complexDemand
Complex demand for a recognition boundary of extent L and integer complexity Z scales the base maintenance demand by the linear factor (1 + |Z| * k_R). Researchers deriving holographic limits on conscious extent cite this when showing how conserved information units tighten the maximum coherent size. The definition is a direct algebraic product of maintenanceDemand L with the complexity multiplier.
claimThe complex demand for a boundary of extent $L$ and complexity integer $Z$ is $d(L,Z) = m(L) (1 + |Z| k_R)$, where $m(L)$ is the maintenance demand over the barrier period and $k_R = ln phi$ is the recognition Boltzmann constant.
background
In the Consciousness Bandwidth module a conscious boundary of spatial extent L persists for a fixed barrier period of 360 ticks. Its base maintenance demand m(L) equals barrierPeriod times the J-cost J(L / lambda_rec), where J is the recognition cost function satisfying the Recognition Composition Law. The integer Z is the complexity count drawn from the anchor map in Masses.Anchor, which assigns to each particle species an integer built from its charge quantum number Q via Z = (6Q)^2 + (6Q)^4 plus sector offsets.
proof idea
This is a one-line definition that multiplies the already-defined maintenanceDemand L by the factor (1 + |Z| * k_R). The absolute value on Z and the real multiplication are supplied directly by Mathlib; no lemmas are invoked beyond the positivity of k_R already established in Constants.BoltzmannConstant.
why it matters in Recognition Science
The definition supplies the Z-dependent scaling required by the downstream theorems complexDemand_ge and higher_Z_more_demand, which prove demand is strictly monotone in |Z|. It implements the module's fourth key result z_complexity_reduces_extent, showing that higher complexity reduces the critical coherent extent L_crit set by equating holographic budget S_holo = L^2 / (4 ell_P^2) to total demand. Within Recognition Science it links the eight-tick octave barrier to the phi-ladder via the J-cost and the recognition Boltzmann constant k_R = ln phi.
scope and limits
- Does not compute a numerical value for any specific L or Z.
- Does not derive the critical extent L_crit or solve demand = budget.
- Does not incorporate the spatial dimension D = 3 or the phi-ladder rung structure.
- Does not address time evolution or decay outside the fixed 360-tick barrier.
Lean usage
unfold complexDemandformal statement (Lean)
167noncomputable def complexDemand (L : ℝ) (Z : ℤ) : ℝ :=
proof body
Definition body.
168 maintenanceDemand L * (1 + |Z| * k_R)
169
170/-- Complex demand ≥ simple demand for any Z. -/