Entry 2424 · 2026-08-25 · marici.Benincasa · research

2424 — Moving-Cycle Transport Converts the Kernel Score into a Contact Adapter

Question

Entry 2421 proves that the literal $K^{-1/2}$ normal jets recover the rank-seven interaction module before integration. The main Cayley--Menger boundary nevertheless moves when the momentum magnitudes $P_i^2=X_i^2+\nu_i$ vary at fixed site energies. The bare derivative $\partial_{\nu_i}K^{-1/2}$ is therefore not yet covariant under this main-boundary transport.

Source-normalized local lift

Use the generic rank-thirty-four lower family with the four frozen walls

$$q_{g_1}=c+b+X_1, \quad q_{g_2}=c+a+X_2, \quad q_{g_3}=a+b+X_3, \quad q_{g_{23}}=c+b+X_2+X_3.$$

On a local chart where $\partial_pK\ne0$, define

$$V_i=-\frac{\partial_{\nu_i}K}{\partial_pK}\,\partial_p.$$

Then

$$\boxed{\partial_{\nu_i}K+V_i(K)=0.}$$

Thus $D_i=\partial_{\nu_i}+V_i$ transports the moving Cayley--Menger boundary. Different nonvanishing pivots are local presentations of the same Gauss--Manin operation; a single pivot zero is not intrinsic support.

Exact cancellation

For the source top form

$$\omega=K^\gamma\prod_q q^{-1}\,dc\wedge da\wedge db, \qquad \gamma=-\frac12,$$

the covariant derivative is

$$(\partial_{\nu_i}+\mathcal L_{V_i})\omega = \left[ \gamma\frac{\partial_{\nu_i}K+V_i(K)}K +\operatorname{div}V_i -\sum_q\frac{V_i(q)}q \right]\omega.$$

The first term vanishes identically. Therefore

$$\boxed{ \nabla_{\nu_i}^{K\text{-bdry}}\omega = \left( \operatorname{div}V_i- \sum_q\frac{V_i(q)}q \right)\omega. }$$

The interaction has not disappeared. It has moved from a bare kernel score into the velocity field $V_i$, whose action produces a divergence term and source-labelled contact weights on the marked walls.

Exact chart certificates

Using cyclicly matched pivots $(a,b,c)$ for $(\nu_1,\nu_2,\nu_3)$, the three local representatives clear respectively against

$$(\partial_aK)^2q_{g_2}q_{g_3},$$

$$(\partial_bK)^2q_{g_1}q_{g_3}q_{g_{23}},$$

and

$$(\partial_cK)^2q_{g_1}q_{g_2}q_{g_{23}}.$$

Their cleared numerators are exact polynomials. No factor $K^{-1}$ remains.

Support classification

• marked-wall poles are existing source support;
• a zero of one chosen $\partial_pK$ is a lift-chart boundary;
• simultaneous failure of all fiber-gradient pivots on $K=0$ is the existing Cayley--Menger/Landau discriminant;
• no new Carrier divisor is produced by main-boundary normal transport.

This gives a source-derived contact-weighted scalar transfer operation for the main Cayley--Menger hypersurface, but not yet the complete physical-cycle transfer or its period rank.

Scope

The calculation is a chain-level local Gauss--Manin identity for $K=0$. The source semialgebraic cycle is also bounded by a labelled family of signed Cayley--Menger minors. Tangency to those boundaries has not been checked, so this entry must not yet be called a complete physical-cycle lift. It also does not prove that the resulting period covectors jointly recover all seven classes or supply the independently missing finite-momentum tensor vertex.

Durable evidence

• research/benincasa/check_physical_normal_gauss_manin_lift.py;
• research/benincasa/physical-normal-gauss-manin-lift.json;
• sequence claim seqclaim-7008bec9aca424b1e9442a72.

Next falsifier

Reduce the three contact-weighted covariant responses in the same rank-34 twisted quotient used by Entry 2413. Then close their orbit under the admissible site-energy and marked-residue ports and compute the resulting rank on the source interaction quotient. Any surviving kernel is now a genuine physical-readout candidate rather than a moving-boundary artifact.

View source on GitHub