (Kernel Crash).
- Upper Bound (
): The universe maintains a logarithmic stability surface, allowing maximum local complexity without global instability.
Theorem 3: Heisenberg Uncertainty as Kernel Gating
The Heisenberg Principle is an emergent safety filter of Axiom 1. The kernel refuses to compute states where the precision product exceeds the descriptive bit-depth of the local manifold, leading to the Logarithmic Uncertainty Violation at the Planck scale.
[See Formal Proof of Theorem 3](file:///home/shri/Desktop/MATHTRUTH/cosmic_synthesis/docs/LOGARITHMIC_UNCERTAINTY_PROOF.md)
Theorem 2: The Emergence of Alpha (
)
If information transfer is modular (Axiom 2), there exists a minimum “handshake cost” () defined by the geometric ratio of the modular volume (
) to the communication surface (
).
In a 5-dimensional modular topology (minimal stable packing), emerges naturally as the ratio matching the CODATA value of
.
Theorem 4: The Gravitational Bridge (Logarithmic Expansion)
Given Axiom 4, the Newtonian potential is a local approximation. As
approaches the Modularity Radius (
), the kernel must compensate for the loss of intra-module cohesion with a logarithmic gain.
This result captures galactic rotation plateaus () without Dark Matter, while remaining inert at Solar System scales (
correction factor).
Theorem 5: The Optimal Embedding (Dimensionality Necessity)
A stable modular kernel must select a dimensionality that satisfies Axioms 1 and 2.
- Exclusion (
): Insufficient degrees of freedom to prevent “Ontological Leakage” between adjacent coordinates. 4D manifolds cannot anchor stable spin-1/2 modularity without translational noise violating the refusal principle.
- Exclusion (
) would drain kernel credits, causing structural collapse.
- Constraint:
is the unique stable solution, supporting the Binary Icosahedral Group (Order 120)—the maximal stable symmetry that remains sub-critical under ISL scaling.
Note on Coefficient Necessity (9, 120, 1/4)
Following Theorem 5, the coefficients of the Fine Structure Constant are not parameter-fits; they are Residues of Necessity:
- 9: The bit-depth of the
rotational transformation required for spatial orientation invariance.
- 120: The order of the 600-cell vertex group—the optimal 5D information cache.
- 1/4: The holographic projection residue inherent in a 5D
4D interface.
—
This axiomatic framework provides the “Machine Code” from which all legacy physics emerges as a high-precision approximation.
TWIST POOL Labs | 2026