Scientific Status

This page presents an authorial research framework within the Science of Fabric Reality program. It is provided for examination, comparison, and further formal validation. It should not be read as external authorial framework consensus unless such validation is explicitly cited.

I. Mathematical Grounding of iDST

The Infinite Digital Structure Theorem (iDST) asserts that any self-organizing digital manifold must converge to a bounded topological invariant state under infinite recursive scaling. Without a rigorous proof of convergence, recursive networks are prone to unboundedly increasing entropy, leading to systemic collapse (e.g., buffer overflows, state drift, or validation failures).

This registry indexes the foundational proofs establishing the convergence bounds, transfinite scaling limits, and core stabilization formulas that govern the entire Digital Fabrica Theory (DFT) ecosystem.

II. The Invariant Stability Limit

The core convergence of the transfinite scale-entropy within a digital manifold is governed by the Invariant Stability Limit:

$$\underset{N\to \mathrm{\infty }}{lim}\sum _{k=1}^N\frac{\mathcal{S}(k)}{k^s}={\mathrm{\Lambda }}_{DFT}$$

where:

• $N$ represents the recursive scaling depth of the manifold.
• $\mathcal{S}(k)$ is the Scale Entropy Function at level $k$, measuring the topological complexity of the relational lattice.
• $s$ is the transfinite complex parameter governing the dimensions of the manifold.
• ${\mathrm{\Lambda }}_{DFT}$ is the Pasev Constant of Stabilization, a transfinite structural limit that guarantees convergence and prevents infinite state expansion.

III. Proof Sketch & Supporting Lemmas

The mathematical validation of the stability limit relies on two key lemmas:

Lemma 1: Sub-Exponential Entropy Growth

The scale complexity $\mathcal{S}(k)$ of the relational manifold is strictly bounded by a sub-exponential growth rate:

$$\mathcal{S}(k)<\mathcal{M}\cdot k^{\theta }(\text{for }\theta <\text{Re}(s)-1)$$

This bound ensures that the terms of the summation decrease fast enough to guarantee absolute convergence.

Lemma 2: Relational Path Conservation

Under transfinite ordinal transformations, the integral of relational invariants along any closed network contour remains invariant:

$$\oint \delta \mathcal{S}(k)=0$$

This conservation law prevents the emergence of unboundedly growing local fluctuations, preserving global stability.

IV. Epistemic Verification Path

These foundational proofs are archived with permanent identifiers (DOIs) under the Universum Knowledge Corpus (UKC) standard. Independent mathematical logic and transfinite systems audits are welcome to formalize the bounds of the Pasev Constant across other non-Archimedean topological spaces.

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Status: Authorial Proof Manuscript — Archived in the UKC Library
Last updated: August 16, 2025