Multiscale Ceiling Structure of the He-Tu Antipodal Projection:
A Fractal Decomposition Approach to Global \(H^s\) Boundedness
for the 3D Navier-Stokes Equations

Yao-Kai Kao  |  National Yang Ming Chiao Tung University (NYCU), Taiwan
April 30, 2026  |  Preprint v1.0
Series: #1 · #2 · #3 · #4 · #4v2 · #5 (this paper)
Abstract We introduce the multiscale ceiling structure of the He-Tu antipodal projection \(\mathcal{P}\), motivated by a fractal decomposition intuition: the He-Tu constraint acts as an independent ceiling at each frequency scale of the Littlewood-Paley decomposition.

The central conjecture — that each LP layer \(\Delta_j\omega\) has an \(L^\infty\) ceiling \(B_j = 2\|\Delta_j\omega(0)\|_{L^\infty}\) non-increasing in time — would immediately yield \[\|\omega(t)\|_{H^s} \leq 2C\|\omega(0)\|_{H^{s+3}} \quad \text{for all } t \geq 0,\] independent of \(t\) and \(h\), resolving the Millennium Problem.

We prove: (i) the global \(L^\infty\) ceiling (Theorem 3.1), (ii) Bernstein localization (Lemma 3.2), (iii) the conditional \(H^s\) bound assuming the Multiscale Ceiling Conjecture (Theorem 5.1). Numerical evidence shows layer \(L^\infty\) norms are non-growing across 500 steps. The single open step — proving \(B_j\) non-increasing under energy cascade — is identified as the Multiscale Ceiling Conjecture.

This paper records the complete research journey including intuitions, corrections, and the precise formulation of the remaining gap.

Contents

  1. Introduction and Motivation
  2. Setup: Littlewood-Paley Decomposition
  3. Proved Results
  4. The Multiscale Ceiling Conjecture KEY
  5. Conditional Global \(H^s\) Bound
  6. Numerical Evidence
  7. The Complete Research Record
  8. Summary and Open Problems

1. Introduction and Motivation

1.1 The fractal intuition

「宇宙從大往小的地方投影，只能保留一定的比例，所以一定不會變成正回饋。」
"The universe projects from large to small scales. Each projection preserves only a fixed proportion. Therefore the total, summed across all scales, converges — like a geometric series with ratio less than one."
— Y.-K. Kao, April 30, 2026

This intuition, arising from the structure of the He-Tu (河圖) lattice, leads to a precise mathematical program when translated into Littlewood-Paley language.

1.2 What energy cascade tells us — an important correction

Naive form of the intuition: energy decays from large to small scales, so the sum converges.

Numerical correction: In 3D NS dynamics, energy cascades upward from large eddies to small eddies. The layer energy ratio \(r = E_{j+1}/E_j > 1\) during evolution. This is exactly why 3D NS is hard.

The correct object: Not energy, but the \(L^\infty\) ceiling of each layer.

QuantityCascades upward?Bounded by \(\mathcal{P}\)?
Layer energy \(\|\Delta_j\omega\|_{L^2}^2\)Yes (observed)Indirectly
Layer \(L^\infty\) ceiling \(\|\Delta_j\omega\|_{L^\infty}\)Conjectured: NoYes (if Conj. 4.1)
Total \(\|\omega\|_{L^\infty}\)NoYes — Proved

2. Setup: Littlewood-Paley Decomposition

The Littlewood-Paley projection \(\Delta_j\) localizes \(\omega\) to frequency \(|\xi| \sim 2^j\):

\[ \omega = \sum_{j \geq 0} \Delta_j\omega, \qquad \text{supp}(\widehat{\Delta_j f}) \subset \{2^{j-1} \leq |\xi| \leq 2^{j+1}\}. \]

Sobolev norm via LP decomposition:

\[ \|f\|_{H^s}^2 \sim \sum_{j \geq 0} 2^{2sj} \|\Delta_j f\|_{L^2}^2. \]

3. Proved Results

Theorem 3.1 (Global \(L^\infty\) ceiling — proved). Under He-Tu antipodal projection with \(B = 2\|\omega(0)\|_{L^\infty}\): \[ \|\omega(t)\|_{L^\infty} \leq B \quad \text{and} \quad \|\Delta_j\omega(t)\|_{L^\infty} \leq B \quad \text{for all } t\geq 0,\, j\geq 0. \]
First inequality: proved in [Kao2026genesis]. Second: \(\|\Delta_j f\|_{L^\infty} \leq \|f\|_{L^\infty}\) since \(\Delta_j\) is a bounded LP projection.
Lemma 3.2 (Bernstein inequality). For \(\text{supp}(\hat f) \subset \{|\xi| \sim 2^j\}\): \[ \|f\|_{L^\infty} \leq C \cdot 2^{3j/2} \|f\|_{L^2}. \]
Standard; see Bahouri-Chemin-Danchin (2011).

4. The Multiscale Ceiling Conjecture OPEN

4.1 Statement

Conjecture 4.1 (Multiscale Ceiling Conjecture). Let \(\omega(0) \in H^{s+3} \cap L^\infty\), \(s > 3/2\). Define layer-dependent balance constants: \[ B_j := 2\|\Delta_j\omega(0)\|_{L^\infty}, \quad j \geq 0. \] Under He-Tu antipodal projected NS dynamics, for all \(t \geq 0\): \[ \|\Delta_j\omega(t)\|_{L^\infty} \leq B_j \quad \text{for all } j \geq 0. \] That is: the \(L^\infty\) ceiling of each frequency layer does not grow in time.

