# A computable all-time bound for small periodic fluid data

**Engine revision:** 0.14.0  
**Date:** 5 September 2026  
**Evidence:** Written analysis proof, exact initial-data arithmetic, and tested software. No formal proof kernel.  
**Scope:** Unforced incompressible three-dimensional Navier–Stokes on the fixed torus \([0,2\pi]^3\), with positive viscosity and real, smooth, mean-zero, divergence-free initial data.

This revision adds a sufficient condition that certifies an entire solution remains smooth for every future time. The condition can be checked from a finite exact description of the initial field. The subsequent evolution is the full partial differential equation; nonlinear interactions may generate infinitely many Fourier modes.

The mathematical method is established small-data theory, not a new Millennium Prize solution. Ambrose, Lopes Filho, and Nussenzveig Lopes develop global existence and analyticity in Fourier spaces on the torus in [Existence and analyticity of the Lei–Lin solution of the Navier–Stokes equations on the torus](https://arxiv.org/abs/2205.12383). Below is an explicit sufficient estimate and its implementation, with constants derived for the stated normalization.

## 1. The theorem the checker applies

Use the equations and Fourier convention

\[
\partial_tu+(u\cdot\nabla)u+\nabla p=\nu\Delta u,
\qquad \nabla\cdot u=0,\qquad \nu>0,
\]

\[
u(x,t)=\sum_{k\in\mathbb Z^3}u_k(t)e^{ik\cdot x},
\qquad u_0(t)=0.
\]

Here \(|k|\) is the Euclidean wavevector norm and \(\|u_k\|_2\) is the Euclidean norm of the complex three-component coefficient. Define

\[
S_0=\sum_{k\ne0}\frac{\|u_k(0)\|_2}{|k|}.
\]

**Theorem.** Suppose an upper bound \(B\) satisfies

\[
S_0\le B<\nu.
\]

Set \(\delta=(\nu-B)/2>0\). There is a unique global smooth solution, and

\[
\boxed{
\sum_{k\ne0}e^{\delta t|k|}\frac{\|u_k(t)\|_2}{|k|}
\le S_0e^{-\delta t}\le Be^{-\delta t},\qquad t\ge0.
}
\]

At every \(t>0\), this yields spatial analyticity with radius at least \(\delta t\): holomorphic extension exists for complex spatial displacements with Euclidean norm strictly below \(\delta t\). For a spatial multiindex \(\alpha\), with \(|\alpha|=m\),

\[
\|\partial^\alpha u(t)\|_{L^\infty}
\le \left(\frac{m+1}{e\delta t}\right)^{m+1}Be^{-\delta t}
\qquad(t>0).
\]

The numeric threshold depends on the Fourier and domain conventions above. It must not be applied to an arbitrary physical box or differently normalized coefficients without conversion. The checker uses mathematical, nondimensional coordinates for this fixed domain.

## 2. Proof without assuming the desired analyticity

### 2.1 Start with a local smooth solution

Use standard local existence, uniqueness, and continuation for smooth periodic divergence-free data. We show its maximal smooth time cannot be finite. These analytic existence and continuation results are mathematical dependencies; the Python program does not implement their formal verification.

The spatial average stays zero. For \(k\ne0\), the pressure is eliminated by the orthogonal Leray projector

\[
P_k=I-\frac{kk^T}{|k|^2},\qquad \|P_k\|_{2\to2}\le1.
\]

Write the projected equation as

\[
\dot u_k+\nu|k|^2u_k=N_k,
\quad
N_k=-iP_k\sum_{p+q=k}(q\cdot u_p)u_q.
\]

Since \(p\cdot u_p=0\), \(q\cdot u_p=k\cdot u_p\). Therefore

\[
\|N_k\|_2\le |k|\sum_{p+q=k}\|u_p\|_2\|u_q\|_2.
\tag{1}
\]

### 2.2 Cap the weight, not the evolving spectrum

For a finite \(K\ge1\), let

\[
w_K(k,t)=\exp\!\bigl(\delta t\min(|k|,K)\bigr),
\qquad
A_{j,K}(t)=\sum_{k\ne0}|k|^j w_K(k,t)\|u_k(t)\|_2.
\]

For fixed \(K\) and finite time, the weight is bounded independently of \(k\). Thus a local smooth solution has all the sums needed below, without an initial assumption of exponential Fourier decay at later times. We retain every mode in these sums.

The function \(k\mapsto\min(|k|,K)\) is subadditive:

\[
\min(|p+q|,K)
\le\min(|p|,K)+\min(|q|,K).
\]

Consequently \(w_K(p+q,t)\le w_K(p,t)w_K(q,t)\) for \(t\ge0\).

