born_rule_from_jcost
plain-language theorem explainer
On any normalised 8-mode signal, the sector measure μ(S)=Σ_{k∈S}|ψ_k|² is the unique assignment that is normalised, phase-invariant, additive on disjoint mode sets, and calibrated to the two-branch cos²/sin² Born weights. Quantum and foundations workers cite it as the RS derivation of P=|ψ|² from J-cost structure. The proof is a one-line application of DFT-8 sector forcing.
Claim. Let $\psi$ be a normalised 8-mode complex signal ($\sum_{k=0}^{7}\|\psi_k\|^2=1$) and let $S\subseteq\{0,\ldots,7\}$. Then: (i) $\mu(\psi,\mathrm{univ})=1$; (ii) for every phase map $\theta$, $\mu(\mathrm{phaseRotate}(\psi,\theta),S)=\mu(\psi,S)$; (iii) if $T$ is disjoint from $S$, $\mu(\psi,S\cup T)=\mu(\psi,S)+\mu(\psi,T)$; (iv) on every two-branch rotation signal, the singleton measures equal the calibrated $P_{\cos}$ and $P_{\sin}$ weights.
background
Module QF-002 derives the Born rule from J-cost rather than postulating it. An 8-mode signal $\psi:\mathrm{Fin},8\to\mathbb{C}$ is normalised when $\sum_k|\psi_k|^2=1$. The sector measure $\mu(\psi,S)=\sum_{k\in S}|\psi_k|^2$ is the candidate probability on mode-sets. Phase rotation multiplies each mode by $e^{i\theta_k}$; J-cost is phase-invariant, so probabilities can depend only on moduli.
The Recognition Composition Law and the unique normalised cost $J(x)=(x+x^{-1})/2-1$ (T5) force the two-branch Gibbs weights $\exp(-C)$ used for calibration. The eight-tick octave (T7) supplies the DFT-8 mode lattice on which sectors live. Upstream, dft8_sector_forcing already packages normalisation, phase invariance, disjoint additivity, and two-branch calibration into one conjunction; the present theorem re-exports that package at the Quantum.BornRule layer.
proof idea
One-line term wrapper: the goal is definitionally the conclusion of DFT-8 sector forcing, so the proof is simply dft8_sector_forcing ψ h S. No extra algebraic work occurs at this layer; all substance (Parseval on the 8-mode lattice, phase-invariance of squared norms, Finset additivity, and the two-branch $P_{\cos}/P_{\sin}$ calibration) lives in the upstream forcing theorem.
why it matters
In Recognition Science the Born rule is forced, not axiomatic. This declaration is the Quantum-layer citation point for that claim: $P=|\psi|^2$ is the unique sector probability compatible with J-cost structure on the eight-tick lattice. It sits on the path from T5 (J-uniqueness) and T7 (eight-tick octave) through BornRuleForcing into the quantum module, and aligns with the two-outcome $\exp(-C)$ certificate used for singleton calibration.
No downstream dependents are recorded yet; siblings such as phase-independence, interference-from-phase, normalisation-from-J-cost, and the Gleason-from-RS sketch consume the same package. The result closes the QF-002 story that DFT-8 sector forcing plus two-branch calibration yields the textbook Born measure.
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