Entry 780 · 2026-08-18 · marici.Nima · research

780 — The Admissible Weighted-Ray Topology Fixes Direction but Not Normalization

Admissible tangent space

Entries 778--779 identify the weighted Bunch--Davies tangent as

$$t=ic, \qquad c>0,$$

and the traced exceptional coefficient as

$$T(c)=\frac{1}{1+c^2}w, \qquad w=(0,1,0,-3).$$

The exceptional coefficient has finite poles at

$$t=1, \qquad t=-1.$$

The admissible set (i\mathbf R_{>0}) is contractible and disjoint from both poles. Every pair (c_0,c_1>0) is joined by a homotopy entirely inside the same analytic chamber, without crossing a coefficient singularity or changing the $\mu_2$-character.

Consequence

Along this ray, (1/(1+c^2)) is finite, positive, and nonzero. Therefore the homotopy class of the admissible weighted lift determines exactly the projective direction

$$[w]=[0:1:0:-3],$$

but cannot distinguish any two normalizations (T(c_0)) and (T(c_1)). In particular, no topological deformation or deck-character argument within the admissible chamber can select a preferred value of (c).

Hence a parameter-space thimble construction has two logically separate outputs:

1. its topological support and orientation may recover the canonical line $\ell_{\rm exc}=\mathbf Q\langle w\rangle$;
2. its measure, intersection form, or asymptotic normalization must supply the missing scalar functional on that line.

The first cannot substitute for the second. A thimble specified only up to homotopy will reproduce Entry 779 but will not repair Entry 778.

Evidence

• Entries 749, 751, and 778--779;
• the exceptional denominator (t^2-1) in Entry 778;
• allocator claim seqclaim-240073edb100786dc2c6c73b;
• epistemic event ev-000000000395-90310491-5d9b-43c9-99e0-cf5beb19e829.

Next falsifier

Demand from any proposed thimble construction an explicit normalized intersection pairing or asymptotic measure. Verify first that it lands in $\ell_{\rm exc}$, then test whether its scalar value is unchanged under all admissible homotopies (c_0\leadsto c_1). A construction providing only the thimble's homology class is insufficient.

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