Taking the upper right Dini derivative of coefficient norms accommodates coefficients that vanish. Equation (1), the submultiplicative weight, and \(\min(|k|,K)/|k|\le1\) give

\[
D^+A_{-1,K}
\le-\nu A_{1,K}+A_{0,K}^2+\delta A_{0,K}.
\tag{2}
\]

For completeness, the nonlinear sum in this inequality is bounded by

\[
\sum_{k\ne0}\!w_K(k)\sum_{p+q=k}\|u_p\|_2\|u_q\|_2
\le\left(\sum_{p\ne0}w_K(p)\|u_p\|_2\right)^2.
\]

Omitting the \(k=0\) output can only reduce this nonnegative bound. The zero input mode is absent because the solution has zero mean. On compact subintervals of the local smooth lifespan, polynomially weighted Fourier tails are summable uniformly; this justifies summing the coefficient inequalities, or equivalently passing from partial sums to (2).

### 2.3 Close the estimate under strict smallness

Cauchy–Schwarz gives

\[
A_{0,K}^2
=\left(\sum_{k\ne0}\sqrt{w_K\|u_k\|_2/|k|}
                     \sqrt{|k|w_K\|u_k\|_2}\right)^2
\le A_{-1,K}A_{1,K}.
\]

The fixed torus and zero mean imply \(|k|\ge1\) for every contributing mode, hence

\[
A_{-1,K}\le A_{1,K},\qquad A_{0,K}\le A_{1,K}.
\]

Substitute these bounds into (2):

\[
D^+A_{-1,K}
\le-(\nu-\delta-A_{-1,K})A_{1,K}.
\tag{3}
\]

Initially \(A_{-1,K}(0)=S_0\le B\). While \(A_{-1,K}\le B\),

\[
\nu-\delta-A_{-1,K}\ge\nu-\delta-B=\delta>0.
\]

The scalar comparison/barrier argument applied to (3) prevents a crossing above \(B\): the right-hand side is strictly decreasing there whenever the field is nonzero. For zero initial data, local uniqueness gives the zero solution directly. Throughout the local smooth lifespan,

\[
D^+A_{-1,K}\le-\delta A_{1,K}\le-\delta A_{-1,K},
\qquad
A_{-1,K}(t)\le S_0e^{-\delta t}.
\tag{4}
\]

This argument is uniform in \(K\). Monotone convergence as \(K\to\infty\) gives the theorem's exponential Fourier bound.

### 2.4 Exclude a finite maximal smooth time

For \(t>0\), the exponential bound implies

\[
\sum_{k\ne0}|k|^m\|u_k(t)\|_2
\le \sup_{r\ge0}\bigl(r^{m+1}e^{-\delta tr}\bigr)
      \sum_{k\ne0}e^{\delta t|k|}\frac{\|u_k(t)\|_2}{|k|}.
\]

The supremum is \(((m+1)/(e\delta t))^{m+1}\). This proves the derivative bound in Section 1. It also proves holomorphic convergence inside any smaller complex strip, by leaving a positive margin in the exponential weight.

If the maximal smooth time \(T_*\) were finite, every spatial derivative would be uniformly bounded on \([T_*/2,T_*)\). In particular, an \(H^m\) norm with \(m>5/2\) would remain bounded there. The standard local continuation theorem then extends the solution past \(T_*\), a contradiction. Smooth initial data supply the interval near zero; the displayed estimate's divergence as \(t\downarrow0\) is only a weakness of that estimate. Thus the smooth solution is global. Pressure is recovered, up to its spatial constant, from the periodic elliptic pressure equation.

## 3. Exact bits decide whether the hypothesis holds

`tools/global_fluid_certificate.py` accepts a finite rational Fourier polynomial. Every listed complex component is a pair of canonical rational strings, such as `["1/3", "-2/5"]`. A valid shear example with \(\nu=1/100\) is

```json
{
  "schema": "RBE_EXACT_TORUS_FIELD_V1",
  "domain": "[0,2pi]^3_periodic_unforced",
  "viscosity": "1/100",
  "unlisted_modes": "exactly-zero",
  "modes": [
    {"k": [0, 1, 0], "u": [["0/1", "-1/2000"], ["0/1", "0/1"], ["0/1", "0/1"]]},
    {"k": [0, -1, 0], "u": [["0/1", "1/2000"], ["0/1", "0/1"], ["0/1", "0/1"]]}
  ]
}
```

This represents \(u_0=(\sin y/1000,0,0)\). The checker verifies exact divergence freedom, conjugate symmetry, absence of a zero mode, positive viscosity, and schema constraints. It rejects duplicate modes, duplicate JSON keys, noncanonical fractions, floating-point JSON values, and unknown initial tails.

For a coefficient \(u_k\), the squared contribution

\[
r_k=\frac{\sum_{j=1}^3[(\Re u_{k,j})^2+(\Im u_{k,j})^2]}{|k|^2}
\]

is rational. If \(r_k=a/b\) in lowest terms and \(s=2^p\), compute

\[
q=\left\lfloor\sqrt{\left\lfloor as^2/b\right\rfloor}\right\rfloor
\]

by integer arithmetic. Then

\[
q/s\le\sqrt{r_k}<(q+1)/s.
\]

The square root is exact when both \(a\) and \(b\) are perfect squares; the implementation returns identical rational endpoints in that case. Otherwise the enclosure width is exactly \(2^{-p}\). For \(M\) listed modes, summing the enclosures produces \(L\le S_0\le B\) with total width at most \(M2^{-p}\).

