WEBVTT

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The finite neutral complex answers an algebraic question. In the generic stratum, every closed direction is

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gauge. The quotient has dimension zero. Exact minors and rank nullity certify this statement. They do not

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certify the continuous operator or a Higgs mass.

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Switch to the restored stratum. One closed class remains after quotienting by gauge directions. This exact

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distinction is useful for testing hypotheses about the scalar sector. It is separate from the continuous

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vector spectrum shown next.

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At the exact Wilson angle one fifth, the light eigenvalue lies between two hundred forty four over eight

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thousand four hundred twenty five, and four over one hundred twenty five. The guaranteed gap is three

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hundred thirty seven over four thousand. These are rational bounds for the continuous operator, with its

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boundary conditions, in the stated five component active block.

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Choose one quarter. The lower endpoint is one over twenty eight and the upper endpoint is one twentieth. The

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fixed model has unit lengths and boundary strengths, with eta zero. Three decoupled zero directions are kept

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separately. Changing the unrelated fermion controls does not change this certified model.

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The written proof first bounds the second eigenvalue using means, traces and the min max principle. A smooth

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trial function satisfies all operator boundary conditions. Its exact Rayleigh quotient and residual enter

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the published Temple inequality. The downloaded proof states these analytic arguments. The symbolic verifier

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checks their algebra and the rational endpoints; software tests alone are not the proof.

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Download the proof and verifier, and replay the certificate with SymPy. The enclosure establishes

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positivity, simplicity and isolation in this conditional model. It does not certify the Wilson derivative, R

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Z W, a vacuum, currents or a physical Higgs direction. Do not differentiate interval endpoints. The live

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demo, manual and certificate preserve these limits.