IndisputableMonolith.Quantum.BornRule
Module packaging the Born rule as forced from J-cost and the eight-tick sector measure, not postulated. It collects nonnegativity of norm-squared weights, phase independence, interference from relative phase, normalization, and a Gleason-style uniqueness link. Quantum facade, BornRuleStructure, entanglement ontology, and the skeleton chapter import it. Proofs reduce to Cost identities plus the BornRuleForcing uniqueness theorem on 8-mode sectors.
claimOn eight-tick mode amplitudes $\psi_k$, Born weights $p_k=|\psi_k|^2$ are nonnegative, independent of $\arg(\psi_k)$, and normalize when $\sum_k|\psi_k|^2=1$. They match the two-branch $\exp(-C)$ rule from J-cost, yield interference cross-terms from relative phase, and are the unique additive phase-invariant probability on 8-mode sectors (Gleason-style from RS).
background
Recognition Science treats quantum probability as derived structure on the discrete 8-tick clock (phases $0,\pi/4,\ldots,7\pi/4$), not as an independent axiom. The Cost layer supplies the J-functional $J(x)=(x+x^{-1})/2-1$ and the two-branch recognition weight $\exp(-C)$. Foundation.BornRuleForcing states the unique probability on 8-mode sectors that is normalised, phase-invariant (depends only on $|\psi_k|$), additive on disjoint mode sets, and consistent with that two-branch rule.
This module sits in the Quantum domain and turns that forcing into concrete Born-rule lemmas: consistency of norm-squared, phase independence, interference from phase, derivation and normalization from J-cost, and a Gleason-from-RS statement. Constants and EightTick fix the RS-native tick and the octave period $2^3$ underlying the sector measure.
proof idea
Not a single theorem: a small library of lemmas. Nonnegativity of norm-squared is elementary real analysis on complex amplitudes. Phase independence and $|e^{i\theta}|=1$ identities discharge arg-dependence. Interference lemmas expand $|\psi+\phi|^2$ cross terms from relative phase. The J-cost bridge lemmas reduce Born weights and normalization to Cost identities plus the unique 8-mode measure from BornRuleForcing. Gleason-from-RS packages that uniqueness as the RS stand-in for Gleason. A named falsifier interface records defeat conditions. No deep tactic search; mostly algebraic wrappers on upstream forcing.
why it matters in Recognition Science
Feeds the Quantum facade (re-exports recognition-first quantum structure), BornRuleStructure (structural nonnegativity of probability weights), EntanglementOntologyStructure (interference cross terms as entanglement content), and Skeleton.Quantum, which states explicitly that the Born rule is forced from J-cost rather than postulated.
In the broader chain this is the quantum-layer landing of T7 eight-tick structure and the Cost/J-uniqueness line: once the sector measure is unique under normalisation, phase invariance, additivity, and $\exp(-C)$ matching, standard Born weights and interference are theorems. Skeleton still flags the canonical commutator and the $\mathbb{Z}/8$ Weyl relation as open; this module closes the probability side, not the full noncommutativity keystone.
scope and limits
- Does not prove the canonical commutator $[x,p]=i\hbar$.
- Does not discharge the eight-tick Heisenberg–Weyl relation (still open upstream).
- Does not construct a full RSHilbertSpace or LedgerToHilbert bridge.
- Does not treat continuous-spectrum POVMs beyond 8-mode sector measures.
- Does not claim experimental numerics; only formal forcing and identities.