semanticGate_implies_attention_cap
plain-language theorem explainer
The theorem extracts the attention upper bound from the semantic condensation gate predicate. Recognition bandwidth researchers cite it when enforcing attention caps inside admissible query regimes. The proof is a one-line wrapper that projects the fourth conjunct of the gate definition.
Claim. If the query-level semantic admissibility region holds (entropy exceeds its floor, signature lies in its closed interval, attention is at most phi cubed, and z meets or exceeds phi to the 45), then attention satisfies attention ≤ φ³.
background
The Critical Recognition Loading module develops structural lemmas for healthy operation inside the sub-saturation band of the load ratio rho = R_dem / R_max, with stability judged on the 360-tick supervisory horizon forced by lcm(8,45). SemanticCondensationGate encodes the query-level admissibility region as the conjunction entropyFloor < entropy ∧ signatureMin ≤ signature ≤ signatureMax ∧ attention ≤ phi³ ∧ phi⁴⁵ ≤ z. Upstream entropy definitions include total defect of a configuration (zero defect yields minimum entropy) and thermodynamic forms S = kβ⟨E⟩ + k ln Z or S = k_B (ln Z + β⟨E⟩) via partition functions.
proof idea
One-line wrapper that applies the fourth conjunct h.2.2.2.1 of the conjunction inside the SemanticCondensationGate definition.
why it matters
It supplies the attention bound to the downstream criticalRecognitionLoading_certificate, which asserts IsSubcritical together with attention ≤ phi³ and phi⁴⁵ ≤ z. The result closes one structural piece of the recognition loading control theorem, consistent with the eight-tick octave and phi-ladder forced by the T0-T8 chain.
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