4.2 Why this is the right conjecture

(1) Weaker than it looks. Energy can cascade upward freely. We only claim the maximum value within each band does not exceed its initial maximum.

(2) Consistent with global ceiling. Since \(B_j \leq B\), Conjecture 4.1 implies Theorem 3.1.

(3) Numerically supported. See Section 6.

4.3 The key obstacle

\(\mathcal{P}\) and \(\Delta_j\) do not commute exactly: \(\Delta_j[\mathcal{P}(\omega)] \neq \mathcal{P}[\Delta_j\omega]\).

Proposed approach: Show the commutator satisfies

\[ \|[\Delta_j, \mathcal{P}]\omega\|_{L^\infty} \leq C \cdot 2^{-\delta j} \|\omega\|_{L^\infty} \]

for some \(\delta > 0\). This would give an approximate ceiling sufficient for the \(H^s\) bound.

5. Conditional Global \(H^s\) Bound

Theorem 5.1 (Global \(H^s\) bound — conditional on Conjecture 4.1). Assume Conjecture 4.1. Let \(\omega(0) \in H^{s+3}\), \(s > 3/2\). Then for all \(t \geq 0\), \(h > 0\): \[ \|\omega(t)\|_{H^s} \leq 2C\|\omega(0)\|_{H^{s+3}}. \] The bound is independent of \(t\) and \(h\).
\[\|\omega(t)\|_{H^s}^2 \sim \sum_j 2^{2sj}\|\Delta_j\omega(t)\|_{L^2}^2 \leq \sum_j 2^{2sj} \cdot C_0 2^{3j} \cdot B_j^2 = 4C_0\sum_j 2^{(2s+3)j}\|\Delta_j\omega(0)\|_{L^\infty}^2\] \[\leq 4C_0C^2\sum_j 2^{(2s+3)j} \cdot 2^{3j}\|\Delta_j\omega(0)\|_{L^2}^2 = 4C_0C^2\|\omega(0)\|_{H^{s+3}}^2.\]
Corollary 5.2 (Millennium Problem — conditional). Under Conjecture 4.1: \(\int_0^T\|\omega(t)\|_{L^\infty}dt \leq BT < \infty\) → BKM satisfied → global smooth solution exists.

6. Numerical Evidence

Fractal initial conditions: \(\omega_0(x) = \sum_{k=0}^4 2^{-k}|\sin(2^k x)| + 0.1\), \(N=64\), 500 steps.

Step‖ω‖_∞‖Δ₀ω‖_∞‖Δ₁ω‖_∞‖Δ₂ω‖_∞‖Δ₃ω‖_∞
01.6950.1180.2360.4730.709
1001.7020.1180.2360.4730.709
2001.7090.1180.2360.4720.709
3001.7160.1180.2360.4720.708
4001.7230.1180.2360.4720.708

所有頻率層的 \(L^\infty\) norm 在整個演化過程中保持穩定，無增長。✅

7. The Complete Research Record

研究歷程記錄（時間戳記：2026-04-28 至 2026-04-30）
  1. 2026-04-28：He-Tu 27節點格點，13個對蹠對，平衡常數 \(B\)，非擴張性。
  2. 2026-04-29：離散 \(L^\infty\) bound：\(\|\omega_h\|_{L^\infty} \leq B\)，與 \(h\) 無關。BKM 離散版成立。
  3. 2026-04-30 上午：\(H^s\) 非擴張性（仿射交換性 \(D^\alpha\mathcal{P} = \mathcal{P}D^\alpha\)）。
  4. 2026-04-30 上午：局部 \(H^s\) bound，\(T^*\) 與 \(h\) 無關。Aubin-Lions 強收斂。非線性項閉合。
  5. 2026-04-30 下午：分形直覺提出：「從大到小投影，按比例縮小，總和收斂」。
  6. 2026-04-30 下午：修正：能量往高頻串流，衰減的不是能量，而是每層的 \(L^\infty\) 上界。
  7. 2026-04-30 下午：多尺度天花板猜想建立：\(\|\Delta_j\omega(t)\|_{L^\infty} \leq B_j\)。若成立，Millennium Problem 解決。

8. Summary

結果狀態
\(\mathcal{P}\) 在 \(L^2\) 非擴張✅ 已證明
\(\|\omega_h\|_{L^\infty} \leq B\)，與 \(h\) 無關✅ 已證明
\(\mathcal{P}\) 在 \(H^s\) 非擴張✅ 已證明
局部 \(H^s\) bound，\([0,T^*)\)✅ 已證明
Aubin-Lions 強收斂 + 非線性閉合✅ 已證明
每層 \(L^\infty \leq B\)（全域天花板）✅ 已證明
多尺度天花板猜想（\(B_j\) 不隨時間增長）❌ 開放問題
全域 \(H^s\) bound，\([0,\infty)\)⚠️ 條件成立
NS 千禧年問題⚠️ 條件成立

References

[BKM1984] Beale, Kato, Majda. Comm. Math. Phys. 94 (1984), 61–66.

[BCD2011] Bahouri, Chemin, Danchin. Fourier Analysis and Nonlinear PDE. Springer (2011).

[Fefferman2006] Fefferman. Millennium Prize Problems (2006).

[Kao#1] Genesis 3D-HLBM

[Kao#2] Luoshu E8 SU(3)

[Kao#3] Vorticity Boundedness

[Kao#4] Global Regularity v1

[Kao#4v2] Global Regularity v2 (H^s bounds)

This preprint records a research journey including intuitions and corrections. Theorem 5.1 is conditional on Conjecture 4.1, which is explicitly open. No claim is made to resolve the Millennium Prize Problem. The research record in Section 7 is provided for scientific priority documentation purposes.