coherenceTimeCert
The definition assembles a certificate asserting that coherence times on the phi-ladder scale exactly by the golden ratio at each successive rung, together with explicit algebraic bounds on the eighth and twelfth powers. Quantum biologists modeling cryptochrome or similar systems would cite the certificate to connect a femtosecond base time to microsecond biological windows. It is assembled as a structure literal that delegates the scaling property to the ratio theorem and imports the Fibonacci equality plus the two inequalities.
claimLet the certificate structure require that coherence time at rung $k+1$ divided by coherence time at rung $k$ equals the golden ratio for every natural number $k$, that the eighth power of the golden ratio equals $21$ times the golden ratio plus $13$, that the eighth power exceeds $46$, and that the twelfth power exceeds $300$. The definition supplies the ratio property from the scaling theorem, the equality from the Fibonacci identity, and the two inequalities from the amplification bounds.
background
The module treats quantum coherence times as arising from the J-cost functional on the phi-ladder. Coherence time at rung $k$ is defined to be the base time multiplied by the golden ratio to the power $k$. The CoherenceTimeCert structure packages the constant-ratio property together with concrete numerical checks on the eighth and twelfth powers that follow from the Fibonacci representation of those powers. The local setting is the BIT hypothesis that biological and artificial systems maintain coherence at specific ladder rungs, with the base time near $7.3$ fs and the eighth harmonic near $8$ Hz.
proof idea
The definition is a one-line structure constructor. It assigns the ratio field to the theorem coherenceTimeRatio, the value field to the Fibonacci identity phi8_fibonacci, the eighth-power inequality to phi8_gt_46, and the twelfth-power inequality to phi12_gt_300.
why it matters in Recognition Science
The certificate supports the claim that biological coherence times sit at rung 12, producing times in the microsecond range from the femtosecond base. It closes the derivation from the J-cost functional through the phi-ladder to observable decoherence scales. Within the framework it aligns with the eight-tick octave and the spatial dimension count of three, providing a concrete interface for downstream calculations of coherence windows.
scope and limits
- Does not derive the numerical value of the base coherence time from the J-cost equation.
- Does not compute explicit coherence times at any chosen rung.
- Does not model environmental or thermal decoherence channels.
- Does not establish that the phi-ladder is the unique scaling compatible with the Recognition Composition Law.
formal statement (Lean)
65noncomputable def coherenceTimeCert : CoherenceTimeCert where
66 phi_ratio := coherenceTimeRatio
proof body
Definition body.
67 phi8_val := phi8_fibonacci
68 phi8_amp := phi8_gt_46
69 phi12_amp := phi12_gt_300
70
71end IndisputableMonolith.Physics.QuantumCoherenceTimeFromJCost