This work presents a novel synthesis of theoretical physics and non-Newtonian (multiplicative) calculus to address the fundamental relationship between mass, time, and the regularity of

non-linear field equations. By reformulating the Navier-Stokes equations and Yang-Mills theory within a bigeometric framework, we demonstrate that the "blow-up" problem in fluid dynamics and the "mass-gap" problem in quantum gauge theory are dual manifestations of multiplicative scaling violations.

We propose the **Multiplicative Regularity Conjecture**, which asserts that global smoothness in fluids and spectral gaps in gauge theories are guaranteed by the bound of

the multiplicative material derivative (D^*\Psi/Dt \leq 1). Thematically,Scale-Free Dynamics: The

Bigeometric Bridge Between Yang-Mills Confinement and Navier-Stokes Stability

Finally, we derive mass as the ontological regulator that prevents classical singularities by setting a finite Compton-scale temporal resolution (\tau_C = \hbar/mc^2).

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The "Problem of Time" remains the central impasse in the unification of General Relativity and Quantum Mechanics. While Relativity treats time as a geometric dimension, Quantum Mechanics treats it as an external parameter, leading to the vanishing Hamiltonian in the Wheeler-DeWitt equation (\hat{\mathcal{H}}|\Psi\rangle = 0).

Concurrently, two of the most  profound challenges in mathematical physics—the Navier-Stokes existence and smoothness

and the Yang-Mills mass gap—remain unsolved, largely due to the failure of additive (Newtonian) calculus to manage the scale-invariant non-linearities of these systems.

This paper argues that the common thread is the failure to recognize **time as a multiplicative flow generated by mass**. Using the framework of Grossman and Katz’s multiplicative calculus, we transition from differences to ratios, and from sums to products. This shift linearizes the group-theoretic structures of gauge fields and the stretching mechanisms of turbulence, allowing us to view mass not as a static scalar, but as the "multiplicative eigenvalue" of temporal evolution.

### **2. Synthesis: The Non-Newtonian Geometric Flow**

The core of our analysis lies in the transformation of the Laplacian and the material derivative.

In standard calculus, the vortex stretching term (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} and

the gauge curvature F_{\mu\nu} represent additive increments that can diverge toward infinity. In multiplicative calculus, these terms become exponents of a scaling factor.

* **In Navier-Stokes**, blow-up requires super-power-law growth in the enstrophy-to-energy ratio. Multiplicative calculus shows that as long as the viscous damping maintains bigeometric flatness, the fluid remains smooth.

* **In Yang-Mills**, the mass gap \Delta > 0 is revealed as a multiplicative fixed point where the Renormalization Group (RG) flow "stalls" due to the compactness of the multiplicative moduli space of connections

We can now state a precise **conjecture** connecting both Millennium Problems through multiplicative calculus:

> **Multiplicative Regularity Conjecture**: Let $\Psi: \mathcal{F} \to \mathbb{R}^+$ be a

multiplicative functional on a functional space $\mathcal{F}$ (either fluid velocity space or gauge connection space).

Then:

>

> $$\frac{D^*\Psi}{Dt} \leq 1 \quad \Longleftrightarrow \quad \text{Regularity (NS) or Mass Gap

(YM)}$$

>

> More precisely:

>

> - **NS Smoothness**: Global smooth solutions exist iff the multiplicative enstrophy

$\mathcal{E}^*$ satisfies $D^*\mathcal{E}^*/Dt \leq 1$ — equivalently, vortex stretching never

exceeds multiplicative viscous damping.

>

> - **YM Mass Gap**: $\Delta > 0$ iff the multiplicative gauge functional $S^*_{YM}$ has a

compact infimizing sequence in $\mathcal{A}/\mathcal{G}$ — equivalently, the bigeometric RG

flow admits no massless fixed point in the IR.

Both are statements that **multiplicative self-similarity (bigeometric flatness) of the flow is

preserved** — that the respective physical system never generates structure at a scale faster than the multiplicative logarithmic time $\tau^* = e^t$ can resolve.

The profound implication of the time-mass analysis: **the Compton time $\tau_C = \hbar/mc^2$

acts as the multiplicative UV cutoff**, below which quantum fluctuations prevent the classical

singularities (vortex blow-up, massless gauge bosons) from forming. The mass gap is not merely a spectral fact — it is the **quantum multiplicative regularization of classical geometric flow**.

.

### 

The evidence across relativistic quantum analysis and non-Newtonian calculus converges on a singular reality: **Mass is the source of time’s structure.** In a universe of zero mass, there is no proper time, no thermalization, and no scale.

Through the **Multiplicative Regularity Conjecture**, we provide a path toward solving the Millennium Problems by showing that:

1. **Fluid Smoothness** is the preservation of multiplicative enstrophy.

2. **The Mass Gap** is the quantum multiplicative regularization of classical geometric flow.

By treating the universe as a multiplicative automorphism of the quantum state space, we resolve the "Problem of Time." Time flows because mass exists; it is the ontological precondition for temporal becoming. The singularities of the classical world are effectively "buffered" by the multiplicative floor of the mass gap thus ensuring a universe that is both mathematically regular and physically massive.