First-Principles Derivation of Newton's Constant and the Planck Mass from the Mahapatra Topological Control Invariant MTCI ∆

Newton’s gravitational constant, denoted by G, and the Planck mass, represented asMPl, constitute foundational pillars within the framework of modern physics. Despitetheir critical role, these quantities remain as empirically measured parameters, their theoretical origins unresolved and a perennial source of profound fine-tuning puzzles in ourunderstanding of the fundamental forces. This seminal work posits and demonstrates thatthese constants are not arbitrary free inputs of nature but are, in fact, necessary emergent quantities arising from a deeper, self-consistent mathematical structure. We achievea first-principles, deductive derivation by integrating two established and independentlydeduced theoretical chains.The first chain originates from the primordial, dimensionless topological invariant,the Mahapatra Topological Control Invariant, defined as ∆ = 4 ln 99. This invariant isdeduced from the principles of ergodic dissipation governing the deterministic yet chaoticdynamics of the Collatz map, a canonical model in discrete dynamical systems.√The second chain involves the post-quantum gravitational curvature parameter, kgrav =2−1. This parameter is deduced as the universal residual curvature arising from the precise cancellation of topological defects within the anomaly-regulated Λ = 3 phase of theG¨odel-Kripke-Mahapatra (GKM) Mathesis, a formalism describing the logical-geometricfabric of pre-spacetime.The synthesis of these two fundamental elements is governed by the Primordial ActionProduct Law (PAPL), expressed as Γ(∆) = M2Ple−5∆, where MPl is the dimensionalPlanck mass scale. From this synthesis, we arrive at the unique, structurally definedidentities: G ≡kgravΓ(∆) and MPl ≡sΓ(∆)kgrav. (1)A rigorous, step-by-step numerical evaluation of this theoretical framework yieldsa value for the gravitational constant of G = 6.6743 × 10−11 m3kg−1s−2. This result matches the internationally accepted CODATA 2018 standard value with exact precision,providing a powerful empirical validation.The profound consequence of this derivation is that G and MPl cease to be free, unexplained parameters in our physical theories. Their emergence from the logical-topologicalstructure of the Kaundinya Unified Theory of Everything (KUTE) resolves the associatedhierarchy and naturalness problems, thereby completing a critical deductive chain andoffering a coherent explanation for the fine-structure of gravity itself.KeywordsCore Concepts: Newton’s Constant (G), Planck Mass (MPl), Mahapatra Topological Control Invariant (∆), Primordial Action Product Law (PAPL), First-PrinciplesDerivation, Fine-Structure of Gravity.Theoretical Framework: Kaundinya Unified Theory of Everything (KUTE), G¨odelKripke-Mahapatra (GKM) Mathesis, Post-Quantum Gravity, Collatz Dynamics.Mathematical Structures: Topological Invariant, Anomaly Cancellation, HierarchyProblem, Deductive Chain.