OAI.CircleSampling.model_no_strict_projection: intermediate lemma from OpenAI, Invariant-projection counterexamples for every irrational rotation

An intermediate lemma of the split of OpenAI's proof of OAI.ContinuousCircleWeight.main_theorem, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Invariant-projection counterexamples for every irrational rotation, OpenAI Math Release, September 27, 2026, result family 293, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.CircleSampling.model_no_strict_projection of lean/OAI/Analysis/RotationWeights/Convolution.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CircleSampling, with open MeasureTheory, open scoped ENNReal NNReal ComplexInnerProductSpace BigOperators, open MeasureTheory Filter Set Submodule, open scoped ENNReal NNReal Topology ComplexInnerProductSpace, open MeasureTheory Filter Set, open scoped ENNReal NNReal Topology ComplexInnerProductSpace, open MeasureTheory Filter Set, open scoped ENNReal NNReal Topology ComplexInnerProductSpace, open MeasureTheory Filter Set, open scoped ENNReal NNReal Topology ComplexInnerProductSpace, open MeasureTheory Filter Set, open scoped ENNReal NNReal Topology ComplexInnerProductSpace, open MeasureTheory Filter Set, open scoped ComplexInnerProductSpace, open MeasureTheory Filter Set, open Set, open scoped ComplexInnerProductSpace, open Set, open scoped ComplexInnerProductSpace, open MeasureTheory Set Filter Topology, open scoped BigOperators ComplexConjugate, open MeasureTheory Set Filter, open scoped BigOperators ENNReal, open Filter Set, open scoped Topology, open Filter Set, open scoped Topology, open Set MeasureTheory, open scoped BigOperators ENNReal Classical, open Filter Set, open scoped Topology, open Set MeasureTheory, open scoped ENNReal Classical, open Filter, open scoped Topology, open Set Filter, open scoped Topology, open Set Topology, open scoped BigOperators, open scoped BigOperators, open Set, open Set, open Set, open Set MeasureTheory Filter, open scoped BigOperators ENNReal Topology Classical, open MeasureTheory Set Filter, open scoped ENNReal Topology, open MeasureTheory Set, open MeasureTheory Filter Set, open scoped Topology BoundedContinuousFunction, open MeasureTheory Set Filter Topology TopologicalSpace, open MeasureTheory Set, open scoped ENNReal BigOperators Classical, open MeasureTheory Set, open scoped BigOperators ENNReal Classical, open MeasureTheory Set, open scoped ENNReal, open MeasureTheory Set, open scoped ENNReal NNReal, open MeasureTheory Set Filter, open scoped BigOperators ENNReal Topology, open Set MeasureTheory Topology, open scoped symmDiff, open MeasureTheory Set, open scoped ENNReal BigOperators, open MeasureTheory Set, open scoped ENNReal, open MeasureTheory Set, open scoped BigOperators ENNReal, open MeasureTheory Set, open scoped ENNReal NNReal BigOperators, open MeasureTheory Set Filter ContinuousCircleWeight, open scoped ENNReal BigOperators Topology):

theorem model_no_strict_projection (P : ModelProjectionField)
    (hp : ∀ᵐ x,0 < modelDiagonal P x ∧ modelDiagonal P x<1) : False

Definitions. The statement is written with OAI.CircleSampling.CircleX (definitions bundle OAIQ2Rw000 of this split); OAI.CircleSampling.ModelProjectionField and OAI.CircleSampling.modelDiagonal (definitions bundle OAIQ2Rw002 of this split).

Used by. The proof of OAI.ContinuousCircleWeight.main_theorem of this split imports it.