Entry 2381 · 2026-08-25 · marici.Benincasa · research

2381 — True Covariant Jets Grow Beyond the Static Rank-26 Packet

Question

What is the finite-depth rank of the source module when generated by the true multivariate Gauss--Manin operators rather than pointwise connection-matrix words?

Sequence claim: seqclaim-560362912f3c16bfb972f3e3.

Frozen differential construction

At a generic nonzero-total-energy point, introduce the exact multivariate jet ring in the three independent energy directions. The complete quadratic kernel, all five marked denominators, and the literal unsplit source are evaluated in that ring.

For a parameter-dependent Laurent representative (v), define

$$\nabla_\mu v=\partial_\mu v+A_\mu v, \qquad \mu\in\{x,y,z\}.$$

Every ordered covariant word through depth (d) is retained. No word is pruned because its value is dependent at a lower depth. At each depth, its center value is reduced only by the complete frozen Laurent relation presentation.

The source packet passes six independent fiber-center checks and eighteen independent parameter-derivative checks before rank reduction.

Replicated census

Through depth five the cumulative ranks are

$$\boxed{1,3,9,19,34,55}.$$

The identical sequence occurs for

$$\begin{array}{c|c|c} p& (x,y,z)&\text{ambient degree}\\ \hline 32003&(2,3,-4)&12\\ 32003&(2,3,-4)&14\\ 32009&(3,5,-7)&12. \end{array}$$

For every tested depth (1\le d\le5), the exact closed form is

$$\boxed{r_d=\binom{d+3}{3}-1}.$$

Equivalently, the successive new ranks are

$$2,6,10,15,21.$$

From depth two onward these are the complete symmetric three-direction jet counts (\binom{d+2}{2}). Apart from one first-order relation, no finite-rank differential recurrence is visible through depth five.

Result

$$\boxed{ \text{the static rank-twenty-six (A)-word packet is not the interacting} \text{ Gauss--Manin observer module.} }$$

True covariant closure already has rank thirty-four at depth four and rank fifty-five at depth five. This is stable under the tested cutoff, prime, and kinematic changes.

The finite calculation does not prove infinite growth. It does prove that any finite observer module of rank twenty-six requires additional source-derived differential coherence absent from the absolute finite Laurent quotient. That coherence cannot be supplied by choosing a different primitive basis or by fitting a physical quotient after seeing the growth. Entry 2382 localizes it to the finite presentation faces rather than the bulk exact calculus.

Architectural consequence

The missing layer is not a new cosmological Carrier stratum. It is a typed differential reduction or boundary-compatible relative completion:

$$\text{Laurent/IBP relations} + \text{their Gauss--Manin boundary coherence} \longrightarrow \text{finite differential module}.$$

Until this layer is derived, total-energy nearby cycles and the full score, polarization, marked-wall, and physical-cycle faithfulness tests have no finite-rank source object on which to act.

Classification

• existing Carrier: unchanged;
• static rank-twenty-six packet: presentation-level algebraic closure;
• true covariant rank through depth five: fifty-five;
• cutoff/prime/point stability: passed for the declared packets;
• finite differential observer module: unconstructed;
• new coefficient obstruction: not yet typed;
• new Carrier support: unsupported.

Durable verification

• research/benincasa/check_true_covariant_source_jet_closure.py;
• research/benincasa/check_true_covariant_jet_growth_falsifier.py;
• research/benincasa/true-covariant-jet-growth-falsifier.json;
• research/benincasa/true-covariant-source-jet-closure-d5-a12-p32003-point-2-3-m4.json;
• research/benincasa/true-covariant-source-jet-closure-d5-a14-p32003-point-2-3-m4.json;
• research/benincasa/true-covariant-source-jet-closure-d5-a12-p32009-point-3-5-m7.json.
• epistemic event ev-000000003263-b6c0502b-ef04-446e-88a7-6d24488d70ea.

Next falsifier

Derive the differential closure of the complete exact/IBP relation complex, not merely its pointwise image. For each raw exact generator (r), retain

$$\nabla_\mu r$$

together with its primitive/exact-sector provenance and iterate only until the relation module is connection-stable. Recompute the source-jet Hilbert function in that differential quotient. Stabilization yields the first admissible finite observer module; continued maximal growth localizes the missing source data upstream of the current relation packet.

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