# A sharper error bound and a nonlinear reference

**Engine:** 0.17.0  
**Date:** 5 September 2026  
**Evidence:** Written sufficient PDE argument, exact rational certificates, standalone checking, and fixed-input comparisons. No formal-kernel verification or mathematical novelty claim.

## 1. The specific barrier crossed

For viscosity one on the periodic torus \([0,2\pi]^3\), consider the exactly specified, real, mean-zero, divergence-free initial field

\[
u_0(x,y,z)=\bigl(2\sin y+\varepsilon\sin z,\;
\varepsilon\sin 2x,\;\varepsilon\sin y\bigr),
\qquad \varepsilon=\frac1{10}.
\]

The frozen comparison holds the initial field, degree-six time polynomials, 32 intervals of width \(1/8\), final time four, and 64-bit outward norm witnesses fixed.

| Reference approximation | Previous error bound | Sharper error bound |
|---|---|---|
| Six initial Fourier wavevectors | Sufficient envelope unresolved | Sufficient envelope unresolved |
| Eighteen wavevectors including nonlinear interactions | Sufficient envelope unresolved | **Independently accepted certificate** |

Thus both changes are necessary **among these four tested configurations** to certify this input. This is not a claim that all other parameter choices or established mathematical techniques fail. The older exact-shear certificate in v0.15 also fails its sufficient condition on this input.

For the accepted certificate, all times through \(T=4\) are covered. The maximum certified \(\chi^{-1}\) error is below \(0.04424\), and the full solution's norm at time four is below \(0.05144<\nu=1\). The small-data theorem in `GLOBAL_FLUID_BOUND.md` then gives global smoothness for this specific initial field, subject to the analytic dependencies stated below.

Its initial \(\chi^{-1}\) norm is

\[
\sqrt{4+\varepsilon^2}+\frac32\varepsilon>2,
\]

so the initial field does not satisfy the older total-smallness condition \(\|u_0\|_{\chi^{-1}}<1\).

The result expands this engine's verified coverage. It neither establishes a new regularity class in the literature nor resolves arbitrary large-data Navier–Stokes regularity. Approximate-solution validation is an established research method; see [Morosi and Pizzocchero, 2011](https://arxiv.org/abs/1104.3832) and their [smooth-solution analysis, 2014](https://arxiv.org/abs/1405.3421).

## 2. Norms and the complete error equation

Use the Fourier convention \(f(x)=\sum_k\widehat f(k)e^{ik\cdot x}\) and define

\[
\|f\|_{\chi^s}=\sum_{k\ne0}|k|^s\|\widehat f(k)\|_2.
\]

The PDE is unforced incompressible Navier–Stokes, with positive constant viscosity \(\nu\). Let \(v\) be a real, mean-zero, divergence-free, continuous, piecewise polynomial reference in time with finite Fourier support. Define

\[
w=u-v,\qquad
r=\partial_tv-\nu\Delta v+P[(v\cdot\nabla)v],
\]

\[
E=\|w\|_{\chi^{-1}},\quad E_0=\|w\|_{\chi^0},\quad
E_1=\|w\|_{\chi^1},\quad V=\|v\|_{\chi^0},\quad R=\|r\|_{\chi^{-1}}.
\]

For divergence-free fields, writing the convection in divergence form and using the Euclidean contraction of the Leray projection gives

\[
\|P[(a\cdot\nabla)b]\|_{\chi^{-1}}
\le \|a\|_{\chi^0}\|b\|_{\chi^0}.
\]

The output factor \(|k|\) cancels the \(\chi^{-1}\) weight. Apply the triangle inequality to the three terms \((v\cdot\nabla)w\), \((w\cdot\nabla)v\), and \((w\cdot\nabla)w\). For a local smooth solution this yields, almost everywhere in time,

\[
D^+E+\nu E_1\le 2VE_0+E_0^2+R.
\tag{1}
\]

Cauchy–Schwarz and the torus's lowest nonzero frequency give

\[
E_0^2\le EE_1,\qquad E_1\ge E.
\tag{2}
\]

No exact-solution Fourier mode is discarded in these estimates.

## 3. Keep a positive amount of dissipation

Choose constants

\[
c>0,\qquad \mu>0,\qquad c+\mu<\nu,
\qquad D=\nu-c-\mu>0.
\]

Here \(c\) is the bootstrap error cap and \(\mu\) is dissipation reserved for the continuation argument. While \(E\le c\), (1)–(2) imply

\[
D^+E+\mu E_1
\le -D E_1+2V\sqrt{EE_1}+R.
\tag{3}
\]

When \(E>0\), put \(x=\sqrt{E_1/E}\ge1\). The scalar optimization is exact:

\[
\Lambda(V,D)=\max_{x\ge1}(-Dx^2+2Vx)
=\begin{cases}
V^2/D,&V\ge D,\\
2V-D,&0\le V<D.
\end{cases}
\tag{4}
\]

To prove it, complete the square:

\[
-Dx^2+2Vx=\frac{V^2}{D}-D\left(x-\frac VD\right)^2.
\]

If \(V/D\ge1\), the vertex is admissible; otherwise the maximum over \(x\ge1\) occurs at one. At \(E=0\), the required inequality follows directly from \(-DE_1\le0\), without dividing by \(E\).

