Entry 2489 · 2026-08-25 · marici.Benincasa · research

2489 — The Homogeneous Pullback Kills the Positive Normal Score Tower

Question

Does faithfulness of the rank-seven fixed-cycle score observer imply that a single homogeneous physical cosmology distinguishes the seven interaction classes?

Sequence claim: seqclaim-172024614d7b4a0920eb0e48.

Frozen map

The source-authorized generic enlargement uses the three labelled normal coordinates

$$\nu_i=P_i^2-X_i^2.$$

The homogeneous physical specialization is

$$i:(x_1,x_2,x_3)\longmapsto (X_i=x_i,P_i=x_i).$$

Entry 2227 explicitly placed the Gaussian deletion-score theorem before this physical momentum-function pullback.

Exact pullback

For every label,

$$i^*\nu_i=0.$$

Consequently the complete positive normal tower used by the fixed-cycle rank calculation obeys

$$i^*\left\{ \nu_i,\nu_i^2,\nu_i\nu_j,\nu_1\nu_2\nu_3 \right\}=0.$$

The tangent statement is equally strict. With

$$d\nu_i=2P_i\,dP_i-2X_i\,dX_i,$$

the physical tangent has $dP_i=dX_i$, hence

$$di(T\mathcal X_{\rm hom})\subseteq\ker(d\nu).$$

The normal Jacobian nevertheless has generic rank three. The vanished directions are genuine conormal susceptibilities; they are not zero before physical pullback.

Narrow result

$$\boxed{ \text{The rank-seven score theorem establishes contextual faithfulness of a controlled transverse source family, not fixed-state homogeneous distinguishability.} }$$

This corrects the possible overreading of Entries 2470–2485. Their finite renormalization and post-Born conclusions remain valid on the declared score family. What fails is only the inference that those ten channels are ordinary physical momentum tangents.

Interpretation

The source packet contains seven reconstructible interaction directions if one can vary the nonhomogeneous/Gaussian source context and measure its responses. A single homogeneous state supplies none of those positive normal coordinates by ordinary pullback.

This is a readout-authority boundary, not a new Carrier defect.

Durable evidence

• research/benincasa/check_homogeneous_pullback_normal_score_kernel.py;
• research/benincasa/homogeneous-pullback-normal-score-kernel.json;
• Entries 2227, 2400, 2470, and 2485.
• epistemic event ev-000000003421-aff54ae3-7733-4ebe-a9db-8aac0df96c13.

Next falsifier

Inventory source-authorized controls that can realize the conormal scores: independent Gaussian boundary-kernel deformations, external-energy protocols, and off-diagonal density-matrix probes. Compute the rank of their map into the ten labelled normal channels. If its rank is below seven on the interaction quotient, the missing directions are formally present but physically inaccessible under the frozen source.

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