vacuum_climate_zero_cost
plain-language theorem explainer
The theorem shows that the all-zero climate state has J-cost exactly zero. Climate dynamics researchers using the Recognition Science framework cite it as the base case establishing the vacuum as the unforced equilibrium. The proof is a direct term-mode reduction: unfold the climateJCost sum and apply the Jcost unit lemma to each component.
Claim. For any natural number $N$, the climate J-cost of the zero state satisfies $J_mathrm{climate}(mathbf{0}) = 0$, where $J_mathrm{climate}(s) = sum_{i=1}^N J(1 + s_i^2)$ and $J(x) = (x-1)^2/(2x)$.
background
climateJCost sums the per-component J-costs over an N-dimensional climate phase point, with each term Jcost(1 + s_i^2). The module sets the local context for climate attractor structure, where RS predicts long-term trajectories converge to a low-dimensional set whose J-cost is the global minimum on the energy/entropy field. The result rests on the upstream lemma Jcost_unit0, which states Jcost(1) = 0 and follows directly from the squared-ratio form of J.
proof idea
The proof unfolds climateJCost to expose the Finset sum, applies sum_eq_zero to reduce the claim to each summand vanishing, and simplifies every term via the Jcost_unit0 lemma that Jcost(1) = 0.
why it matters
This supplies the zero-cost base case for the master theorem vacuum_is_global_minimum, which asserts the vacuum is the global J-cost minimum in climate phase space. It is assembled into attractorStructureCert, confirming the module's claim that the climate attractor's J-cost is the global minimum. The result sits inside the Recognition forcing chain via the J-uniqueness property (T5) and the eight-tick structure that calibrates the cost function.
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