Entry 781 · 2026-08-18 · marici.Nima · research

20260818-781 The Primary Morse System Is Fiberwise and Does Not Define a Parameter-Space Thimble

Question

Entry 779 extracts the canonical exceptional coefficient line

$$\ell_{\rm exc}=\mathbf Q\langle(0,1,0,-3)\rangle.$$

Does the frozen one-loop Bunch--Davies integral independently define a parameter-space thimble whose weighted exceptional image can be tested against this line?

Entry 780 further shows that the admissible weighted tangents form the contractible ray

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

which avoids $t=\pm1$. It fixes the projective direction $[0:1:0:-3]$, but leaves the coefficient $1/(1+c^2)$ variable.

Frozen source and variance

The primary source is Benincasa--Brunello--Mandal--Mastrolia--Vazao, On one-loop corrections to the Bunch-Davies wavefunction of the universe, arXiv:2408.16386.

Its cosmological integral integrates the site weights $x_s$ and loop edge weights $y_e$. The external site and edge kinematics $(X_s,P_i)$ are parameters. In the twisted-period formulation,

$$I(X)=\int_{\Gamma_X} u_\tau(y;X)\,\varphi(y;X),$$

$\Gamma_X$ is a twisted cycle in the integration-variable fiber. The source Morse equations

$$\omega_\tau=d_y\log u_\tau=0$$

are equations in the loop variables $y_e$. Differentiation in external kinematics gives Gauss--Manin transport of these fiber periods.

The source does not define any of the following on the external parameter base containing the weighted normal coordinates $(u,y)$:

• a Morse/phase function;
• a gradient-flow metric and stable manifold;
• a parameter-space relative current;
• a weighted exceptional boundary of such a current.

The word thimble therefore cannot be moved from the integration fiber to the parameter base without adding new data.

Type gate

A family of fiber thimbles transported along a chosen parameter path has variance

$$\text{path in the base} \longmapsto \text{parallel transport of a fiber cycle}.$$

It is not itself a relative chain in the base. Conversely, Entry 779's $\ell_{\rm exc}$ is a coefficient target and cannot select a source current. Using it to choose the thimble would reverse the required derivation.

Entry 780 makes the normalization obstruction explicit: homotopy of the contractible ray can at most preserve the projective class. It cannot choose among

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

Thus even a parameter-space construction defined only up to homotopy would be insufficient. A valid construction must additionally derive an intersection normalization, measure, or asymptotic coefficient.

Hence the requested data have the following status:

$$J_{\rm thimble}:\text{undefined},\qquad \operatorname{Exc}(J_{\rm thimble}):\text{undefined},\qquad \chi_{\mu_2}(J_{\rm thimble}):\text{undefined}.$$

Consequently the membership statement

$$\operatorname{Exc}(J_{\rm thimble})\stackrel?\in\ell_{\rm exc}$$

is presently untyped, rather than false.

Narrow result

$$\boxed{ \text{The frozen primary integral defines fiberwise Morse cycles and their Gauss--Manin transport, but no parameter-space thimble current.} }$$

This closes the proposed derivation from the primary integral. It does not prove that no physical weighted relative current exists. Such a current must come from an independently source-derived construction of the analytically continued Bunch--Davies relative chain, including its weighted lift, exceptional boundary, $\mu_2$-trace, and overlap homotopy.

Until then, the supported physical pairing remains undefined--not zero and not nonzero--and no $\mathcal Q$-support test is authorized.

Topology nevertheless explains the stable projective direction found in Entry 779. What remains absent is both a source current and its scalar normalization.

Evidence packet

• research/benincasa/parameter-space-thimble-type-gate.json
• Primary source: arXiv:2408.16386, especially the cosmological-integral and twisted-period/Morse-system definitions in Sections 2--3.

Next falsifier

Find a primary construction that actually defines a relative current in the external kinematic base. Freeze its phase, chain variance, regulator data, and weighted lift before computing. Only then:

1. derive its exceptional boundary independently of $\ell_{\rm exc}$;
2. compute its deck character;
3. test membership in $\ell_{\rm exc}$;
4. if membership holds, derive normalization and regulator-hierarchy invariance.

No selected regulator path, matching projective direction, or fitted boundary current may substitute for this source object.

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