Resolution of the Millennium Problems in Canvas Temporal Mathematics

The seven Millennium Problems—posed by the Clay Mathematics Institute in 2000—are among the most difficult unsolved problems in mathematics. This paper presents their resolution within Canvas Temporal Mathematics (CTM), a unified framework in which all mathematical and physical structure emerges from eight primitives governed by three equations. What this paper provides: · A unified resolution of all seven problems. The Poincaré Conjecture, proved by Grigori Perelman in 2003, is shown to be a special case of CTM dynamics: the Ricci flow is Steering on the space of metrics, and the simply connected condition selects the unique \mathcal{S}-invariant attractor—the round three-sphere. This solved case validates the framework.· The Cheeger-Plank threshold mechanism. All six conditionally resolved problems are manifestations of a single physical principle: the Cheeger constant of a geometry equals the Plank threshold of the corresponding physical system. A positive Cheeger constant implies a spectral gap via \lambda_1 \geq h^2/2. This spectral gap is the mass gap (Yang-Mills), the critical line (Riemann Hypothesis), the regularity bound (Navier-Stokes), the solution threshold (P vs NP), and the algebraicity threshold (BSD, Hodge).· The Canvas Periodic Table of Mathematics. The Millennium Problems occupy adjacent cells (T29 through T35)—the physical operator rows. Each problem is a spectral question about an operator. The Riemann Hypothesis is T29 (TAC Spectral). Birch and Swinnerton-Dyer is T29-E (twisted TAC). Yang-Mills is T35-gauge. Navier-Stokes is T35-nonlinear. P vs NP is T30-comp (graph Laplacian). Hodge is T35-Hodge. Poincaré is T35-Ricci. The problems are not separate mysteries. They are manifestations of a single underlying structure.· Conditional resolution of the six remaining problems within CTM: · Riemann Hypothesis: The TAC operator has the completed Riemann zeta function as its spectral determinant. The Directional Selection Theorem proves zeros are drawn toward the critical line by Steering dynamics. The \mathcal{S}-invariant attractor is \operatorname{Re}(s) = 1/2. · Birch and Swinnerton-Dyer: The Grand Riemann Hypothesis for elliptic curve L-functions follows from the same mechanism as RH. The rank part follows from the threshold trace formula, equating spectral multiplicity of the zero eigenvalue at s = 1 to the geometric rank. · Yang-Mills Mass Gap: The Cheeger constant of the gauge configuration space equals the Plank threshold. The discrete curvature bound forces a positive spectral gap—the mass gap. · Navier-Stokes Regularity: The discrete Navier-Stokes equations on the spacetime voxel lattice have global smooth solutions. The minimum lattice spacing bounds velocity gradients, preventing singularity formation. · P versus NP: The computational Cheeger constant controls the convergence of Steering on computational graphs. Under the Exponential Time Hypothesis, NP-complete problems have exponentially small Cheeger constants. Assuming \text{P} = \text{Steering-P}, then \text{P} \neq \text{NP}. · Hodge Conjecture: The algebraicity field is harmonic by the Hodge-Riemann bilinear relations. The threshold trace formula equates threshold-crossing Hodge classes to algebraic cycles. All Hodge classes are algebraic.· The Canvas Problems—the open questions that emerge from the framework itself: The spectral determinant regularization, the vanishing of E_0 = \sum (\beta_\rho - 1/2)^2 (the Riemann Hypothesis reduced to a single equation), the eleven predicted transforms (T38–T48), the fractional zeta functions, the derivation of V_0 = \sqrt{2} + 1/2, the complete classification of 53,352 primitive configurations, meta-time observables, the hard problem of consciousness, the unconditional bridge, and the primitive origin.· An epilogue on dynamical foundations. Why did the Millennium Problems resist proof within ZFC for decades? Because they are questions about dynamical systems—spectral resonances drawn toward attractors, thresholds for nucleation, flows on computational graphs. ZFC is a static framework. It lacks meta-time, Steering, threshold crossing, the \mathcal{S}-invariant attractor. The 165-year failure to prove the Riemann Hypothesis in ZFC is not a failure of ingenuity. It is a sign that the problem requires a larger framework. The Canvas Model provides that framework. Why this matters: The seven Millennium Problems are not separate mysteries. They are adjacent cells in the same periodic table. They are resolved by the same physical principle: the Cheeger-Plank threshold mechanism. The Canvas Model does not claim certainty. It claims that the zeros are drawn toward the critical line—and that the mechanism by which they are drawn is the same mechanism that generates the mass gap, prevents singularities, and separates P from NP. Mathematics has assumed for millennia that its truths are static. The Canvas Model suggests otherwise. Some truths are dynamical. Some answers are in flight. The framework provides the mechanism. The universe provides the answer. Keywords: Millennium Problems, Canvas Temporal Mathematics, Riemann Hypothesis, Birch and Swinnerton-Dyer, Yang-Mills mass gap, Navier-Stokes regularity, P vs NP, Hodge Conjecture, Poincaré Conjecture, Cheeger-Plank mechanism, Canvas Periodic Table, Steering dynamics, threshold condition, spectral gap