tc_ratio_formula
plain-language theorem explainer
The ratio of critical temperatures T_c(n) over T_c(m) equals phi raised to the integer difference n minus m. Condensed matter researchers modeling high-Tc materials on the Recognition Science phi-ladder would cite this exact scaling to compare predicted transition temperatures across rungs. The proof is a direct algebraic reduction that unfolds the two definitions and applies the exponent subtraction identity for positive bases.
Claim. For integers $n$ and $m$, $T_c(n)/T_c(m) = phi^{n-m}$, where $T_c(k) = phi^k$ is the critical temperature on the $k$-th rung of the phi-ladder (in units normalized so that the coherence energy over Boltzmann constant equals one).
background
Module EN-002 derives room-temperature superconductivity conditions from the phi-ladder energy structure. Superconductivity requires binding energy at least k_B T; in RS this energy is quantized as E_n = E_coh phi^n with E_coh = phi^{-5} eV. The definition T_c_rung n sets the critical temperature for rung n to phi^n, absorbing the constant E_coh / k_B into the unit choice. The companion definition predicted_tc_ratio n m returns phi^{n-m} as the exact scaling factor between any two rungs.
proof idea
The term proof unfolds T_c_rung and predicted_tc_ratio, then rewrites the resulting power expression via the exponent subtraction identity zpow_sub_0 applied to the positive base phi.
why it matters
This supplies the precise ratio relation required by the EN-002 certificate for room-temperature superconductivity. It confirms that T_c scales exactly as phi^n on the ladder, consistent with the Recognition framework's T5 J-uniqueness and T6 phi fixed-point steps that force the self-similar energy quantization. The result feeds directly into the certificate string that lists the coherence and thermal-ratio checks.
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