1. Starting point: a shared tension across very different problems The original motivation was not to “solve the Millennium Problems,” but to understand a recurring difficulty: mathematically unrelated problems appear to fail at the same structural point. Across topology, analysis, algebraic geometry, gauge theory, arithmetic, and computational complexity, the same pattern emerges: local equations are well posed; local solutions exist; global realizability either fails or remains inaccessible. This suggests that the core difficulty is not local, but concerns global realizability. 2. A key insight: Poincaré as prototype, not exception A decisive conceptual step was to reinterpret the Poincaré Conjecture. Rather than treating it as a singular topological result, we view it as a resolved instance of a general phenomenon. Perelman’s contribution reveals that: the obstruction is real and global; it is not detectable locally; it can be eliminated only through a coercive global dynamics (Ricci flow); the obstruction is not denied, but rendered dynamically unsustainable. This exposes a third conceptual category that was previously unclear: an obstruction may exist formally, yet fail to be dynamically realizable. Before Poincaré, this distinction was not sharply visible. 3. Generalization: the obstruction is consistently “degree three” A comparative analysis across problems reveals a striking regularity: the obstruction is not first-order (local); not second-order (linear or bilinear); but consistently appears as a third-order global obstruction. The label H³-type obstruction is used in a structural sense:not restricted to classical cohomology, but denoting a global realizability obstruction separating local consistency from global realization. 4. Why introduce the Unified Fabric (Tessuto Unico) At this stage, a deeper question arises: why do such obstructions exist at all? why do they resist local elimination? why do they recur in such different mathematical settings? The Unified Fabric is introduced as an explanatory framework, not as an axiom: space as a geometric, atemporal substrate; dynamics via discrete local updates; finite capacity / bounded throughput; time as a process, not a background parameter. Within this framework, it becomes natural that: global coordination carries a cost; some configurations are locally consistent but dynamically unrealizable. The Unified Fabric does not replace continuum mathematics;it clarifies which properties should emerge in the continuum limit. 5. From explanation to precise formulation The focus then shifts from explanation to formulation. The guiding question becomes: What is the single mathematical statement whose truth would resolve the problem? This leads to a systematic reduction: For each Millennium Problem: the relevant H³-type obstruction is identified; a natural coercive dynamics is formulated; a Lyapunov functional is defined; the problem is reduced to one precise “Bridge Lemma.” Thus, each problem is distilled to a single technical core. 6. Two guiding case studies Hodge Conjecture problem: global realizability of rational Hodge classes; dynamics: realization-cost minimization; obstruction: diffuse, atemporal representatives; conclusion: algebraicity is forced under coercivity. Navier–Stokes problem: vorticity concentration; dynamics: Navier–Stokes evolution itself; obstruction: accumulation without boundary throughput; conclusion: blow-up is incompatible with finite capacity. In both cases, the full problem reduces to one coercive inequality. 7. Extension to the remaining Millennium Problems Using the same structural template: Yang–Mills → spectral coercivity and mass gap; Riemann Hypothesis → essential self-adjointness of a canonical generator; Birch–Swinnerton–Dyer → arithmetic coercivity (height / Selmer structure); P vs NP → non-liftability under bounded computational resources. Crucially: these are not claimed as solutions,but as honest structural reductions.