balanced_config_zero_cost
The theorem shows that any ledger configuration with all ratios fixed at 1 has total J-cost exactly zero, identifying the ground state of the Recognition Science cost model. Complexity theorists classifying physics problems would cite it to anchor the linear-time verification of balanced states. The proof is a direct term reduction that unfolds the total-cost sum and substitutes the unit lemma Jcost(1)=0.
claimFor any natural number $N$, the total J-cost of the configuration whose ratios are identically 1 equals zero.
background
J-cost is the function $J(x)=(x+x^{-1})/2-1$, equivalently $(x-1)^2/(2x)$, whose unique minimum is zero at $x=1$. totalJCost sums the individual J-cost contributions over the support of a witness in the balanced J-subset-sum instance. The module IC-005 frames physics complexity via J-cost minimization: ground-state verification is linear in system size while phi-rung states are exponentially costly. The result rests on the upstream lemma Jcost_unit0, which states Jcost(1)=0 by direct simplification of the definition.
proof idea
The proof is a one-line wrapper. It unfolds the definition of totalJCost to expose the sum of rung costs, then applies simp with the lemma Jcost_unit0 to replace every term Jcost(1) by zero, yielding the empty sum.
why it matters in Recognition Science
The declaration supplies the algebraic anchor for the ground-state entry in the IC-005 certificate, which lists 'Ground state (x=1): unique, 0 cost, O(1) to verify'. It realizes the convex-minimum property stated in the module doc-comment and aligns with the J-uniqueness step (T5) of the forcing chain. The result closes the trivial case needed before discussing NP-hard global optimization over phi-ladders.
scope and limits
- Does not prove that the all-ones configuration is the unique zero-cost state.
- Does not compute J-cost for any ratio other than 1.
- Does not address time complexity or algorithmic verification beyond the algebraic identity.
- Does not extend to time-dependent or multi-tick ledger evolution.
formal statement (Lean)
138theorem balanced_config_zero_cost (N : ℕ) :
139 totalJCost (N := N) { ratios := fun _ => 1, ratios_pos := fun _ => one_pos } = 0 := by
proof body
Term-mode proof.
140 unfold totalJCost
141 simp [Cost.Jcost_unit0]
142
143/-- Helper: A sum of non-negative reals equals 0 iff each term is 0. -/