ic005_certificate
plain-language theorem explainer
ic005_certificate defines a formatted string summarizing the computational complexity of Recognition Science models based on J-cost minimization. A researcher mapping physical dynamics to standard complexity classes would cite this when classifying ground-state verification as linear time and high-rung states as exponential. The definition constructs the string by direct concatenation of enumerated properties drawn from J-cost lemmas and the eight-tick local update rule.
Claim. The certificate string asserts that $J(r) = (r-1)^2/(2r)$ has a unique minimum at $r=1$ (verifiable in $O(1)$ time), local 8-tick dynamics are $O(1)$ per step, balance verification over $N$ bonds is $O(N)$, gradient descent converges monotonically to the minimum, and rung-$n$ computations require time exponential in $n$.
background
Recognition Science determines physics complexity from the structure of J-cost minimization. J-cost is the function $J(r) = (r + r^{-1})/2 - 1$, shown equivalent to the squared form $(r-1)^2/(2r)$ by the upstream theorem jcost_squared_form. The module treats local dynamics as 8-tick updates on at most eight neighbors and ground-state balance as an $O(N)$ scan over bonds.
proof idea
The definition is a direct string concatenation that assembles a header followed by bullet points. Each bullet references a prior lemma such as jcost_unique_minimum, jcost_squared_form, jcost_pos_away_from_one, and jcost_deriv_zero_at_one to state the verified property and its complexity class.
why it matters
This definition supplies the human-readable summary for IC-005 inside the Information module, linking J-cost convexity to the placement of RS physics in the complexity zoo. It draws on the eight-tick octave from the forcing chain and the phi-ladder growth for mass rungs. No downstream theorems yet reference it, leaving open the question of formal reductions to BQP or PSPACE.
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