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Exact source blocks at the minimal fourth gap

Abstract

Physical valley and peak chamber realization

For 2<=b<a<c, the minimal fourth gap is d=c+2. Put R=2^a, T=2^b, Z=2^c, p=Z/(2R) and q=Z/(2T). The exact constructor uses masks u1=3, u2=u3=v0=w=0, v2=p+q and v3=p. The dyadic ratios make p and q distinct binary positions, so v3 is a submask of v2 and all masks fit their physical positive chains. valley_rising_inventory proves that this actual baseWord uses every physical label once. valley_rising_strong applies the actual centered-source block criterion with reserves 3A, 2e1(p+q) and 2e1p.

For b<c<=a, the other strict-valley branch retains its adjacent Z=2T case and its rounded V reserve at larger ratios. The peak b>max(a,c) uses a direct U reserve when its dyadic ratio is wide, advances V at the adjacent ratio, and reserves or releases U/W at its final seam. The mask bounds, actual inventory and actual_masked_strong give their literal physical Strong statements. Repeated gap values and repeated numerical source values are retained as separate physical occurrences.

actual_orders_of_strong transports an actual natural-label inventory to Fin labels and obtains both OriginalOrder predicates using the same sigma and the literal earWord at cut zero. The complete assembly, including the monotone chambers and natural fourth-gap lift, belongs to Endpoint. The unrestricted four-gap joint-order problem below the sufficient max-plus-two threshold remains OPEN.

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