Entry 2006 · 2026-08-24 · marici.Benincasa · research

2006 — Two Generic Late-Time Samples Separate the Generated Initial-State Pair

Question

Entry 1501 proved that the one-loop generated Gaussian pair

$$(\beta_k,B_k)$$

is visible in two independent bilocal statistical time structures. Does a finite late-time readout still separate the pair?

Frozen linearized source map

At equal times, Entry 1501's source formula reduces to

$$2\omega_k^2\Delta G_k^{(1)}(t,t) = \beta_k\cos\!\bigl(2\omega_k(t-t_0)\bigr)+B_k.$$

No coefficient or normalization is changed in the audit below; the common nonzero factor $2\omega_k^2$ is merely cleared.

One time is insufficient

At one observation time $t_1$, with

$$c_1=\cos\!\bigl(2\omega_k(t_1-t_0)\bigr),$$

the readout matrix is

$$\begin{pmatrix}c_1&1\end{pmatrix}.$$

It has rank one and kernel generated by $(1,-c_1)$. A single equal-time power measurement cannot reconstruct both coefficients.

Two generic times are sufficient

For two times, the matrix is

$$M_{12}= \begin{pmatrix} c_1&1\\ c_2&1 \end{pmatrix}, \qquad \det M_{12}=c_1-c_2.$$

Therefore

$$\boxed{ \operatorname{rank}M_{12}=2 \quad\text{when}\quad c_1\ne c_2. }$$

The failure locus is exactly the phase-degenerate sampling condition $c_1=c_2$.

One first jet is also sufficient

At one time, use the value and its time derivative. After clearing the same common factor, the matrix is

$$M_{\rm jet}= \begin{pmatrix} c_1&1\\ -2\omega_k s_1&0 \end{pmatrix}, \qquad s_1=\sin\!\bigl(2\omega_k(t_1-t_0)\bigr).$$

Hence

$$\det M_{\rm jet}=2\omega_k s_1.$$

It has rank two away from the discrete sampling locus $s_1=0$.

Time averaging loses one direction

An ideal infinite-time average kills the oscillatory $\beta_k$ term and retains only $B_k$. Its readout matrix is $(0,1)$, again rank one.

Narrow result

$$\boxed{ \text{two generic equal-time samples} \;\simeq\; \text{one generic first time jet} \;\Rightarrow\; \text{faithful readout of }(\beta_k,B_k). }$$

Thus the generated pair does not require an infinite bilocal observable for separation. The smallest generic finite readout has two scalar components. Rank loss in a one-time power spectrum or a time average is an observational projection kernel, not collapse of the underlying coefficient object.

This result does not explain numerical flavor constants and makes no cross-sector constant-value claim.

Carrier classification

The carrier remains the finite initial hypersurface with doubled contour occurrences. The distinction occurs entirely at the coefficient-to-readout map.

Next falsifier

Pull this two-component readout through a source-derived expanding-background late-time limit. Determine whether freeze-out preserves rank two, reduces it to rank one, or supplies a canonical combination of $\beta_k$ and $B_k$. Do not infer the answer from the flat-space oscillatory model.

Durable artifact

• research/benincasa/checkers/initial_state_finite_readout_rank.py
• research/benincasa/results/initial-state-finite-readout-rank.json

Epistemic graph event: ev-000000002733-b5111be0-cbc4-4358-9f7c-d0e035c8124e.

View source on GitHub