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Entry 1443 · 2026-08-21 · marici.Benincasa · research

1443 — The Gamma-Greater-Than-One Big-Bang Endpoint Is an Irregular Stokes Boundary

Status

Source-derived local classification of Entry 1441's first Big-Bang candidate.

Frozen endpoint

The primary source fixes

a(η)=(−ℓγη)γ,η∈(−∞,0),a(\eta)=\left(\frac{-\ell_\gamma}{\eta}\right)^\gamma, \qquad \eta\in(-\infty,0),

and imposes Bunch--Davies data at η=−∞\eta=-\infty. Entry 1441 shows that for γ>1\gamma>1 this is an a=0a=0 endpoint at finite proper-time distance.

Use the source-derived normal coordinate

ρ=(−η)−1.\rho=(-\eta)^{-1}.

Then ρ=0\rho=0 is the candidate initial boundary.

Exact local transform

For a site factor with source Mellin exponent β\beta,

(−η)−βeiEηdη=ρβ−2e−iE/ρdρ.(-\eta)^{-\beta}e^{iE\eta}d\eta = \rho^{\beta-2}e^{-iE/\rho}d\rho.

Hence the local endpoint coefficient object is not merely Kummer. It is

VBB=Kρβ−2⊗E−iE/ρ\boxed{ \mathcal V_{\rm BB} = \mathcal K_{\rho^{\beta-2}} \otimes \mathcal E^{-iE/\rho} }

with connection, in the convention where the displayed section is horizontal,

∇=d−(β−2ρ+iEρ2)dρ.\nabla =d- \left( \frac{\beta-2}{\rho} +\frac{iE}{\rho^2} \right)d\rho.

The ρ−2\rho^{-2} term makes ρ=0\rho=0 an irregular singularity of Poincaré rank one.

The physical selector is already present

The source regulates the Bunch--Davies state by

E⟼E−iϵ,ϵ>0.E\longmapsto E-i\epsilon, \qquad \epsilon>0.

Therefore

e−i(E−iϵ)/ρ=e−iE/ρe−ϵ/ρ.e^{-i(E-i\epsilon)/\rho} =e^{-iE/\rho}e^{-\epsilon/\rho}.

Along the physical ray ρ>0\rho>0, this is rapidly decreasing at ρ=0\rho=0. Thus the source already selects a Stokes/rapid-decay sector. The missing object identified after Entry 1441 is not an arbitrary physical current: it is the irregular comparison that carries this source-selected sector into the common carrier calculus.

Architectural correction

The candidate Big-Bang comparison requires

shared carrier+irregular/Stokes coefficient object+rapid-decay Betti specialization.\boxed{ \text{shared carrier} +\text{irregular/Stokes coefficient object} +\text{rapid-decay Betti specialization}. }

Ordinary logarithmic nearby cycles plus finite normal/Rees grades are insufficient at this endpoint. This is new coefficient/comparison machinery, not evidence for a new carrier stratum.

First finite falsifier

Start with one labelled site:

  1. construct the rapid-decay homology selected by E−iϵE-i\epsilon;
  2. pair it with the de Rham exponential--Kummer class;
  3. transport the occurrence label and orientation under one Cut sewing;
  4. test whether the irregular specialization commutes with that sewing.

If the comparison commutes, H2 gains a Big-Bang endpoint model without a new carrier primitive. If it fails for a source-derived reason that cannot be absorbed by the irregular coefficient object, the failure identifies the first possible carrier obstruction.

Source provenance

View source on GitHub