hbar_exponent_eq_configDim
plain-language theorem explainer
The RS-native action quantum equals φ to the power of minus the configuration dimension of a recognition event. At forced spatial dimension D=3 this recovers ħ=φ^{-5}, so the exponent 5 is D+2 rather than a free parameter. Anyone tying Planck's constant in RS units to the forcing chain would cite this. The proof evaluates configDim at D=3 and rewrites through the known identity ħ=φ^{-5}.
Claim. In RS-native units, $\hbar = \varphi^{-\mathrm{configDim}(D)}$, where $D=3$ is the forced spatial dimension and $\mathrm{configDim}(D)=D+2$ counts the degrees of freedom of a recognition event ($D$ spatial, one temporal tick advance, one ledger-balance mode). Equivalently $\hbar=\varphi^{-(D+2)}$, hence $\hbar=\varphi^{-5}$ at $D=3$.
background
Module GapDerivation closes boundary item B-22: the coherence-energy exponent equals the configuration dimension of a recognition event. That dimension is $D+2$: $D$ spatial modes from the lattice (T8), one temporal mode from tick advance (T2), and one balance mode from ledger neutrality $J(x)=J(x^{-1})$ (T3). Coherence energy is counted as $\varphi^{-1}$ per degree of freedom, so $E_{\mathrm{coh}}=\varphi^{-(D+2)}$.
In RS-native units the action quantum is $\hbar=E_{\mathrm{coh}}\cdot\tau_0$ with $\tau_0=1$. The upstream lemma hbar_eq_phi_inv_fifth records the native identity $\hbar=\varphi^{-5}$. The present theorem identifies that exponent $5$ with $\mathrm{configDim}(D)$ at the forced $D=3$, rather than treating $5$ as a free parameter.
Sibling configDim_at_D3 supplies the arithmetic $\mathrm{configDim}(3)=5$. Spatial $D$ itself is the constant $3$ forced by T8.
proof idea
Tactic proof in two moves. First prove the integer equality $-(\mathrm{configDim},D)=-5$ by invoking the sibling evaluation configDim_at_D3 and discharging with omega. Then rewrite the goal by that equality and by Constants.hbar_eq_phi_inv_fifth, realign the real power with the integer cast via Real.rpow_intCast, and close with norm_num. No new analytic content: it is a bridge from the configuration-dimension count to the already-recorded native identity $\hbar=\varphi^{-5}$.
why it matters
Pins the exponent in the primer identity $\hbar=\varphi^{-5}$ to configuration dimension $D+2$, closing B-22 as stated in the module doc: the exponent is forced, not chosen. Sits inside the Gap-45 package, where the same $D+2$ factor appears in gap_at_D3 ($D^2(D+2)=45$ at $D=3$). Downstream native-unit lemmas such as hbar_pos and tau0_pos live in the same C-004 package this identity supports.
Framework landmarks: T8 forces $D=3$; T2 and T3 supply the two extra degrees of freedom; the RS-native constant $\hbar=\varphi^{-5}$ is thereby read as configuration dimension rather than postulated. Complements the coprimality arguments in the same module that $D$ must be odd.
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