IndisputableMonolith.Quantum.BornRuleStructure
Module collecting structural consequences of the Born rule once it is forced from the ledger: probability weights are nonnegative, phase factors cancel in the squared modulus, and the rule is compatible with the two-branch exp(−C) form. Quantum and foundations workers cite it when they need the nonnegativity and phase-invariance layer without reopening the uniqueness argument. Content is mostly short lemmas and structure bundles over the upstream Born-rule derivation.
claimStructural layer for the Born rule $P=|\psi|^2$ on 8-mode sectors: probability weights are nonnegative, global phases cancel in $P$, and the measure is compatible with the two-branch form $P\propto e^{-C}$ from the $J$-cost ledger.
background
Recognition Science forces the Born rule from the $J$-cost on the eight-tick (DFT-8) sector rather than postulating it. The upstream module BornRule states the uniqueness claim: $P=|\psi|^2$ is the unique probability measure on 8-mode sectors that is normalised, phase-invariant, additive over disjoint mode-sets, and consistent with the two-branch $\exp(-C)$ rule.
This module sits one layer above that uniqueness theorem. It packages the structural properties that follow once the measure is fixed: nonnegativity of weights at each mode, cancellation of pure phases in the squared modulus, and a small structure record that bundles those facts for downstream quantum constructions. The local setting is the RS quantum sector (QF-002), not a general Hilbert-space axiomatization.
proof idea
Not a single theorem: a short structural module. Typical pattern is to take the forced Born weight $P=|\psi|^2$ (or its ledger form via $e^{-C}$), expand the complex modulus, and read off nonnegativity and phase cancellation by elementary algebra on $\mathbb{C}$. Named siblings (born_rule_nonnegative_at, born_rule_phase_cancels, born_rule_implies_nonnegative, born_rule_structure, born_rule_from_ledger) are thin lemmas or structure constructors over that algebra; no deep new forcing is done here.
why it matters in Recognition Science
Closes the structural gap between the uniqueness theorem for $P=|\psi|^2$ (QF-002 / upstream BornRule) and any later use that needs nonnegativity or phase invariance as named facts. In the RS chain this sits after T7 (eight-tick octave) and the $J$-cost recognition composition law, which together force the two-branch $\exp(-C)$ seed that uniqueness promotes to full Born weights. No downstream edges are recorded yet in the mirror graph; the module is infrastructure for quantum probability statements that must not re-prove $|z|^2\ge 0$ or phase cancellation inline.
scope and limits
- Does not re-prove uniqueness of $P=|\psi|^2$; that lives in the upstream BornRule module.
- Does not derive the Born rule from scratch or reopen the DFT-8 sector forcing.
- Does not treat continuous-spectrum or infinite-dimensional Hilbert spaces beyond 8-mode sectors.
- Does not establish measurement dynamics, collapse postulates, or decoherence rates.
- Does not claim experimental bounds; only structural nonnegativity and phase cancellation.