Complete technical report with the comparative analysis of the seven Millennium Problems, the convergence study of the Poincaré Conjecture and the extensive development of the P = NP exploration through the Deterministic Geometric Oracle (DGO).
Formal peer-review preprints and technical reports registered with Digital Object Identifiers (DOI) issued by Zenodo / CERN (Geneva, Switzerland). Incorporating dual-path verification (True / False), analytic formalization, and reproducible empirical audits:
Each Millennium Problem with dedicated dual-path verification: the resolving path and the non-resolving path, confirming the mathematics is correct.
Main results of Program M-1: the convergence function c(N), the three geometric phases, the Runge model, the Deterministic Geometric Oracle and the Quark-Quaternary mega-scale experiment.
The central finding of the program is that the minimum number of iterations for the statistical convergence of the algoritmo de la Consultoría satisfies:
where c(N) is not a universal constant but a function that evolves with scale, governed by three distinct geometric phases.
| N (nodes) | c observed | Phase | Source |
|---|---|---|---|
| 20 – 100 | 6.000 | I — Local geometry | Phase 0 |
| 200 – 500 | 10.000 | Transition | Phase 1 |
| 1.000 | 23.165 | Critical transition | P1 |
| 5.000 | 18.786 | II — Macro | P1 |
| 7.000 | 18.072 | II — Macro | P1 |
| 12.000 | 17.035 | II — Macro | P1 |
| 18.000 | 16.330 | II — Macro | P1 |
| 25.000 | 15.800 | II — Macro | Mega-scale v2 |
c = 6.000 = K₂ (Kissing Number of the Euclidean plane). Local geometric structure dominates. Each node has 6 neighbors and SA needs exactly c = K₂ "rotations" per Voronoi neighborhood.
c rises from 6 to ~23. High-density clusters emerge. SA must navigate between clusters, breaking the local planarity approximation. The energy landscape becomes rugged.
c decreases following c(N) — modelo propietario. Uniform distribution generates statistical homogeneity. The KNN graph is nearly regular and SA exploits this regularity: every search move carries predictable statistical information.
As a unified functional form over all N we propose:
The model qualitatively captures the three phases but the Two-Phase Model M2F (c₀ = 6 for N ≤ 100 · linear-log transition · 160/log(N) for N > 1.000) offers a better empirical fit.
The DGO is a proprietary deterministic algorithm based on structured search. Results are verifiable and reproducible. Outperforms stochastic SA in 4 of 5 instances.
| N | DGO (dist.) | MSA (dist.) | Result | Pruning |
|---|---|---|---|---|
| 20 | 433.27 | 555.93 | ✅ DGO −22% | 74.2% |
| 30 | 554.00 | 498.74 | ❌ MSA better | 81.6% |
| 50 | 724.00 | 912.03 | ✅ DGO −21% | 88.8% |
| 75 | 916.61 | 1369.20 | ✅ DGO −33% | 92.6% |
| 100 | 1132.37 | 1548.32 | ✅ DGO −27% | 94.3% |
N = 50,000 nodes distributed in two independent blocks of 25,000. 8,000,000 total iterations. Architecture: dedicated compute nodes and proprietary engine. Duration: 47 minutes.