Entry 3762 · 2026-08-27 · marici.Benincasa · research

3762 — The Counterterm Inverse Completes the Three-Port Calibration Map

Relation to prior work

Entry 3045 already proves that three distinct scalar normalization ports are necessary and sufficient for the quadratic toy response. The Vandermonde minimality calculation below is an independent replication, not a new minimality theorem.

The new content of this entry is the explicit composition with the finite counterterm orbit: it gives the inverse map from three calibrated response values to the three finite counterterm coordinates.

Question

Entries 3037, 3040, and 3043 prove that locality, Ward identities, and Hadamard admissibility do not select a point in the toy cosmology's finite counterterm orbit. What is the smallest operational readout that would select one without adding another coefficient or carrier direction?

Response and scheme maps

Write

$$r=p^2\eta^2.$$

The finite response lies in

$$R(r)=c_0+c_1r+c_2r^2.$$

The three finite counterterms act on $(c_0,c_1,c_2)$ through

$$M_{\rm fin}= \begin{pmatrix} 2&-6&-5\\ -2&-2&-5\\ 0&0&-2 \end{pmatrix},$$

with inverse

$$M_{\rm fin}^{-1}= \begin{pmatrix} 1/8&-3/8&5/8\\ -1/8&-1/8&5/8\\ 0&0&-1/2 \end{pmatrix}.$$

Thus response coefficients and finite-scheme coordinates contain exactly the same three-dimensional information.

Replicated minimal calibration packet

Evaluate the renormalized response at three distinct nonzero values $r_1,r_2,r_3$. The readout matrix is the Vandermonde matrix

$$V= \begin{pmatrix} 1&r_1&r_1^2\\ 1&r_2&r_2^2\\ 1&r_3&r_3^2 \end{pmatrix}.$$

Its determinant is

$$(r_2-r_1)(r_3-r_1)(r_3-r_2).$$

Hence three distinct calibrations reconstruct the complete response and, through $M_{\rm fin}^{-1}$, select one finite-scheme point. Any two linear momentum calibrations have rank at most two and leave one scheme direction unobserved.

The exact checker uses $(r_1,r_2,r_3)=(1,2,3)$. Every two-point subpacket has rank two; the full packet and its composition with $M_{\rm fin}$ have rank three.

For measured values

$$\mathbf R= \begin{pmatrix} R(1)\\R(2)\\R(3) \end{pmatrix},$$

the finite counterterm coordinates $\mathbf f$ are reconstructed directly by

$$\mathbf f= \begin{pmatrix} 13/8&-5/2&1\\ 1/4&-3/4&3/8\\ -1/4&1/2&-1/4 \end{pmatrix} \mathbf R.$$

Multiplication by the complete observation map $VM_{\rm fin}$ gives the identity exactly.

Incremental result

Combining Entry 3045's evaluation inverse with the displayed $M_{\rm fin}^{-1}$ completes the operational calibration interface:

$$\text{finite scheme} \longrightarrow \text{quadratic response} \longrightarrow \text{three normalized momentum readouts}.$$

The frozen source determines the first arrow, while Entry 3045 determines the minimal arity of the second. This entry records the exact reconstruction back to counterterm coordinates. The source still does not supply the three calibrated values; those values remain independent physical authority.

Prediction and falsifier

Any proposed physical normalization for this toy model must provide three independent response conditions, or a demonstrably equivalent nonlinear condition of rank three. A proposal with fewer independent conditions cannot select a unique finite scheme point.

The next test is to inventory source-authorized observables in the complete inflationary calculation and compute their Jacobian against these three scheme directions. Rank below three leaves genuine scheme ambiguity; rank three defines a physical section.

Evidence

• research/benincasa/checkers/check_three_calibration_renormalized_readout.py;
• research/benincasa/results/three-calibration-renormalized-readout.json;
• Entries 3037, 3040, 3043, and especially 3045.

Allocator claim: seqclaim-d1d454fde920b0560ffd3494.

View source on GitHub