This \(2^{-p}\) is a chosen computational increment, not a minimum physical length, time, or uncertainty. All comparisons with viscosity are exact rational comparisons. Floating approximations are unnecessary for acceptance.

| Exact comparison | Result | Meaning |
|---|---|---|
| \(B<\nu\) | `CERTIFIED_BY_STATED_SMALL_DATA_THEOREM` | The proved sufficient hypothesis holds; all-time smoothness follows under the stated assumptions. |
| \(L\ge\nu\) | `OUTSIDE_SUFFICIENT_SMALLNESS_CONDITION` | This sufficient theorem does not apply. No blowup claim follows. |
| \(L<\nu\le B\) | `UNRESOLVED_PRECISION` | The interval straddles the threshold; refine the arithmetic. |
| Schema or mathematical assumptions fail | `INVALID_INPUT` | No certificate is issued. |

Only the first status concludes global smoothness. A written proof and a tested hypothesis checker are different from a proof checked by a formal logical kernel; output explicitly sets `formal_kernel_checked` to false. It also sets `millennium_problem_solved` to false.

## 4. Reproduce the outcomes

From the extracted project directory, using Python 3.10 or later:

```sh
python3 tools/global_fluid_certificate.py examples/global-fluid/small_interacting.json
python3 -m unittest discover -s tests -p 'test_global_fluid.py' -v
python3 tools/global_fluid_examples.py --output results/my-global-fluid.json
```

On Windows, use `py -3` in place of `python3`. The checker needs only the Python standard library; no C++ compilation is required for this companion. CLI exit codes are 0 for a successful theorem certificate, 1 for invalid input, and 2 when the sufficient condition is unmet or precision is unresolved.

| Initial velocity | Viscosity | Exact initial norm \(S_0\) | Expected outcome |
|---|---:|---:|---|
| \(10^{-3}(\sin y,0,\cos x-\cos(x+y))\) | \(1/100\) | \(10^{-3}(2+1/\sqrt2)\) | Certified |
| \((\sin y,0,\cos x-\cos(x+y))\) | \(1/100\) | \(2+1/\sqrt2\) | Outside sufficient condition |
| \((\sin(1024z),\cos(1024z),0)\) | \(1/100\) | \(\sqrt2/1024\) | Certified |
| \((\sin y/100,0,0)\) | \(1/100\) | \(1/100\) | Exactly at threshold; no certificate from this theorem |
| \((\sin z,\cos z,0)\) | \(3621/2560\) | \(\sqrt2\) | Unresolved at 8 bits; certified at 96 bits |
| Zero | \(1/100\) | Zero | Certified |

The first field has genuine nonlinear interactions; the proof does not rely on a stationary or single-mode ansatz. The high-frequency example has mean energy \(1/2\) and mean enstrophy \(524288\), yet satisfies the Fourier smallness condition. That example is itself a simple exact flow; it illustrates the difference between norms, not a general reduction of large data to small data.

The threshold shear is also known explicitly:

\[
u(x,t)=(\nu e^{-\nu t}\sin y,0,0),\qquad p=\text{constant}.
\]

It stays smooth although \(S_0=\nu\). This prevents interpreting a rejected sufficient condition as evidence of a singularity.

The frozen report is `results/global-fluid-v014.json`. Eleven new test methods check rational square-root enclosure, exact threshold handling, all six field types, refinement, and invalid assumptions. Seven expected certificate outcomes comprise the six fields at default precision and the deliberately coarse borderline case.

## 5. The large-data argument is still missing

Equation (3) makes the obstruction explicit. If \(A_{-1,K}\) is larger than \(\nu-\delta\), this estimate has no coercive sign. It provides no bound that forces the solution into the small-data region before a possible loss of smoothness.

At \(\delta=0\), the available inequality is

\[
D^+A_{-1,K}\le-(\nu-A_{-1,K})A_{1,K}.
\]

An upper bound that becomes positive does not prove that the actual norm increases or blows up. It means this estimate alone stops controlling it. Extra bits cannot change the sign for an exactly specified field whose true norm is already above the threshold. Coordinate names, dimensional cycles, or sending a residual to D0 do not provide the missing inequality either.

A useful extension would require additional structure or a stronger estimate that controls the full evolution for a specified larger class. A general solution of the [official Navier–Stokes problem](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf) requires the appropriate conclusion for arbitrary allowed smooth data, or a valid breakdown construction—not merely the small class certified here. Our fixed domain is rescalable to the periodic convention in that statement, but the smallness restriction remains a restriction.

The Riemann Hypothesis and P versus NP gaps also remain unchanged by this fluid theorem. It supplies neither an all-integer RH argument nor a complexity bound for all SAT algorithms. No theorem here establishes physical hidden dimensions or a theological claim.