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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Pratt's primality certificates establish coNP membership for PRIMES only because of deep number-theoretic structure; analogous certificates for composite numbers with *all* factor pairs are not obviously constructible in polynomial-length form.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
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    • 1.A single prime factor of n already constitutes a polynomial-length certificate for compositeness; the claim conflates difficulty with non-existence.
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    • 2.The asymmetry may reflect current proof techniques, not fundamental structure—coNP-completeness of COMPOSITES suggests certificates likely exist but aren't yet discovered.
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    • 3.Pratt's certificate works because PRIMES is in coNP; this doesn't prove analogous structures are impossible for composites, only that we haven't found them.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Pratt certificates leverage Fermat's Little Theorem and primitive roots—deep algebraic structures unavailable for composite factorization proofs.
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    • 2.Composite numbers lack a universal structural property analogous to primality; each composite requires individual factor verification, not a unified certificate.
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    • 3.Polynomial-length certificates for composites would need to encode all factor pairs compactly, but no known encoding exploits number-theoretic structure to achieve this.
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