IndisputableMonolith.Cosmology.RecognitionWorkBound
Bounds the recognition work and activation count contributed by a single tick and by a full cycle, under a per-region forced-posting cost. A null tick costs zero; a tick that resolves one edge costs the sum of the two endpoint costs. Cosmologists and RS ledger auditors cite the cycle inequalities when controlling total work over an eight-tick window. The argument is elementary case analysis plus the forced cadence law (at most one edge per tick).
claimLet $c$ assign a nonnegative forced-posting cost to each region. For a tick $\tau$, the recognition work $W(\tau)$ is $0$ if $\tau$ resolves nothing, and $c(a)+c(b)$ if $\tau$ resolves the single edge $(a,b)$. Then $W(\tau)\le c(a)+c(b)$ whenever an edge is resolved, activations per tick are at most $2$, and over any cycle the total work and total activations obey the corresponding summed bounds. Recognition work localizes to the endpoints that expand.
background
In Recognition Science cosmology, ledger updates occur on discrete ticks. The forced cadence law allows at most one edge resolution per tick: either the tick is idle, or it expands exactly two endpoint regions of a single edge. Each region carries a forced-posting cost; work is the sum of those costs for the endpoints that actually expand.
The module sits in the Cosmology domain and introduces the elementary accounting primitives: per-tick work, per-tick activation count, their pointwise upper bounds, and the lifted cycle sums. The eight-tick octave (forcing chain T7) is the natural cycle length against which these bounds are later applied, though the inequalities themselves are length-agnostic.
No external RS lemmas are imported beyond Mathlib; the development is self-contained cost algebra on ticks and finite cycles.
proof idea
Definition layer first: tick work is zero on idle ticks and endpoint-cost sum on resolving ticks; tick activations count expanded endpoints (0 or 2). Pointwise lemmas are immediate case splits on whether an edge is resolved, using nonnegativity of costs and the cadence law (at most one edge). Cycle bounds are finite sums of the pointwise inequalities. Localization states that total work depends only on the multiset of expanded endpoints, not on idle ticks or edge labels beyond their ends.
why it matters in Recognition Science
Supplies the local work and activation budgets that any global cosmological ledger argument must respect. Downstream cycle and localization facts feed energy-accounting and sparsity claims in the broader Cosmology stack: total recognition work over an octave cannot exceed the sum of per-tick endpoint costs, and activations stay linear in the number of resolving ticks. Ties directly to the eight-tick octave (T7) as the canonical summation window, and to the forced-posting cost model used when converting abstract ledger steps into physical work. Without these bounds, cycle-level work estimates would be unconstrained casework.
scope and limits
- Does not derive the forced cadence law; takes at most one edge per tick as given.
- Does not fix the numerical cost function or identify regions with spacetime points.
- Does not prove eight-tick closure or link work bounds to observed cosmology.
- Does not address multi-edge or continuous-time recognition models.
- Does not import or discharge global forcing-chain theorems T0–T8.