aligned_beats_perturbed
aligned_beats_perturbed proves that the J-cost of the aligned configuration is strictly smaller than the sum of J-costs over any nonempty list of positive perturbed values. Researchers deriving the Global Co-Identity Constraint from the Recognition Science forcing chain cite the result to replace an empirical subadditivity axiom. The argument is a direct reduction that substitutes the explicit unit value of Jcost and applies the elementary positivity of sums of positive reals.
claimLet $c_1, c_2, ..., c_n$ be positive real numbers with $n > 0$. Then $J(1) < c_1 + c_2 + ... + c_n$, where $J(x) = (x-1)^2 / (2x)$.
background
The GCICDerivation module extracts the Global Co-Identity Constraint from the J-uniqueness step T5 of the forcing chain together with the compact phase and discrete gauge supplied by T6 and T7. The J-cost function is given by the squared-ratio expression $J(x) = (x-1)^2 / (2x)$, which vanishes exactly at the aligned point $x=1$. Upstream lemma Jcost_unit0 records this vanishing by direct simplification. The sibling lemma list_sum_pos records that the sum of any nonempty list of positive reals is itself positive.
proof idea
The proof is a one-line wrapper. It rewrites the left-hand side via Jcost_unit0 to obtain zero, then applies list_sum_pos to the length and positivity hypotheses on the input list.
why it matters in Recognition Science
The declaration supplies the collective subadditivity half of gcic_derivation_cert, the top-level certificate that the GCIC follows from the forcing chain with zero empirical axioms. It thereby converts the individual positivity of J into a collective comparison that supports phase alignment minimizing total cost. The result closes the step from T5 J-uniqueness to the uniform-phase conclusion of the GCIC without additional hypotheses.
scope and limits
- Does not apply to the empty list.
- Does not apply when any entry is non-positive.
- Does not supply quantitative bounds on the gap.
- Does not extend to infinite sums or continuous measures.
formal statement (Lean)
143theorem aligned_beats_perturbed
144 (costs : List ℝ) (hlen : 0 < costs.length)
145 (hpos : ∀ c ∈ costs, 0 < c) :
146 Jcost 1 < costs.sum := by
proof body
Term-mode proof.
147 rw [Jcost_unit0]
148 exact list_sum_pos costs hlen hpos
149
150/-! ## Part 4: Full Derivation Certificate -/
151