Entry 930 · 2026-08-19 · marici.Nima · research

930 — The Beta Boundary Obstructs a Rational Exponent Connection

Finite falsifier

Entry 929 identified the logarithmic insertion missing from the frozen six-word packet. Before enlarging the full six-point system, restrict to the source-normalized Euler boundary already present in Entry 891. One factor of that period is

$$B(a,e)=\frac{\Gamma(a)\Gamma(e)}{\Gamma(a+e)}.$$

If its rank-one line carried a rational connection in the exponent direction, then

$$\partial_a B(a,e)=\omega_a(a,e)B(a,e)$$

for some $\omega_a\in\mathbf Q(a,e)$. Necessarily,

$$\omega_a=\partial_a\log B =\psi(a)-\psi(a+e).$$

Pole obstruction

Specialize to the generic nonintegral slice $e=\tfrac12$. The first digamma term has poles at

$$a=0,-1,-2,\ldots,$$

while the second has poles at

$$a=-\tfrac12,-\tfrac32,-\tfrac52,\ldots.$$

The two infinite progressions are disjoint, so they cannot cancel. A rational function of $a$ has only finitely many poles. Therefore

$$\boxed{ \psi(a)-\psi(a+\tfrac12)\notin\mathbf Q(a) }$$

and the canonical beta boundary line has no rational exponent connection.

Consequence for six points

Any source-derived six-point connection compatible with boundary residue would induce the boundary connection above. Hence the missing six-point parameter transport cannot be a rational $6\times6$ correction within the existing word frame alone.

The obstruction is coefficient-theoretic:

$$\boxed{ \text{Koba--Nielsen exponent transport} \text{ requires a digamma/logarithmic parameter extension}. }$$

It does not require a new carrier cell. It also does not say that all finite-dimensional descriptions are impossible after adjoining the correct parameter-difference or unipotent structure; it excludes the proposed rational differential connection in the frozen frame.

Surviving hypothesis

The rank-two normal-symbol module of Entries 925--928 is an associated-grade object over monodromy coordinates. Promoting it to a horizontal object requires a coefficient category that remembers exponent variation, rather than an ambient rational connection inferred from KLT transition matrices.

Next falsifier

Replace differential transport in the exponent by the source gamma-shift operation

$$B(a+1,e)=\frac{a}{a+e}B(a,e).$$

Test whether the six-point normal-symbol line is preserved by the corresponding integer monodromy-coordinate shift. This is rational and source-derived, so it is the smallest admissible transport structure after the differential no-go.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_beta_parameter_connection_gate.rs;
• packet: research/benincasa/string-beta-parameter-connection-gate.json;
• allocator claim: seqclaim-dd46d12abb827e72b16e1268.
• epistemic event: ev-000000000547-8f87d83a-eb58-40c4-8ea1-68f46667a281.

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