curvature_tuple_uniqueness_bundle_for_delta_kappa
plain-language theorem explainer
Bundles three uniqueness statements for the legacy curvature correction δ_κ = −103/(102 π⁵): the π-power must be 5, the denominator coefficient must be 102, and the numerator must be 103. Cited by the α-resolution closure-status certificate and the structural-primitives wrapper that rewrites the same facts in seam primitives. Proof is a one-line product of three already-proved forcing lemmas.
Claim. Let $\delta_\kappa = -103/(102\,\pi^5)$. For natural numbers $d$, $k$, and $n$: $-\frac{103}{102\,\pi^d}=\delta_\kappa$ if and only if $d=5$; $-\frac{103}{k\,\pi^5}=\delta_\kappa$ if and only if $k=102$; and $-\frac{n}{102\,\pi^5}=\delta_\kappa$ if and only if $n=103$.
background
Alpha Resolution Pass 2 turns the residual between the symbolic RS inverse-fine-structure formula and the CODATA anchor into an explicit additive closure target. It does not yet derive a new geometric correction from first principles; it defines the exact shift needed for CODATA alignment and proves the corrected value sits in band, leaving a formal target for later geometry.
The legacy curvature correction is the real constant $\delta_\kappa = -103/(102,\pi^5)$, retained for older reports. The configuration-space dimension used for curvature integration is fixed at 5. The three atomic forcing facts packaged here say that, inside the natural one-parameter families that vary only the $\pi$-exponent, only the denominator coefficient, or only the numerator, equality with $\delta_\kappa$ pins those parameters to 5, 102, and 103 respectively.
proof idea
Term-mode packaging: the goal is a triple conjunction of biconditionals. The proof supplies the three conjuncts by applying, in order, the exponent-forcing lemma in the pure power family, the denominator-forcing lemma at fixed $\pi^5$, and the numerator-forcing lemma at fixed $\pi^5$. No extra algebra is performed at this layer.
why it matters
Gives the verification layer a single handle on curvature-tuple uniqueness for $\delta_\kappa$, so downstream certificates need not reassemble the three forcing facts. It is consumed by the structural-primitives wrapper that restates the same uniqueness in seam numerator/denominator and configSpaceDim, and by the module-level closure-status theorem, which records geometric-closure flags, exact CODATA alignment of the corrected $\alpha^{-1}$, uniqueness of the additive shift, and the exponent-forcing clause among its conjuncts.
In the broader RS picture this sits inside the $\alpha^{-1}$ band work (target window near 137.03–137.04): the curvature correction is still a legacy surface, and Pass 2 explicitly leaves open a first-principles derivation of an equivalent geometric term. The bundle makes that open target precise by locking the discrete tuple $(n,k,d)=(103,102,5)$.
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