The generalized Collatz map Ca, b (n) = n/2 for even n and Ca, b (n) = an + b for odd ncontains three coefficients: the algebraic multiplier a (odd positive integer), the paritysustaining shift b (odd integer), and the starting integer n (positive integer). This paperdemonstrates, from first principles and without external assumptions, that these threecoefficients determine a topological classification of the map. The derivation proceeds unidirectionally from the parity rule. The 2-adic valuationdistribution P (k = j) = 2−jis derived from the equidistribution of odd residues modulopowers of two. The expected logarithmic dissipation per parity block is computed asEX = ln (a/4), establishing that the map is dissipative if and only if a < 4. Theminimal control layer m = 4 is identified as the smallest aggregation scale satisfyingtrajectory independence, statistical stability, and scale invariance. The Dalvi Dictact – the principle of local-to-global topological completion – is appliedthrough the renormalization operator R. The closure equation ln B = 16 ln (a/4) forcesa unique integer base: B (a) = 4a16%. The primordial topological invariant is then defined as: ∆ (a) = 4 ln B (a) = 4 ln 4a16%. For dissipative multipliers (a < 4), the invariant is finite and positive. For expansivemultipliers (a ≥ 5), no finite invariant exists. The parity-sustaining condition b odd isnecessary for the system to exist at all; without odd b, the 2-adic distribution collapses. The starting integer n dissipates under iteration, converging to the unique attractor cycle1 → 4 → 2 → 1 when EX < 0. Explicit invariants are computed: for a = 3 (the original Collatz map), B = 99 and∆ = 4 ln 99 ≈ 18. 3804794; for a = 1, B = 232 = 4294967296 and ∆ = 128 ln 2 ≈88. 7228391. The minimal control layer m = 4 is shown to correspond structurally to thefourth state (Turiya) recognized in the Mandukya Upanishad, providing ancient validation of the derived mathematical necessity without reliance on philosophical presupposition. The paper provides rigorous Python code for all derivations: ergodic dissipation, 2-adic distribution verification, Dalvi Dictact closure, integer base computation, and SpiralStrap visualization. The code is deterministic, uses only standard scientific libraries (NumPy, SciPy, Matplotlib), and is fully reproducible. The MDC-X Theorem establishes a first-principle, unidirectional derivation of topological invariants from the generalized Collatz map. The three coefficients correspond tothree structural roles: a selects the invariant (Creation), b sustains parity (Maintenance), and n dissipates to the attractor (Dissolution). Zero remains the Majorana fixed point (x = −x ⇒ x = 0) – the unmoving center around which all dynamics unfold.
Dillip Kumar Mahapatra (Thu,) studied this question.