closure_selects_no_value
plain-language theorem explainer
Any two distinct positive targets for α⁻¹ are realized by two distinct positive U(1) normalizations κ_γ, each keeping the forced Q₃ closure intact. Cite when arguing that forced closure alone never selects a unique inverse coupling. The proof applies the surjectivity lemma twice and recovers κ-distinctness from target-distinctness by rewriting.
Claim. For all $t_1,t_2>0$ with $t_1\neq t_2$, there exist $\kappa_1,\kappa_2>0$ with $\kappa_1\neq\kappa_2$ such that the $\kappa_\gamma$-scaled inverse coupling equals $t_1$ at $\kappa_1$ and equals $t_2$ at $\kappa_2$ (i.e. $\kappa_i\cdot\alpha^{-1}_{\mathrm{RS}}=t_i$).
background
In Alpha Genesis the U(1) kinetic normalization $\kappa_\gamma>0$ multiplies the RS inverse-coupling stiffness linearly: $\mathrm{alphaInvK}(\kappa)=\kappa\cdot\mathrm{Constants.alphaInv}$, recovering the construction value at $\kappa=1$. The module upgrades “$\alpha^{-1}$ is a boundary datum” from measured status to a structural theorem via the $\kappa_\gamma$-scaling test.
Forced free-energy closure on the 3-cube $Q_3$ (octahedral face-adjacency Laplacian, spectrum ${0,4,4,4,6,6}$, $\det'M=2304$) was computed directly. The Gaussian log-det and Green-diagonal terms neither supply the $+1/(4\pi)$ counterterm a derivation would need, nor a $-7\times10^{-4}$ tail, so no forced condition pins $\kappa_\gamma$.
Upstream, alphaInv_irreducible_under_closure states: for every $t>0$ there is $\kappa>0$ with ForcedClosure $\kappa$ and $\mathrm{alphaInvK},\kappa=t$. ForcedClosure itself is $\kappa$-independent (photon cycle rank $b_1=5$, seed channel count $11\neq5$).
proof idea
Term-mode proof. Apply alphaInv_irreducible_under_closure to $t_1>0$ to obtain $\kappa_1>0$ with ForcedClosure and $\mathrm{alphaInvK},\kappa_1=t_1$; likewise for $t_2$ to obtain $\kappa_2$. Package the two witnesses. The remaining goal $\kappa_1\neq\kappa_2$ is by contradiction: if $\kappa_1=\kappa_2$, rewrite both scaled equalities to get $t_1=t_2$, contradicting the hypothesis. ForcedClosure witnesses are discarded (underscores); only positivity and the two equalities are kept.
why it matters
Corollary to the κ_γ-scaling irreducibility theorem: forced closure does not select any preferred positive $\alpha^{-1}$. The assignment of targets to normalizations is bijective on $\mathbb{R}_+$, so every candidate is equally compatible with the $Q_3$ invariants. This places $\alpha^{-1}$ parallel to a renormalization-scheme input, not a derived constant like $\hbar=\varphi^{-5}$ from the forcing chain.
The module’s load-bearing soundness point is that a unit-normalization $\ln Z=0$ would fix $\kappa_\gamma$ via the $+(5/2)\ln\kappa_\gamma$ free-energy term, but $\ln Z=0$ is not a forced RS closure. No downstream consumers are wired yet; the result closes the “no fixed point pins it” branch of Alpha Genesis and underwrites the panel ruling that $\alpha^{-1}$ stays a boundary datum inside the reported band.
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