canonicalThreshold
plain-language theorem explainer
The canonical threshold used when comparing the ISO A4 pitch standard to the Recognition Science φ-ladder is the real constant φ − 3/2. Acoustics and metrology workers checking whether 440 Hz sits inside the RS tolerance band would cite it. The declaration is a bare definitional abbreviation with no proof body.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio (the unique self-similar fixed point of the Recognition Composition Law).
background
The module studies the ISO 16:1975 standard A4 = 440 Hz against the Recognition Science φ-ladder. The module note records the numerical proximity φ^17 · 0.123 ≈ 439.2 Hz ≈ 440 Hz, framing A4 as a near-exact ladder rung rather than an arbitrary convention.
The imported Cost layer supplies a non-negative domain cost that measures mismatch between a laboratory frequency and its nearest RS-native value. The constant φ itself is forced uniquely by the Recognition Composition Law (forcing-chain step T6) and appears throughout RS-native units. The threshold φ − 3/2 is the numerical cutoff against which that domain cost is later compared.
proof idea
Definitional abbreviation only. The real constant is set equal to φ − 3/2; there are no lemmas, tactics, or computational steps.
why it matters
Gives the concrete cutoff that the A4 exactness certificate and its positivity lemma compare against. In the broader framework it translates the abstract φ-ladder (T6) into a laboratory frequency tolerance for the eight-tick octave structure. Without a fixed threshold the structural claim that 440 Hz is RS-exact would remain purely numerical rather than certificate-ready. Sibling positivity and certificate inhabitants depend on this value being a well-defined positive real.
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