The Modulus: a Machine-Checked Operator Bound on the Fluctuation Sector of the Coupled Z_2 Slice

Two companion papers were left carrying the same debt from opposite sides. Thegap paper proved that every eigenvalue of the coupled slice other than thePerron eigenvalue is STRICTLY smaller in modulus, and said plainly that thisprovides NO MODULUS of separation. The bridge paper proved that the Gibbscorrelations of the spatial system are matrix elements of a self-adjointtransfer operator, and obtained geometric decay only UNDER A CONTRACTIONHYPOTHESIS IT DID NOT DISCHARGE. The missing step is identical in both: finitelymany strict inequalities are not an operator-norm bound.PROVED. We construct specGap, the largest |mu| over the eigenvalues differentfrom the Perron eigenvalue lambda, and prove specGap < lambda. We then prove theoperator bound: for every observable u orthogonal to the Perron vector,||Ku|| <= specGap*||u||. That is exactly the hypothesis the bridge papercarried, so its bound becomes unconditional. The bound is SHARP: whenever thestate space has at least two points, some nonzero fluctuation observable attainsit. The argument splits at specGap = 0, where the maximising index need notsupply a NON-PERRON eigenvector (it does supply the Perron one), hence none inthe fluctuation sector. Stated about an object too: the set of Rayleighnorm ratios on the fluctuation sector has a greatest element, equal to specGap(same two-point hypothesis: with fewer, that set is empty).A WARNING WE STATE BEFORE ANYONE ELSE HAS TO. specGap < lambda is NOTspecGap < 1: the kernel is unnormalised, so both are typically far above one andspecGap^N GROWS. On its own the unnormalised bound controls growth, it does notexhibit decay. The rate that is below one is the RELATIVE one,specRatio = specGap/lambda < 1, and the decay statement is that the fluctuationcontribution is suppressed by specRatio^N RELATIVE to the Perron scalelambda^N. Both forms are proved; only the second is called decay.The step that does not follow from the inequalities is the one abouteigenvectors AT lambda: geometric simplicity, proved in the Perron paper for anarbitrary eigenvector rather than a positive one, makes them INVISIBLE to afluctuation observable, so the top term of the spectral sum vanishes instead ofmerely being bounded.NOT PROVED. specGap DEPENDS ON THE EXTENT, and nothing here bounds it away fromlambda uniformly. Direct diagonalisation gives specGap/lambda = 0.9205, 0.9829,0.9964, 0.9992 at L = 2,3,4,5 for one parameter pair: a geometric bound whoserate tends to 1 is empty in the limit, and that is reported, not hidden. Thenormalised Gibbs EXPECTATION is now bounded too: splitting the dressed constantobservable along the Perron direction bounds the partition function below withno eigenbasis index identified, so at a fixed extent the two-point function isbounded by C*specRatio^N past an explicit threshold. That rate depends on theextent, so this is not clustering. The bound and its attainmentare both proved, which is what it means for specGap to be the operator norm onthe fluctuation sector; what is NOT done is introducing that norm as a definedobject and proving an equation about it. And the threshold N_0 does not dependon the observable: it is built from the dressed CONSTANT observable, so ONE N_0serves every fluctuation observable at once and only C sees A. Reflectionpositivity is untouched, and nothing in this paper is a claim about SU(N), thecontinuum limit, or the Yang-Mills mass gap.