barrier_cert
plain-language theorem explainer
The barrier_cert assembles the coprimality of 8 and 45, primality of beat frequency 37, and window non-alignment properties into the BarrierCert structure. Researchers on the Gap-45 module cite it when linking these non-alignment facts to the consciousness hypothesis. The definition is a direct field assignment from sibling lemmas already established in the same module.
Claim. The barrier certificate satisfies $8$ and $45$ are coprime, $37$ is prime, $37$ and $360$ are coprime, no window $w$ with $0 < w < 360$ is divisible by both $8$ and $45$, both $8$ and $45$ divide $360$, and the aliasing ratio $37/45 < 1$.
background
The Gap-45 Recognition Barrier module formalizes the mathematical core from coprimality of $8$ and $45$. No finite sub-period of the $8$-tick cycle aligns with the $45$-fold phase structure, so any bounded decision window encounters incompatible constraints. The module doc states that this forces experiential navigation as a hypothesis connecting the barrier to phenomenology.
proof idea
The definition is a one-line wrapper that populates the BarrierCert structure. It assigns coprime_barrier to the coprime field, beat_is_prime to the beat_prime field, beat_fraction_irreducible to the beat_irreducible field, window_insufficient to the window_min field, minimal_resolution_window to the full_resolution field, and aliasing_ratio_lt_one to the aliasing field.
why it matters
This supplies the mathematical core that feeds directly into the consciousness_hypothesis definition downstream. The module labels the theorems as proved while keeping consciousness emergence as a hypothesis. It touches the eight-tick octave by showing that the $8$-tick cycle cannot align with the $45$-fold structure inside any window shorter than $360$ ticks.
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