T99 provides a geometric interpretation of the Born rule within a real symplectic-metric framework. Rather than treating quantum probabilities as fundamental postulates, the theorem derives them as normalized projection weights onto an observable subspace of a larger real state space. The construction begins with a real vector space \ equipped with a compatible metric-symplectic-complex structure \ (R, g, , J), \ where ²=-I, (Ju, Jv) =g (u, v). \ The state space is assumed to be split into complementary components =O', \ where \ (O\) is the observable subspace and '=JO\ is its conjugate counterpart. The theorem introduces the observable projection operator \O: R O\ and an orthogonal decomposition of the observable sector into measurement directions =ᵢ Oᵢ, \ with associated projectors \ (ᵢ\). Every state R\ admits a unique decomposition =o+h, O, O'. \ The observable component is =O v, \ while the conjugate component lies outside the directly observed sector. A central result is that the conjugate sector is metrically equivalent to the observable sector. Because (Jv, Jv) =g (v, v), \ The conjugate component possesses the same norm structure as the observable component. It is therefore not a lower-order correction or negligible remainder, but a geometrically equivalent part of the larger real state space. The theorem then derives the Born readout formula as a projection law. For each measurement direction \ (Oᵢ\), ᵢ (v) =\|ᵢ v\|²\|v\|². \ The probabilities satisfy \0 Pᵢ (v) 1, \ and \ᵢ Pᵢ (v) =PO (v) =\|O v\|²\|v\|². \ Whenever the state is normalized entirely within the observable sector, \\|O v\|=\|v\|, \ The probabilities satisfy \ᵢ Pᵢ (v) =1. \ The Born rule, therefore, emerges as the normalized squared projection of the full state onto observable measurement directions. Choosing the complex structure \ (J\) recovers the familiar complex formulation. The real metric and symplectic structures combine into the Hermitian inner product (, ) +i\, (, ), \ and the projection formula becomes ᵢ=| i||². \ The standard Born rule is thus recovered as the complex representation of the underlying projection geometry. Several consequences follow. Components lying entirely in the conjugate sector contribute no direct observable readout when measurements are restricted to the observable slice. The observable sector nevertheless remains a complete probability space, since normalized states within \ (O\) satisfy the usual probability-sum rule. Standard quantum mechanics, therefore, appears as the special case in which measurements access only the observable sector and no probability weight leaks from the conjugate component into observable readout. T99 provides a geometric foundation for the Born rule within the reduced Q5 framework. The theorem interprets quantum probabilities as observable-slice projections of a larger real symplectic structure and identifies the standard complex Born rule as the closed-observable-sector limit of that broader geometric description.
Craig Edwin Holdway (Wed,) studied this question.