Holographic Lattice Theory: A Conjectural Framework for Mersenne Prime Distribution — Pre-registered Coordinate Predictions for M53-M60

This pre-registration document presents a conjectural mathematical framework termed "Holographic Lattice Theory" for analyzing and predicting the distribution of Mersenne primes (Mp = 2^p - 1). CORE HYPOTHESIS:The theory proposes that Mersenne prime exponents exhibit fractal self-similarity when decomposed into normalized 5-point clusters. By mapping consecutive prime clusters onto dimensionless phase coordinates φᵢ ∈ [0,1], the framework hypothesizes the existence of a universal "fractal gene" that governs prime placement across different scales. PRE-REGISTERED PREDICTIONS:Based on this hypothesis, specific coordinate predictions are timestamped for undiscovered Mersenne primes: - M53: n = 214,357,007 (64.5 million digits)- M54: n = 1,120,219,861 (337 million digits)- M55: n = 1,618,549,043 (487 million digits)- M56: n = 2,338,559,557 (704 million digits)- M57: n = 3,378,866,489 (1.02 billion digits)- M58: n = 4,881,953,303 (1.47 billion digits)- M59: n = 7,053,687,349 (2.12 billion digits)- M60: n = 10,191,516,059 (3.07 billion digits) Each prediction is accompanied by a complete SHA-256 hash identifier for tamper-proof verification. QUANTITATIVE ANALYSIS:The document includes residual analysis comparing the "Ancestral Gene" (M6-M8) with "Deep-Space Replica" (M50-M52), demonstrating:- Mean Squared Error (MSE): 9.05 × 10⁻⁴- Geometric feature preservation: 94.8%- Twin gap compression: 41% IMPORTANT DISCLAIMER:This is a PRE-REGISTERED CONJECTURE, not a peer-reviewed proof. The underlying mathematical framework has not been formally proven. Statistical validation showed mixed results (holdout validation: 18.6 percentile). This document serves as a timestamped record for future falsification or verification upon the discovery of M53. SUPPLEMENTARY MATERIALS:- 11 Holographic Slice diagrams (Plate 1)- Residual analysis visualization- Complete SHA-256 hash registry