Consequently

\[
\boxed{D^+E+\mu E_1\le\Lambda(V,D)E+R.}
\tag{5}
\]

Both branches of \(\Lambda\) are increasing in \(V\) and agree at \(V=D\). With interval-wide reference and residual caps \(V\le V_j\), \(R\le R_j\), (2) and (5) imply

\[
\boxed{D^+E\le b_jE+R_j,\qquad
b_j=\Lambda(V_j,D)-\mu,\quad E\le c.}
\tag{6}
\]

The release uses \(c=\mu=\nu/8\), so \(D=3\nu/4\). Its large-\(V\) coefficient is \(4V^2/(3\nu)-\nu/8\), compared with the prior \(4V^2/\nu-\nu/4\). The new cap is also stricter, \(\nu/8\) rather than \(\nu/2\), so it is the full certificate comparison that establishes an improvement on any particular input.

“Sharper” means the constrained scalar quadratic is maximized exactly. It does not mean that the underlying Fourier triangle/Cauchy–Schwarz estimates, choice of norm, or PDE criterion are optimal.

## 4. Why the certificate implies continuation and global smoothness

The checker verifies nonnegative, continuous, piecewise affine \(q\) with

\[
E(0)\le q(0),\qquad q(t)<c,
\]

and, on each affine segment of width \(\Delta\),

\[
\frac{q_R-q_L}{\Delta}\ge b_jq_L+R_j,
\qquad
\frac{q_R-q_L}{\Delta}\ge b_jq_R+R_j.
\tag{7}
\]

Because the right-hand side interpolates affinely, (7) proves the differential supersolution inequality at every interior time. Comparison and the strict cap close the bootstrap, giving \(E\le q<c\) throughout the checked interval of existence.

The reserved dissipation matters. Integrating (5), using \(E\ge0\), bounds

\[
\mu\int_0^t E_1(s)\,ds
\le E(0)+\int_0^t
\left[\max\{\Lambda(V_j,D),0\}\,q(s)+R_j\right]ds.
\tag{8}
\]

There are finitely many bounded reference intervals, so the right-hand side is finite for \(t\le T\). The smooth finite reference also has finite \(\int_0^T\|v\|_{\chi^1}dt\). Therefore

\[
\int_0^T\|\nabla u\|_{L^\infty}dt
\le\int_0^T(\|v\|_{\chi^1}+E_1)dt<\infty.
\]

Standard local smooth existence and the Sobolev energy/continuation estimate extend the solution through \(T\); a putative earlier breakdown contradicts this bound. These analytic facts are explicit dependencies, not facts established solely by executing Python.

Finally the checker requires

\[
B_T=\|v(T)\|_{\chi^{-1}}^{\rm upper}+q(T)<\nu.
\tag{9}
\]

The all-time Fourier small-data theorem in `GLOBAL_FLUID_BOUND.md` applies to the smooth exact solution at \(T\). That theorem does not require its evolved spectrum to remain finite. Its related periodic Fourier background is [Ambrose, Lopes Filho, and Nussenzveig Lopes](https://arxiv.org/abs/2205.12383).

The reference's derivative may jump at interval boundaries. Its checked continuity prevents impulses in its distributional derivative, and the almost-everywhere inequalities integrate across the finitely many joins. Unknown initial tails, external forcing, other spatial domains, and unverified viscosity models are outside this certificate's contract.

## 5. Retaining actual nonlinear interactions

The old reference uses the six signed wavevectors present initially. For this example their retained projected convection is zero, although the full quadratic interaction is not zero. The new reference adds the twelve nonzero wavevectors generated by the initial projected convection, producing eighteen retained wavevectors.

These extra modes have zero initial value and nonzero subsequent polynomial coefficients. Their evolution therefore retains nonlinear interactions rather than merely renaming the heat approximation. Higher generated modes are still omitted from the reference, and their full residual is still bounded.

Exact unrounded nonlinear Taylor recurrences can produce very large denominators. The generator instead:

1. Computes each Taylor coefficient by exact rational Galerkin arithmetic.
2. Rounds one vector from each conjugate pair to a declared dyadic grid, applies the exact rational Leray projection, and restores conjugate symmetry.
3. Rounds the interval endpoint in the same way.
4. Adds a highest-degree polynomial correction to land exactly on that endpoint; the next interval starts at the identical rational state.

The main runs use 48-bit rounding for these intermediate coefficients. Leray projection can introduce non-dyadic denominators; the resulting numbers remain exact rationals. The 64-bit setting separately controls the generator's outward norm and envelope witnesses.

The checker does not assume that these recurrences or rounding steps are correct. It receives only the final rational polynomials and reconstructs

\[
r=\partial_tv-\nu\Delta v+P\nabla\cdot(v\otimes v)
\]

over their entire time intervals and full quadratic output support. Rounding, endpoint correction, and discarded interactions all contribute to this residual. Exact Bernstein convex-hull bounds cover every time, not only sample nodes. All of the corresponding argument in `TRAJECTORY_PROOF.md` remains in force.

