The precise contribution supported by the checked comparisons

Research author: Rushikesh Murlidhar Walke
Manuscript: Global heat-seeded Picard convergence near a nonlinear periodic Navier–Stokes flow, version 1.1, 17 October 2026.

1. Contribution statement

For the explicit nonlinear periodic datum W=(sin z, sin x, sin y) at viscosity one, the manuscript supplies a computer-assisted certificate, uniform over smooth, real, mean-zero, divergence-free perturbations satisfying A₋₁(h)≤10⁻⁴, that the actual ordinary heat-seeded Picard sequence enters a common invariant set at its third iterate and converges on the whole time half-line in L²ₜA₀, with error at iteration n≥3 bounded by 6(9/10)ⁿ⁻³. Separate regularity results enlarge a specified previously stated cyclic-shear class.

This describes the manuscript’s claimed theorem and the difference established by this targeted comparison. The theorem’s validity remains dependent on its analytic proof and finite certificates. This audit verifies the comparison and elementary consequences; it does not independently prove the complete theorem. The class-inclusion argument was already present in manuscript §1.1: the new work here checks it against the accessible primary source rather than presenting it as a newly invented theorem.

2. Exact scope of the main theorem

On the normalized 2π-periodic torus, with no external forcing and viscosity one, define A_s(f)=Σ_k≠0|k|^s|f̂(k)| and X=L²((0,∞);A₀). Theorem 2.1, printed page 5, assumes u₀=W+h, W=(sin z,sin x,sin y), and A₋₁(h)≤10⁻⁴. The perturbation remains smooth, real, mean-zero and divergence-free; it may have infinitely many Fourier modes.

The implementation burden in §§2.4–2.5 is a 17-block estimate retaining the full causal history, an invariant region centred on the unperturbed third iterate, and verified entry of the actual perturbed third iterate. The sequence is the prescribed ordinary global iteration, without an independently substituted seed or time restart. A rougher controlling norm does not make this a theorem for arbitrary nonsmooth distributions.

3. Direct comparison with primary sources

Derived comparison, not a quotation: for the global Fourier–Wiener space Y used by Ambrose and coauthors, A₀(f)²≤A₋₁(f)A₁(f) implies ‖f‖_L²ₜA₀≤√(‖f‖_𝓧⁻¹‖f‖_𝓧¹)≤‖f‖_Y/2. Thus global L²ₜA₀ convergence is not new in isolation. The identified contribution is the particular certified W-centred family, actual-entry condition and uniform numerical bounds together.

4. An exact example beyond the named elementary test

Take h=(1/5000)sin(2y)eₓ and u₀=W+h. This h is smooth, real, mean-zero and divergence-free. The two nonzero coefficients each have magnitude 1/10000 at wave numbers ±(0,2,0). Hence A₋₁(h)=1/10000 exactly and A₋₁(u₀)=3.0001. It satisfies the main theorem’s stated radius.

The complete heat seed satisfies A₀(eᵗΔu₀)=3e⁻ᵗ+(1/5000)e⁻⁴ᵗ, so its X-norm squared is exactly 4.500240005. Using the manuscript’s elementary bilinear constant c=1/(2√2), the origin-centred Banach sufficient test has

Therefore that specific global origin-centred norm estimate cannot certify this example, while the stated W-centred theorem does cover it. This does not establish divergence of the iteration or failure of every prior method. It also avoids comparing an ℝ³ Fourier-integral threshold to periodic Fourier coefficients.

For h_m=(m/10000)sin(my)eₓ, integer m≥2, one has A₋₁(h_m)=10⁻⁴ for every m, but A₂(h_m)=m³/10000 and ‖h_m‖_Ḣˢ=m^s+1/(10000√2). This is a uniform smooth family with no uniform higher-Sobolev bound when s>−1. These deductions expose the role of the chosen topology; they do not create new hypotheses.

5. Strict enlargement of one stated regularity class

Let P interchange x and y and Tf(x)=P f(Pᵀx). Then TU=W and T preserves the equation and every A_s norm. Since |k|≥1 on nonzero integer modes, A₋₁(Th)≤A₂(h). Keeping the same smooth, real, mean-zero and divergence-free perturbation conditions, the earlier cyclic class therefore maps into the manuscript’s Theorem 3.1 region C(8/7,1/100).

Strictness is independently checked in two ways. W lies in the latter set but its minimum A₂ distance from bW, 0≤b≤0.7, is 0.9, exceeding 0.01. At the same amplitude 0.7, take h=0.01sin(2y)eₓ: A₋₁(h)=0.005, whereas A₂(h)=0.04. Its separate Fourier support prevents another amplitude decomposition from removing that discrepancy.

