IndisputableMonolith.Cosmology.RecognitionWorkBound
Bounds the recognition work and activation count contributed by single ticks and by full cycles, under the forced-cadence rule that at most one edge resolves per tick. Cosmology arguments that need a global work budget or a localization statement cite these inequalities. The module packages elementary cost accounting with the two-endpoint expansion rule and the at-most-one-edge cadence law.
claimFor a per-region forced-posting cost, the work of a tick is $0$ if nothing resolves and $\mathrm{cost}(a)+\mathrm{cost}(b)$ if edge $(a,b)$ resolves. At most one edge resolves per tick. The module bounds tick work, tick activations ($\le 2$), cycle work, cycle activations, and states that recognition work localizes.
background
Recognition Science treats posting and resolution on a discrete ledger whose elementary time unit is the tick. The eight-tick octave (forcing chain T7) is the global cadence; locally, the forced cadence law says a single tick resolves at most one edge. When an edge $(a,b)$ is resolved, both endpoints are expanded, so the tick's recognition work is the sum of the two per-region forced-posting costs; an idle tick contributes zero.
This module sits in the cosmology layer and turns that accounting into explicit bounds. Sibling definitions introduce tick work and tick activations, then prove the elementary inequalities (work controlled by the endpoint costs; activations at most two) and lift them to full cycles. A localization statement records that the work can be read off region-wise rather than only globally.
proof idea
Definition-plus-bound module. Tick work and tick activations are defined by cases on whether the tick resolves an edge; the cadence law supplies the at-most-one-edge hypothesis. The tick-level inequalities are immediate from the two-endpoint cost rule and the activation count of an edge. Cycle bounds sum (or otherwise aggregate) the tick bounds over a cycle. Localization is a reassociation of the same sum over regions rather than over ticks.
why it matters in Recognition Science
Cosmology in RS needs a finite recognition-work budget per tick and per cycle so that large-scale bookkeeping (horizon structure, ledger growth, forced posting) stays controlled. These bounds supply that budget under the forced cadence law and the two-endpoint expansion rule, and they feed any parent argument that integrates local posting cost into a global or regional work total. They sit downstream of the eight-tick octave (T7) as the local cost counterpart of the global period, without yet invoking the mass ladder or the alpha band.
scope and limits
- Does not derive the forced cadence law; it assumes at most one edge resolves per tick.
- Does not fix a numerical value of the per-region cost function.
- Does not bound continuum or multi-edge simultaneous resolution models.
- Does not connect work bounds to mass-ladder rungs or alpha.
- Does not prove global cosmological dynamics beyond work and activation counts.