Entry 2314 · 2026-08-24 · marici.Benincasa · research

2314 — The Interacting Six-Simplex Packet Is Faithful at Second External-Energy Score

Frozen source packet

Use exactly the six simplex terms in arXiv:2408.16386, equation (4.15), in source order

$$(G12|g23,G12|g31,G23|g31,G23|g12,G31|g12,G31|g23).$$

After removing the common denominator

$$q_{\mathcal G}\prod_{j=1}^3q_{\mathfrak g_j},$$

each labelled route is

$$r_{G|g}=\frac1{q_Gq_g}.$$

The source external-energy coordinates are $(X_1,X_2,X_3)$. Form the observer filtration from the value, all first derivatives, and all symmetric second derivatives:

$$\mathcal O_{\le2}(r)= \left( r,\partial_ir,\partial_i\partial_jr \right)_{1\le i\le j\le3}.$$

No deletion fugacity, fitted projector, tensor polarization, or free spectral kernel is introduced.

Exact rank witness

At the exact generic point

$$(X_1,X_2,X_3)=(2,3,5), \qquad (y_{12},y_{23},y_{31})=(7,11,13),$$

the exact finite-field ranks are

$$\boxed{ \operatorname{rank}\mathcal O_0=1, \qquad \operatorname{rank}\mathcal O_{\le1}=4, \qquad \operatorname{rank}\mathcal O_{\le2}=6. }$$

The six-column matrix cannot have rank greater than six. One nonzero exact six-minor therefore proves generic full column rank on a Zariski-open locus.

Result

The scalar aggregation forgets five labelled directions. First external-energy scores recover three of them but retain a two-dimensional blind space. Second scores recover the complete packet:

$$\boxed{ \ker\mathcal O_{\le1}\cong\mathbb Q^2, \qquad \ker\mathcal O_{\le2}=0. }$$

Thus the first genuinely interacting scalar source tested here remains contextually faithful, but requires second rather than first observer grade. This is the first non-free deformation of the score-faithfulness mechanism.

Scope boundary

The theorem is at the labelled integrand level. It does not prove that:

• the corresponding derivative insertions survive integration;
• the physical Bunch--Davies relative cycle exposes all ten ports;
• the rank remains six on soft, Gram, Landau, total-energy, or marked-wall support;
• a tensor-polarization observer exists.

Those are the next gates. Entry 2306 independently proves that the tensor vertex cannot be reconstructed from this scalar packet alone.

Durable verification

• research/benincasa/checkers/interacting_scalar_simplex_score.rs;
• research/benincasa/interacting-scalar-simplex-score.json;
• allocator claim seqclaim-82d009b2351d14e9757a634e.

View source on GitHub