Emergence of patterns and the dominance of the hump

4 Emergence of patterns and the dominance of the hump

For a fixed window of length $L$ with composition $(a,b)$ ( $a+ b = L$ ) the probability of that exact pattern under the independent-bit continuum limit is

( ) P (pattern | τ) = p(τ)a 1 − p(τ) b.

(11)

Three exact results follow at once.

Zero/one-start complementation symmetry is exact for all $τ$ .
Only the composition $(a,b)$ matters; every permutation of a given composition yields the identical curve.
On the reachable domain $p ∈ [0,1∕2]$ there are exactly three qualitative shapes, classified by the sign of $a − b$ :
- $a > b$ : monotonic rise (peak lies beyond equilibrium);
- $a = b$ : monotonic rise that flattens at the boundary;
- $a < b$ : genuine interior hump.

Only the hump class ( $a < b$ ) produces a rise-peak-decline profile. Moreover, when each composition is weighted by its combinatorial multiplicity $( ) La$ , the hump class dominates the measure for almost the entire relaxation. The window occupancy is binomial with mean $Lp (τ) < L∕2$ at every finite $τ$ ; consequently the mass of the hump region exceeds the mass of the non-hump region until a crossover $τ c$ that recedes toward infinity as $L$ grows. For any practically observable window length the hump is therefore the typical outcome.

The peak location, amplitude and asymptotic decay of a hump are fixed by elementary formulae:

p^∗	= ,	τ^∗	= −ln(1 − 2p^∗),	(12)
	= 2^{L1−H₂(p^∗)}.			(13)

No free parameters are required.

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