## 6. Executed results

| Input and configuration | Outcome | Maximum error bound | Full norm bound at \(T=4\) |
|---|---|---:|---:|
| \(\varepsilon=0.01\), initial support, sharper bound | Independently accepted | \(<0.026679\) | \(<0.044346\) |
| \(\varepsilon=0.01\), nonlinear reference, sharper bound | Independently accepted | \(<0.002382\) | \(<0.037488\) |
| \(\varepsilon=0.1\), nonlinear reference, sharper bound | Independently accepted | \(<0.04424\) | \(<0.05144\) |
| \(\varepsilon=0.5\), same nonlinear configuration | Envelope unresolved on the second interval | No accepted global bound | No conclusion |

Decimal entries are outward comparisons against the exact rational records. For \(\varepsilon=0.01\), retaining interactions reduces the certified maximum error bound by more than a factor of eleven. This compares two upper bounds, not measured exact trajectory errors.

The stronger \(\varepsilon=0.1\) certificate uses 32 polynomial intervals and 76 affine error segments. Every interval has a nonzero full residual. It includes 1,792 occurrences of residual modes outside the retained support: 56 per interval, not 1,792 distinct wavevectors. The complete solution can contain still more modes; the global error norm covers them.

The comparison and the unsuccessful \(\varepsilon=0.5\) result are frozen. Failure of a sufficient estimate is not a blowup observation or proof that the input is nonsmooth. This is a declared finite set of tests, not a globally optimal search or an unbounded theorem.

## 7. Independent checking and attempted refutation

`verify_sharp_trajectory.py` is standalone and imports only Python standard libraries. Its residual calculation uses divergence of the velocity tensor, its Bernstein conversion uses Horner degree elevation, and its norm checks use exact squared inequalities. These differ from the generator's implementations.

For the new scalar bound, the generator uses the piecewise expression (4), while the checker evaluates the quadratic at \(x=\max(1,V/D)\). It independently checks \(c>0\), \(\mu>0\), \(c+\mu<\nu\), the strict envelope cap, all joins, and the final small-data bound.

Twelve new test methods passed, including exact solenoidality and conjugate symmetry after rounding, nonlinear mode generation, exact endpoint joins, distinct residual reconstructions, both growth branches and equality cases, nonzero initial error, unsupported inputs, isolated standalone acceptance, optimized-Python rejection, and replacement of a stale successful output after a failed run.

The accepted \(\varepsilon=0.01\) nonlinear certificate was challenged with 23 deliberate corruptions. All were rejected. They include missing residual modes, undersized norm witnesses, discontinuous references, interior polynomial bumps invisible to endpoint values and derivatives, scope inflation, zero reserved dissipation, an exhausted viscosity budget, and equality at the strict error cap.

The counterexample record also rejects an attractive but false shortcut: using the small-\(V\) branch \(2V-D\) for every \(V\). At \(D=1,V=2,x=2\), the actual quadratic is four while the proposed bound is three. This refutes the scalar shortcut, not global fluid smoothness.

Independent program agreement and corruption tests do not formally verify the analytic bridge in a logical kernel. That work remains separate. No result here establishes seven physical dimensions, a universal language of nature, RH, or P versus NP.

## 8. Reproduce on a PC

Extract the engine archive. Python 3.10 or later and its standard library suffice; these research companions require no C++ compilation or third-party package. On Windows, replace `python3` with `py -3`.

```sh
python3 tools/sharp_trajectory.py
python3 tools/verify_sharp_trajectory.py results/sharp-certificate-v017.json
python3 tools/sharp_trajectory.py --epsilon 1/10 --output results/sharp-larger-input-v017.json
python3 tools/verify_sharp_trajectory.py results/sharp-larger-input-v017.json
python3 tools/compare_sharp_trajectory.py results/sharp-larger-input-v017.json
python3 tools/challenge_sharp_trajectory.py results/sharp-certificate-v017.json
python3 -m unittest discover -s tests -p 'test_sharp_trajectory.py' -v
```

To reproduce the next unresolved test:

```sh
python3 tools/sharp_trajectory.py --epsilon 1/2 --output results/sharp-next-limit-v017.json
```

An unresolved bound returns exit code 2 and writes a failure record, not a candidate certificate. Invalid input returns 1. A generated candidate returns 0 but still needs independent checking; the verifier returns 0 only on acceptance.

For an already generated certificate, only `verify_sharp_trajectory.py` and that JSON file are needed. The new schema adds exact rational `error_cap` and `dissipation_reserve` to the prior trajectory contract. Mode, time, degree, arithmetic-string, and file-size budgets remain unchanged. `--initial-field` accepts other supported exact inputs; success is not guaranteed.

The next concrete research target is the frozen \(\varepsilon=0.5\) failure. Possible changes include a larger retained interaction support, tighter interval norm bounds, or a less conservative error norm. Any proposed improvement must be assessed on the same input with full residual coverage and independently checked inequalities. General large-data regularity remains a much broader mathematical problem.