IndisputableMonolith.Quantum.BornRule
Module collecting the Born-rule layer of Recognition Science: squared-modulus weights forced from J-cost on the eight-tick sector measure, with phase independence, non-negativity, and interference cross terms. Quantum facade, BornRuleStructure, entanglement ontology, and the Skeleton quantum chapter import it. Argument shape is a thin bridge from Foundation.BornRuleForcing plus elementary complex-norm identities.
claimOn eight-tick mode amplitudes $\psi$, the unique normalised, phase-invariant, additive probability measure consistent with the two-branch $\exp(-C)$ rule is the Born weight $P(\psi)=|\psi|^2$. In particular $|e^{i\theta}\psi|^2=|\psi|^2$, interference arises from relative phase, and the weights are nonnegative.
background
Recognition Science runs on a discrete eight-tick clock (phases $k\pi/4$ for $k=0,\ldots,7$). Foundation.BornRuleForcing states that the unique probability measure on 8-mode sectors that is (1) normalised on normalised states, (2) phase-invariant (depends only on $|\psi_k|$), (3) additive over disjoint mode sets, and (4) consistent with the two-branch $\exp(-C)$ Born rule, is the squared-modulus measure.
The cost side enters through IndisputableMonolith.Cost and the J-functional of the forcing chain ($J(x)=(x+x^{-1})/2-1$). Constants supplies the RS-native tick $\tau_0=1$. This module sits above that foundation and packages the quantum-facing consequences: consistency of $|\cdot|^2$, phase independence, interference from relative phase, normalisation and Born weights from J-cost, and a Gleason-style uniqueness statement from the RS axioms.
Sibling content includes born_rule_consistent (normSq nonnegative), phase and interference lemmas, born_rule_from_jcost / normalization_from_jcost, gleason_from_rs, and a named falsifier interface.
proof idea
Not a single theorem: a small library. Core uniqueness and forcing are imported from Foundation.BornRuleForcing (DFT-8 sector measure). Local lemmas are mostly short analytic facts: $|e^{i\theta}|=1$ implies norm-squared phase invariance; nonnegativity of $|z|^2$; cross terms that produce interference from relative phase; and wrappers that specialise the forced measure to J-cost normalisation and Born weights. Gleason-from-RS packages the uniqueness hypotheses into the RS quantum interface. Falsifier is an interface Prop, not a proved theorem.
why it matters in Recognition Science
Feeds the Quantum facade (re-exports recognition-first quantum structure), BornRuleStructure ("probability weights are nonnegative"), EntanglementOntologyStructure (interference cross terms as structural entanglement content), and Skeleton.Quantum, which records that "the Born rule is forced from J-cost (not postulated)".
In the framework this is the quantum landing of T7's eight-tick octave together with the cost law: measurement weights are not an extra axiom but the unique measure compatible with phase invariance and $\exp(-C)$ on the 8-mode clock. It closes the Born half of the skeleton quantum chapter; the still-open keystone named downstream is the $\mathbb{Z}/8$ clock-shift Weyl relation and the canonical commutator, not the Born weights themselves.
scope and limits
- Does not prove the canonical commutator $[x,p]=i\hbar$.
- Does not construct a full RSHilbertSpace or LedgerToHilbert bridge.
- Does not discharge the eight-tick Weyl relation (open D3 keystone).
- Does not treat continuous-spectrum POVMs beyond the 8-mode sector measure.
- Does not claim experimental numerics; only the forced measure and norm identities.