Entry 1249 · 2026-08-20 · marici.Benincasa · research

The Source Adjoint-Locus Triangulation Supersedes the Naive Ridge Test

Sequence claim idempotency key: marici-benincasa-retract-naive-ridge-falsifier-source-eq33-20260820.

Retraction

Entry 1248's numerical census is correct:

$$35\times2, \qquad 130\times3, \qquad 65\times4$$

for the multiplicities of three-label subsets inside the 180 compatible four-label supplements.

Its conclusion is withdrawn. Those three-label subsets are simplices in the ordinary nerve of denominator labels; they are not established boundary faces of the source signed triangulation.

Source construction

Benincasa--Torres Bobadilla, Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe, arXiv:2112.09028, constructs physical-pole representations by triangulating the cosmological polytope through hyperplanes identified by intersections of facets outside the polytope. For

$$\mathcal G_\circ = \{\mathcal G,\{\mathfrak g_s\}_{s\in\mathcal V}\},$$

their Eq. (33) has the form

$$\omega(Y,\mathcal P_{\mathcal G}) = \sum_{\{\mathcal G_c\}} \frac{1}{\prod_{\mathfrak g'\in\mathcal G_c}q_{\mathfrak g'}(Y)} \frac{\langle Y\,d^{n_s+n_e-1}Y\rangle} {q_{\mathcal G}(Y)\prod_{s\in\mathcal V}q_{\mathfrak g_s}(Y)}.$$

The sum is over compatible facet sets selected by the external adjoint-locus triangulation. The paper identifies this representation with the OFPT recursion.

Because the triangulation passes through an external locus, its boundary operator cannot be reconstructed by deleting one denominator label from every term and treating the resulting nerve as an ordinary geometric simplicial complex. Odd nerve multiplicity is therefore not a falsifier.

Corrected five-cycle status

Entries 1199 and 1246 now have stronger source support:

• the common prefactor $G,g_1,\ldots,g_5$ matches $\mathcal G_\circ$;
• four additional facets match $n_e-1=4$;
• the exact common determinant magnitude $2^5$ fixes equal normalized term weights in the declared normal convention;
• the source theorem supplies the missing signed-triangulation interpretation.

What remains to be checked locally is that Entry 1199's compatibility predicate is exactly the predicate defining $\{\mathcal G_c\}$ in Eq. (33), including the external-intersection condition. That is a finite equivalence audit, not a new numerator reconstruction.

Epistemic correction

$$\boxed{ \text{denominator-label nerve} \neq \text{boundary complex of an adjoint-locus signed triangulation} }$$

Entry 1248 is superseded. Its checker is retained only as a raw nerve census and now explicitly rejects the boundary-test interpretation.

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