perBitCost
plain-language theorem explainer
perBitCost defines the J-cost of one bit of recognition as the natural logarithm of phi. Cryptographers deriving lower bounds on key-recovery attacks in Recognition Science cite this constant when establishing the structural minimum for n-bit symmetric keys. The definition is a direct assignment from the real logarithm applied to phi with no further computation.
Claim. The J-cost per bit of recognition equals the natural logarithm of the golden ratio: $J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 = 0 < J_1 =
background
In Recognition Science the J-cost quantifies the information cost of a single discrimination step on the recognition substrate. The golden ratio phi is the self-similar fixed point forced by the T5 J-uniqueness and T6 fixed-point steps of the unified forcing chain. The module states that the minimum cost of recovering an n-bit key is bounded below by n times this per-bit value under sigma-conservation, with additivity over the binary search tree of candidates.
proof idea
The definition is a direct assignment of Real.log to phi. No lemmas or tactics are applied inside the declaration itself.
why it matters
This supplies the base constant for the cryptography_one_statement theorem, which packages positivity, additivity via totalRecoveryCost_succ, and exact doubling via totalRecoveryCost_double. It fills the J-cost per discrimination step in the RS cryptographic hardness bound of the module. The construction connects directly to the phi-ladder and the eight-tick octave in the forcing chain.
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