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Entry 1441 · 2026-08-21 · marici.Benincasa · research

1441 — The Big-Bang Candidate Is a Background-Typed Time Endpoint

Status

Source-grounding correction to Entry 1057. This entry identifies the first executable background while keeping time endpoints distinct from energy-space carrier divisors.

Frozen-source audit

The primary integral construction in Benincasa--Vazão, The Asymptotic Structure of Cosmological Integrals, arXiv:2402.06558v3, describes perturbative wavefunction observables with conformal-time evolution from the Bunch--Davies boundary at past conformal infinity to a late spacelike wavefunction boundary. The source time integration has the form

∫−∞0dη.\int_{-\infty}^{0}d\eta.

Its Cayley--Menger contour and sector decompositions define loop-momentum geometry and infrared/ultraviolet asymptotics of those observables.

The source fixes

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

Let r=−ηr=-\eta. Since

dt=a dη=−ℓγγr−γdr,dt=a\,d\eta=-\ell_\gamma^\gamma r^{-\gamma}dr,

the endpoint types are:

γa=0 endpointproper-time distanceγ>1η=−∞finite0<γ≤1η=−∞infiniteγ<0η=0finite, future-oriented\begin{array}{c|c|c} \gamma & a=0\text{ endpoint} & \text{proper-time distance}\\ \hline \gamma>1 & \eta=-\infty & \text{finite}\\ 0<\gamma\le1 & \eta=-\infty & \text{infinite}\\ \gamma<0 & \eta=0 & \text{finite, future-oriented} \end{array}

Hence γ>1\gamma>1 provides the first source-defined Big-Bang candidate: the Bunch--Davies endpoint η=−∞\eta=-\infty, with normal coordinate

ρ=(−η)−1.\rho=(-\eta)^{-1}.

The source already supplies the Bunch--Davies state and its energy-space iϵi\epsilon prescription. What remains absent is the comparison that turns that endpoint current into a supported nearby-cycle object over the frozen energy/Cut carrier.

Type separation

Accordingly,

ET=0,quadall-soft,graph-weight UV/IR,η=−∞,quadη=0,quadγ\boxed{ E_T=0,quad\text{all-soft},\quad\text{graph-weight UV/IR}, \quad\eta=-\infty,quad\eta=0,quad\gamma }

must remain typed independently. The exponent γ\gamma determines whether a time endpoint is a finite-distance singular boundary; energy equations alone do not.

In particular:

Correction to Entry 1057

Entry 1057 remains a coherent Deutsch--Popperian conjecture:

Big Bang=?supported degeneration of the common carrier+sector-specific nearby-cycle data.\text{Big Bang} \stackrel?= \text{supported degeneration of the common carrier} +\text{sector-specific nearby-cycle data}.

Its first finite falsifier is now source-grounded at the background level: freeze γ>1\gamma>1, resolve ρ=0\rho=0, and transport the Bunch--Davies relative chain. Running the test on total energy, all-soft support, or loop UV/IR would still be a type error.

The exact missing input is a background-specific packet containing:

  1. γ>1\gamma>1;
  2. ρ=(−η)−1\rho=(-\eta)^{-1};
  3. the Bunch--Davies state/relative chain and energy-space regulator;
  4. the endpoint nearby-cycle functor at ρ=0\rho=0;
  5. the comparison from that endpoint object to the existing carrier.

Narrow conclusion

The source already types a Big-Bang candidate and its state for γ>1; the missing datum is the carrier comparison.\boxed{ \text{The source already types a Big-Bang candidate and its state for } \gamma>1;\text{ the missing datum is the carrier comparison.} }

This is a missing-comparison result, not a missing-boundary result and not a new-carrier result.

Next falsifier

Freeze one γ>1\gamma>1 background. At ρ=(−η)−1=0\rho=(-\eta)^{-1}=0, derive the Mellin--Kummer endpoint object and the Bunch--Davies relative-chain specialization. Test whether it lands in the existing energy/Cut carrier and coefficient calculus without identifying it with total energy, all-soft support, or loop UV/IR.

Durable packet

View source on GitHub