This is a comparison for global PDE regularity. The main ordinary-Picard theorem is not extended to that larger class. No comparison is made between different action quantities, constants or cutoff geometries. The strict containment concerns the published explicit sets, not the maximum possible reach of either general method.

6. A usable consequence of the stated convergence rate

The geometric upper bound gives the following stopping indices. These are exact deductions from the theorem, not measured iteration errors, runtime results or finite-time L² velocity errors.

The earlier external-solver experiment used a different finite-time normalized L² energy estimate. It does not establish this main theorem or its novelty. These two pieces of work must retain separate claims and attribution.

7. Decision and remaining limits

Supported framing: a specified, quantitatively certified extension within established Navier–Stokes analysis. The relevant evidence is the complete tuple of datum family, perturbation norm, prescribed iteration, actual-entry certificate, global norm and explicit constants. The strict inclusion above provides a particularly clear comparison for the separate regularity result.

Not decided: exhaustive historical novelty, independent development or priority, full proof correctness, optimal constants, acceptance, financial value or legal entitlement. The checked results do not directly provide the whole stated tuple, but absence from this targeted comparison is not proof of world-first priority. Source attribution and AI-assistance disclosure remain appropriate.

The concrete next use of this result is a precise contribution statement, supported by the comparison and explicit witnesses, when discussing or documenting the manuscript. This audit supplies that statement. No additional numerical solver benchmark is needed to establish the elementary set inclusion.

from fractions import Fraction as F
delta=F(1,5000); m=2
assert delta/m == F(1,10000)
heat_squared=F(9,2)+6*delta/F(1+m*m)+delta**2/F(2*m*m)
assert 2*heat_squared == F(900048001,100000000) > 1
assert 3*(1-F(7,10)) == F(9,10) > F(1,100)
assert F(1,100)/2 == F(1,200) <= F(1,100)
assert F(1,100)*2**2 == F(1,25) > F(1,100)
for goal,answer in [(F(1,100),64),(F(1,1000),86),(F(1,10**6),152)]:
    n=3
    while 6*F(9,10)**(n-3)>goal:
        n+=1
    assert n==answer
print("Exact arithmetic checks passed.")

{
  "scope": "Exact elementary deductions from stated theorem hypotheses; not a rerun or a new proof of the manuscript theorem.",
  "manuscript_sha256": "74af3faa5cb535ff641bca0f6cb0db1ca4df9f11b90353a57b48bf1e9174009b",
  "main_example": {
    "h": "(1/5000) sin(2y) e_x",
    "A_minus1_h": "1/10000",
    "A2_h": "1/1250",
    "A_minus1_u0": "30001/10000",
    "heat_seed_X_squared": "900048001/200000000",
    "elementary_zero_center_contraction_factor_squared": "900048001/100000000",
    "E0": "75000001/50000000",
    "X0": "18750001/12500000",
    "E0_X0_over2": "1406250093750001/1250000000000000"
  },
  "rate_deductions": [
    {
      "X_error_target": "1/100",
      "least_iteration_from_stated_bound": 64,
      "bound_exact": "48519278097689642681155855396759336072749841943521979872827/5000000000000000000000000000000000000000000000000000000000000"
    },
    {
      "X_error_target": "1/1000",
      "least_iteration_from_stated_bound": 86,
      "bound_exact": "47780373265559358009192445772039644256794138579444188363337645975725833060824587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000"
    },
    {
      "X_error_target": "1/1000000",
      "least_iteration_from_stated_bound": 152,
      "bound_exact": "45630493019529458663775342460696105322154565208445812157160063359456332392499692197933402051896313796083494813532664680937837577588565258188667/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
    }
  ],
  "coordinate_change": "P swaps x and y, (Tf)(x)=P f(P^T x); checked T U=W",
  "strict_class_witnesses": {
    "amplitude": {
      "u0": "W",
      "min_A2_distance_from_prior_segment": "9/10",
      "prior_radius": "1/100"
    },
    "same_amplitude_perturbation": {
      "u0": "(7/10)W+(1/100)sin(2y)e_x",
      "A_minus1_h": "1/200",
      "min_A2_distance_from_prior_segment": "1/25",
      "prior_radius": "1/100"
    }
  },
  "high_frequency_family": {
    "h_m": "(m/10000)sin(my)e_x, integer m>=2",
    "A_minus1": "1/10000",
    "A2": "m^3/10000",
    "Hdot_s": "m^(s+1)/(10000*sqrt(2))"
  },
  "checks_passed": true
}