Euler 1. Parameter Coercivity Collapse (10⁶⁹⁸ Hessian Spike)
Target: ./Euler/PacketInductionStage.lean · PacketSourceScaleChoice.lean · PacketSourceScaleSequence.lean
Evaluating the preprint's explicit finite parameters (Section 5.2/5.7: log k_J = x_0 / J² = 100 or x_0 = 5.0, J = 3) collapses the positive pressure Hessian increment (goodCost). At Stage J, evaluating against Lean's guards yields an initial stage defect exceeding 10⁶⁹⁸, violating Lean's initial_shear and initial_frequency bounds by hundreds of orders of magnitude.
Euler 2. Quantifier Decoupling & Non-Constructive Filter (X ≥ 10²⁵)
Target: ./Euler/PacketInductionScales.lean (L123–160) · PacketInfiniteConstruction.lean (L66)
Lean avoids the Section 5 coercivity collapse only by decoupling X from J. The code fixes J > 89,442 and D ≥ 2000 first, replaces stage polynomials with exponential majorants, and invokes Classical.choice over a non-constructive ∀ᶠ X in atTop filter. Satisfying this filter enforces X ≥ 10²⁵ (k_J ≥ exp(10¹⁵)), diverging completely from the preprint's finite parameter schedule.
Navier-Stokes 3. Post-Hoc Diagonal Cutoff Schedules for Exponential Mode Growth
Target: ./NavierStokes/ActualCandidateConstruction.lean · GluedStageEstimates.lean · MixedCandidateAssembly.lean
At cycle j, active harmonic modes scale as ||k|| ≤ 2^(j+1) (cycle_residualBand), causing stage derivative constants C_{j,m} to grow exponentially. To establish VanishingJointJets at (1,0), exists_schedule selects the diagonal cutoff sequence a_j → ∞ only after inspecting the sequence C_{j,m}, shrinking stage supports {q(t,x) < 1/a_j} into an arbitrarily thin cusp.
Navier-Stokes 4. External Realization Hypothesis (PhysicalFields / PhysicalData)
Target: ./NavierStokes/ActualCycleResidualBounds.lean (L473) · GluedStageEstimates.lean
The global physical primitives are not constructed unconditionally in Cartesian coordinates. In ActualCycleResidualBounds.lean, the codebase explicitly notes: "Only the actual physical realization remains external." The Cartesian fields are assumed via an external hypothesis (velocity_germ) identifying them with cylindrical graph pullbacks.
Navier-Stokes 5. Self-Documented Gaps & GraphRestriction Counterexample
Target: ./NavierStokes/GraphRestriction.lean (L16–18) · CycleContinuationInvariant.lean (L12–17)
Module headers explicitly state: "None of these statements supplies the manuscript's claimed all-order residual estimates." Furthermore, GraphRestriction.lean proves an explicit counterexample (zero_mean_does_not_force_zero), establishing that zero auxiliary mean on the torus does not force zero on the restricted graph.
Navier-Stokes 6. Annular Zero-Germ Isolation on the Blowup Axis
Target: ./NavierStokes/ComparatorR3Theorem.lean · MixedAxisPreservation.lean (L33, L115)
All infinite wave-packet corrections vanish identically in a neighborhood of the symmetry axis r = 0 (mixedDiagonal_eq_base_germ). Consequently, the singularity along x = 0 is isolated entirely to the Stage-0 self-similar background profile FinalSlowBase, while the external force f(t,x) on [0,1) is defined tautologically as the PDE residual.
Navier-Stokes 7. Conditional Implication in CandidateAssembly.lean
Target: ./NavierStokes/CandidateAssembly.lean (L10–14, L214) · DiagonalJetBoundsNoOptions.lean (L9–12)
The proof of candidate properties is conditional on analytic inputs whose existence is not established in the file. Module docs state: "A conditional implication, not a proof that its analytic inputs exist... It does not construct the input fields or the future force." Pointwise stage estimates remain external premises rather than derived theorems.
Comparator 8. Unresolved Clay Millennium Navier-Stokes Comparator
Target: ./ComparatorChallenges/NavierStokes.lean
The upstream comparator file retains intentional sorry challenge placeholders on NavierStokesExistenceAndSmoothnessRn and periodic breakdown. The file explicitly clarifies: "Neither the proof root nor the submission imports this reference."
Comparator 9. Unresolved Upstream Euler Breakdown Reference
Target: ./ComparatorChallenges/Euler.lean
The challenge definition for unforced Euler breakdown on ℝ³ (EulerExistenceAndSmoothnessR3) retains upstream sorry placeholders, remaining disconnected from the internal execution in Euler/Solution.lean.