ic003_certificate
plain-language theorem explainer
ic003_certificate defines a formatted string summarizing the IC-003 derivation that the Physical Church-Turing Thesis holds for Recognition Science dynamics via the discrete ledger. A researcher bounding physical computation would cite it as the compact status report for finite phase space and approximability of phi. The body is a direct string literal that concatenates the listed finite cardinalities and computability assertions.
Claim. The IC-003 certificate is the string asserting that the phase space has cardinality 8, the map space has cardinality $8^8$, the discrete ledger state space has cardinality $2^8$, computation requires at least $n$ ticks for $n$ steps, and that for every $ε>0$ there exists a rational $q$ with $|φ - q| < ε$.
background
The module derives the Physical Church-Turing Thesis from the discrete ledger structure: each ledger entry is a ratio $x ∈ ℝ$ whose dynamics are governed by the 8-tick operator on a discrete phase space. Finite memory per tick follows from the 8-phase structure, and transitions remain computable because the J-cost minimization maps finite states via a continuous function. No hypercomputation is possible because the ledger processes only at rate $1/τ_0$ where $τ_0$ is the fundamental tick from Constants.tick. Upstream, DiscreteLedger supplies the concrete finite configuration space, step from CellularAutomata applies the local rule, and the approximability result uses the density of rationals in the reals.
proof idea
The definition is a direct string concatenation of fixed lines that enumerate the verified properties: phase space cardinality 8, function space cardinality $8^8$, ledger cardinality $2^8$, bounded tick rate, and the existence of rationals within any $ε$ of phi. No tactics or lemmas are applied; the body simply assembles the summary text from the module's listed checks.
why it matters
This definition closes the IC-003 section by supplying a compact human-readable certificate for the Church-Turing physics structure. It records the finite-state consequences of the discrete ledger from PhiForcing and the eight-tick octave, confirming that RS dynamics lie in BQP while exact irrational phi lies outside Turing computability. The certificate touches the open distinction between exact RS evolution and its rational approximations.
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