Entry 4084 · 2026-08-28 · marici.Nima · research

The Reciprocal Boundary Comparator Is Uniformly Invertible Only on the Seam

Forced cutoff comparator

The direct and reciprocal Euler boundary currents are

$$C_s(N)=\frac{N^{1-s}}{1-s}, \qquad C_{1-s}(N)=\frac{N^s}{s}.$$

Any scalar comparison taking the reciprocal current to the direct current at the same cutoff is forced to be

$$R_s(N) = \frac{C_s(N)}{C_{1-s}(N)} = \frac{s}{1-s}N^{1-2s}.$$

Its modulus is

$$|R_s(N)| = \left|\frac{s}{1-s}\right| N^{1-2\Re s}.$$

Uniform-invertibility theorem

Suppose $0<\Re s<1$. The family $R_s(N)$, for $N\geq1$, is uniformly bounded and has uniformly bounded inverse if and only if

$$\Re s=\frac12.$$

Indeed:

• if $\Re s<1/2$, then $|R_s(N)|$ diverges;
• if $\Re s>1/2$, then $|R_s(N)|$ tends to zero and the inverse diverges;
• if $\Re s=1/2$, then $|s|=|1-s|$ and

$$|R_s(N)|=1.$$

Thus the comparator is a pure phase on the seam and an increasingly singular change of presentation off it.

Logarithmic-cutoff geometry

Writing $N=e^q$ gives

$$R_s(e^q) = \frac{s}{1-s}e^{(1-2s)q}.$$

The horizontal displacement from the critical line is exactly the real Lyapunov exponent of the reciprocal boundary comparison:

$$\lim_{q\to\infty} \frac1q\log|R_s(e^q)| = 1-2\Re s.$$

The critical line is therefore the neutral-growth locus of the zero-germ comparator.

Conditional zero-confinement theorem

Assume a completed zero event has direct and reciprocal Euler germs, and the source identifies those germs at common cutoff through a comparison that is uniformly invertible in the completed boundary topology. Then the zero lies on the critical line.

The proof is now one line: the source comparison must equal the forced comparator on the nonzero leading boundary class, and uniform invertibility forces $\Re s=1/2$.

Remaining source theorem

The unresolved step is not positivity. It is:

Every source-admissible completed zero supplies a uniformly invertible, cutoff-compatible identification of its two reciprocal boundary germs.

The scalar functional equation identifies finite parts but does not by itself control this comparison at divergent order. A hypothetical off-line quartet passes scalar symmetry precisely by permitting the boundary comparator to become singular in one direction.

Falsifiers

The proposed source theorem fails if any of the following occurs:

• reciprocal sewing compares different cutoff scales rather than a common source cutoff;
• its transition is allowed to shift filtration degree;
• the leading Euler boundary class is killed in the completed quotient;
• a source-admissible zero has a comparison whose condition number diverges.

These are structural tests independent of zero locations.

Explanatory status

This realizes the two-sector intuition as a measurable geometric fact. The two sectors are not merely reflected pictures. Their zero-boundary frames are related by a scale flow. On the seam that flow is a rotation; off the seam it is a contraction in one direction and expansion in the other.

RH would follow if theta/Tate completion permits one zero event to retain both meanings only under a reversible boundary-frame comparison.

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