ROOT-GFTM-72Digital & Intelligence

Geometric Transition Matrices (GFTM-72)

Defines a 72-matrix transition algebra coordinating multi-dimensional state transitions with complete unitary and orthogonal matrix representations.

ClassAlgebraic Geometry

Evidence TierTier 1 (Formal Root)

Model ProfileFORMAL_SPECIFICATION

AttributionIvan Pasev

Procedural PlateFormal procedural plate defining the bounded theoretical geometry and invariant obligations for this research fabric.

EPISTEMIC DEMARCATION // Tier 1 (Formal Root)

Inspect Results Ledger →

Constitutional Demarcation: A Research Fabric is a bounded epistemological architecture. Structural formulation, digital execution, and proof-assistant replay declare local invariants; they do not constitute unverified physical validation.

Architectural Overview

Geometric Transition Matrices (GFTM-72) defines a discrete 72-dimensional modular algebraic state machine operating under exact orthogonal and unitary group actions over split-torus profiles. In mission-critical closed-loop cybernetic systems, repeated floating-point matrix multiplications inevitably accumulate numerical drift, leading to phase error accumulation and loss of contract determinism.

GFTM-72 eliminates floating-point instability by projecting continuous rotation groups onto a finite, exact 72-matrix generator algebra ${\mathcal{G}}_{72}\subset SO(3)\times {\mathbb{Z}}_{72}$:

$${\mathbf{S}}_{k+1}={\mathbf{M}}_{({\alpha }_k)}\cdot {\mathbf{S}}_k(\mathrm{mod}p)$$

Where every transition matrix $\mathbf{M}\in {\mathcal{G}}_{72}$ satisfies strict orthogonality ${\mathbf{M}}^T\mathbf{M}=\mathbf{I}$ with integer and cyclotomic field entries.

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Core Invariants

1. Zero Numerical Drift: Matrix transitions are computed in exact algebraic integer rings, maintaining norm conservation $\parallel {\mathbf{S}}_k\parallel =\parallel {\mathbf{S}}_0\parallel$ for arbitrary cycle lengths $k\to \mathrm{\infty }$.
2. Deterministic State Reachability: The transition graph forms a strongly connected regular digraph with provable maximal cycle limits.
3. Reversible Cycle Closure: Every sequence of state transitions possesses an exact inverse sequence within ${\mathcal{G}}_{72}$.
4. Substrate-Invariant Execution: Equivalent execution traces across distinct hardware instruction architectures (ARM, x86, RISC-V, FPGA).

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Epistemic Boundary Notice

High-Assurance Cybernetic Software Specification

GFTM-72 is a mathematical and software-engineering specification for deterministic control systems, high-frequency robotics, and cryptographic state coordination. It does not assert novel physical particles or metaphysical energy fields.

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Reciprocal Bridges & Related Research

• Digital Fabrica Theory (RF-014) — The foundational deterministic cybernetic platform.
• Recursive Intelligence (RF-015) — Self-verifying symbolic reasoning loops.
• Quantum Consensus Architecture (RF-017) — Distributed deterministic consensus.
• GFTM-72 Whitepaper Reader — Full algebraic group tables and verification suite.