ic001_certificate
plain-language theorem explainer
This definition constructs a certificate string that consolidates the derivation of information as the physical ledger under Recognition Science. A physicist or information theorist working on foundational unifications would reference it to confirm the status of IC-001 after the J-cost and entropy results are in place. The construction is a direct string literal that enumerates non-negativity, symmetry, and equivalence properties without invoking further lemmas.
Claim. The certificate asserts that the information cost function satisfies $J(x) = 0$ if and only if $x=1$, $J(x)>0$ for $x≠1$, $J(x)=J(1/x)$, $J(x)≥0$ for all $x>0$, and that Shannon entropy equals the expected cost $H=∑p_i J(p_i)$ with $H≥0$, while Landauer erasure requires positive cost $k_B ln(2)$ and the golden ratio satisfies $J(φ)>0$.
background
The module establishes that every recognition ratio $x>0$ carries a J-cost given by the shifted form $H(x)=J(x)+1=½(x+x^{-1})$, which obeys the Recognition Composition Law. J-cost vanishes only at perfect balance $x=1$ and diverges as $x→0$, forcing existence. Shannon entropy is recovered as the expectation of this cost over a probability distribution, with the deterministic case yielding zero entropy and the uniform case maximizing it at $log n$ bits. Upstream results supply the reparametrized $H$ from Cost.FunctionalEquation and the list structures from Aesthetics and Anthropology modules that feed into the broader ledger interpretation.
proof idea
The declaration is a plain definition that builds the certificate via string concatenation of fixed lines. Each line records a prior result already established in the same module (non-negativity, symmetry, entropy equivalence). No tactics or external lemmas are applied beyond the built-in string append operator; the body simply lists the verified properties and ends with an #eval directive.
why it matters
The certificate marks completion of the IC-001 claim that information constitutes the ledger rather than describing it, directly implementing the module's core list of eight theorems. It sits downstream of the J-cost functional equation and the Recognition Composition Law, and it closes the loop on Wheeler's it-from-bit by equating physical facts with recognition events of positive cost. No downstream uses are recorded yet, leaving open whether later modules will invoke the certificate as a status check.
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