bosonJCostAverage
plain-language theorem explainer
This definition supplies the mean J-cost for bosonic states by averaging cosine evaluations at 0, π/2, π, and 3π/2. Researchers modeling supersymmetry breaking in Recognition Science cite it to quantify the spontaneous asymmetry between boson and fermion sectors arising from 8-tick phases. The definition evaluates to zero by direct substitution of the four trigonometric values.
Claim. The average J-cost for bosons is defined as $ (1 + 0 + (-1) + 0)/4 = 0 $, or equivalently $ (1/4) (cos 0 + cos(π/2) + cos π + cos(3π/2)) $.
background
In the Recognition Science treatment of the Standard Model, J-cost tracks the phase-dependent contribution to recognition cost under the eight-tick octave. Even phases (multiples of 2π/4) yield the sequence cos(nπ/4) = 1, 0, -1, 0, while odd phases yield 1/√2, -1/√2, -1/√2, 1/√2. The module sets this asymmetry as the mechanism that spontaneously breaks supersymmetry: bosons and fermions occupy distinct phase classes, so their average J-costs cannot match even if superpartners exist. The local setting is SM-010, which starts from the standard SUSY proposal (boson-fermion pairing to solve the hierarchy problem) and shows why exact SUSY is incompatible with the J-cost difference enforced by the 8-tick structure.
proof idea
The declaration is a direct definition that computes the arithmetic mean of the four cosine terms at even multiples of π/2. No lemmas are invoked; the value follows immediately from the known values cos 0 = 1, cos(π/2) = 0, cos π = -1, and cos(3π/2) = 0.
why it matters
This supplies the explicit numerical parameter that realizes spontaneous SUSY breaking inside the Recognition Science framework. It directly supports the module target of deriving the breaking scale from J-cost asymmetry rather than from an external soft term. The definition sits at the interface between the eight-tick octave (T7) and the Standard Model sector, providing the concrete quantity that later statements on SUSY viability and LHC limits would reference.
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