Jcost_mono_on_Ici_one
plain-language theorem explainer
J-cost is monotone non-decreasing on [1, ∞). Researchers closing the local-cache gap in graph models of cognition cite this to guarantee that cost grows with separation or scale factor. The argument splits on equality versus strict inequality, dispatching the former by reflexivity and the latter by the companion strict-monotonicity theorem.
Claim. Let $J$ be the J-cost function. For real numbers $a,b$ with $1 ≤ a ≤ b$, $J(a) ≤ J(b)$.
background
The Local Cache Forcing module proves that J-cost minimization on connected graphs forces hierarchical local caching and thereby closes Gap G1 in the brain-holography argument. J-cost is the real-valued cost derived from the Recognition Science J-function; the module first establishes its strict increase on [1, ∞) and then uses monotonicity to compare costs at different distances or phi-powers. The proof imports the reflexivity of ≤ from the arithmetic-from-logic foundation.
proof idea
Case analysis on eq_or_lt_of_le splits the hypothesis a ≤ b into a = b or a < b. The equality case is immediate from le_refl. The strict case applies the sibling strict-monotonicity theorem to obtain a strict inequality and then weakens it to non-strict via le_of_lt.
why it matters
Monotonicity supplies the ordering needed for the module’s later results on collocation minimizing cost and caching being forced. It also prepares the phi-power comparisons that appear in the same file. The result sits inside the Recognition Science forcing chain after J-uniqueness and before the eight-tick octave and D = 3 conclusions.
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