This paper constructs a finite role-side readout package over the $A_{5}$ substrate. It does not claim to derive the general Hilbert-space Born rule, solve the measurement problem, complete Bell/contextuality analysis, or supply quantum gravity. Its aim is more limited and architectural: to isolate the role-side data required by a later Material–Role Separation framework. Starting from the regular address container ${\mathbb{C}}{\lbrack A_{5}\rbrack}$, the paper fixes the irreducible channel labels ${Irr}{(A_{5})}$ and the finite Address–Born channel-capacity measure This is the finite-group Plancherel weight specialized to the $A_{5}$ address container. Equivalently, ${({\dim\rho})}^{2}$ is the dimension of the $\rho$-sector $V_{\rho} \otimes V_{\rho}^{\ast}$ of the regular address algebra, so $\mu_{AB}$ is a normalized sector capacity rather than a postulated quadratic law. The terminology “Address–Born” refers only to this square channel accounting induced by regular address capacity. It does not mean that the general Hilbert-space Born rule, has been derived. The latter belongs to a downstream projector/effect readout bridge. The novelty claimed here is therefore not the representation-theoretic formula itself, but its placement inside an $A_{5}$ readout architecture: regular address accounting, channel weights, binary-icosahedral readout bits, nonnegative observation kernels, signed signal kernels, and localized admission failures are separated and exported as one role-side package. The paper closes four finite role-side statements. First, once the regular address container ${\mathbb{C}}{\lbrack A_{5}\rbrack}$ is adopted, the channel-capacity measure on ${Irr}{(A_{5})}$ is fixed as ${\mu_{AB}{(\rho)}} = {{({\dim\rho})}^{2}/60}$. Second, this measure is a capacity measure over irreducible channels, not an observed probability law by itself. Third, observed probabilities require an admissible setup-dependent kernel $K_{\Sigma,P}$ satisfying stochasticity, address factorization, symmetry compatibility, finite-protocol origin, and predictive compression. Fourth, the $2I$ readout bits do not replace the container ${\mathbb{C}}{\lbrack A_{5}\rbrack}$ by ${\mathbb{C}}{\lbrack{2I}\rbrack}$; they enter only as lift-admitted signed-signal labels over the retained $A_{5}$ channel support. Observation probabilities are not identified with $\mu_{AB}$ alone. They arise only after admission of a nonnegative setup-dependent kernel, Signed readout data, including the skeleton ${\Phi{(\varepsilon_{sig},s)}} = {\varepsilon_{sig}s}$, are assigned to a signal or contrast layer rather than to the probability kernel. The deliverable is the package ${\mathsf{A}\mathsf{B}}_{Role}$, designed for export to the Hodge–McKay comparison and to the later ${MRS}^{\prime}$ selection grammar.

QeiosJun 23, 20260 citations

International Journal of Theoretical Physics

Probability Weights and Measures on Finite Effect Algebras

David J. Foulis, R. J. Greechie

Published Dec 1, 1999• 3 citations

Publisher

DOI: 10.1023/a:1026630231694

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This paper constructs a finite role-side readout package over the $A_{5}$ substrate. It does not claim to derive the general Hilbert-space Born rule, solve the measurement problem, complete Bell/contextuality analysis, or supply quantum gravity. Its aim is more limited and architectural: to isolate the role-side data required by a later Material–Role Separation framework. Starting from the regular address container ${\mathbb{C}}{\lbrack A_{5}\rbrack}$ , the paper fixes the irreducible channel labels ${Irr}{(A_{5})}$ and the finite Address–Born channel-capacity measure This is the finite-group Plancherel weight specialized to the $A_{5}$ address container. Equivalently, ${({\dim\rho})}^{2}$ is the dimension of the $\rho$ -sector $V_{\rho} \otimes V_{\rho}^{\ast}$ of the regular address algebra, so $\mu_{AB}$ is a normalized sector capacity rather than a postulated quadratic law. 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They arise only after admission of a nonnegative setup-dependent kernel, Signed readout data, including the skeleton ${\Phi{(\varepsilon_{sig},s)}} = {\varepsilon_{sig}s}$, are assigned to a signal or contrast layer rather than to the probability kernel. The deliverable is the package ${\mathsf{A}\mathsf{B}}_{Role}$, designed for export to the Hodge–McKay comparison and to the later ${MRS}^{\prime}$ selection grammar.

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This paper constructs a finite role-side readout package over the

A_{5}

substrate. It does not claim to derive the general Hilbert-space Born rule, solve the measurement problem, complete Bell/contextuality analysis, or supply quantum gravity. Its aim is more limited and architectural: to isolate the role-side data required by a later Material–Role Separation framework. Starting from the regular address container

{\mathbb{C}}{\lbrack A_{5}\rbrack}

, the paper fixes the irreducible channel labels

{Irr}{(A_{5})}

and the finite Address–Born channel-capacity measure This is the finite-group Plancherel weight specialized to the

A_{5}

address container. Equivalently,

{({\dim\rho})}^{2}

is the dimension of the

\rho

-sector

V_{\rho} \otimes V_{\rho}^{\ast}

of the regular address algebra, so

\mu_{AB}

is a normalized sector capacity rather than a postulated quadratic law. The terminology “Address–Born” refers only to this square channel accounting induced by regular address capacity. It does not mean that the general Hilbert-space Born rule, has been derived. The latter belongs to a downstream projector/effect readout bridge. The novelty claimed here is therefore not the representation-theoretic formula itself, but its placement inside an

A_{5}

readout architecture: regular address accounting, channel weights, binary-icosahedral readout bits, nonnegative observation kernels, signed signal kernels, and localized admission failures are separated and exported as one role-side package. The paper closes four finite role-side statements. First, once the regular address container

{\mathbb{C}}{\lbrack A_{5}\rbrack}

is adopted, the channel-capacity measure on

{Irr}{(A_{5})}

is fixed as

{\mu_{AB}{(\rho)}} = {{({\dim\rho})}^{2}/60}

. Second, this measure is a capacity measure over irreducible channels, not an observed probability law by itself. Third, observed probabilities require an admissible setup-dependent kernel

K_{\Sigma,P}

satisfying stochasticity, address factorization, symmetry compatibility, finite-protocol origin, and predictive compression. Fourth, the

2I

readout bits do not replace the container

{\mathbb{C}}{\lbrack A_{5}\rbrack}

{\mathbb{C}}{\lbrack{2I}\rbrack}

; they enter only as lift-admitted signed-signal labels over the retained

A_{5}

channel support. Observation probabilities are not identified with

\mu_{AB}

alone. They arise only after admission of a nonnegative setup-dependent kernel, Signed readout data, including the skeleton

{\Phi{(\varepsilon_{sig},s)}} = {\varepsilon_{sig}s}

, are assigned to a signal or contrast layer rather than to the probability kernel. The deliverable is the package

{\mathsf{A}\mathsf{B}}_{Role}

, designed for export to the Hodge–McKay comparison and to the later

{MRS}^{\prime}

selection grammar.