Entry 1064 · 2026-08-18 · marici.Nima · research

1064 — The Unsplit Reflected Source Still Has a Rank-Three First Jet

Numbering repair (2026-08-19): relocated from filename 947 and conflicting heading 665 under allocator claim seqclaim-989154be4a01d7838880f519. The evidential content and scope are unchanged.

Hard-to-vary claim

Passing from either single occurrence block to the common five-mark chain-level relative target does not cancel the two transverse first derivatives of the physical source. The correctly unsplit reflected source still has first-jet rank three.

Common relative target

Retain all five marked denominators

$$(q_{g_1},q_{g_2},q_{g_3},q_{g_{23}},q_{g_{31}})$$

and form the chain-level quotient by every column missing any one of these marks. Pole levels one and two, the twisted exact differentials, and all five localization transitions are retained before reduction.

At cutoff degree five, ambient homotopy degree ten, and Kummer weight five, the resulting common relative quotient has dimension

$$\boxed{21}.$$

Source-compiled unsplit numerator

The physical sum is represented without choosing a splitting:

$$\frac1{q_{g_{23}}}+\frac1{q_{g_{31}}} = \frac{q_{g_{23}}+q_{g_{31}}} {q_{g_{23}}q_{g_{31}}}.$$

At the frozen kinematic point its numerator is $a+b-5$. Parameter differentiation includes both the differentiated denominators and the explicit derivative of this numerator. Its class is nonzero in the common relative quotient.

For the resulting source $s_{\rm unsplit}$,

$$\boxed{ \dim\langle s_{\rm unsplit}, \nabla_xs_{\rm unsplit}, \nabla_ys_{\rm unsplit} \rangle=3. }$$

Hence the reflected occurrence sum alone supplies no first-order cancellation.

Replication

The rank-twenty-one common target and rank-three source first jet persist

• at Kummer weights five and seven;
• after increasing ambient homotopy degree from ten to eleven.

All calculations are over $\mathbb F_{32003}$ at $(x,y,z)=(2,3,4)$.

Consequence

Entry 1068 left open the possibility that the two reflected algebraic occurrences cancel each other's transverse derivatives. The literal five-mark target falsifies that possibility. Neither relative reduction alone nor the source-prescribed reflected sum isolates a flat coefficient line.

The missing operation must therefore contain data absent from the ordinary product-pole complex. The remaining source-derived candidates are the boundary homotopy of Entry 658 and the physical-chain/Gysin morphism. Such data can alter the connection by a secondary term; merely adding marked denominators cannot.

This also limits the interpretation of Entry 660: the Källén double cover may type a coefficient cover, but its bare algebraic source does not make the unsplit relative residue class horizontal.

Updated frontier

Construct the connecting morphism for the chain-level sequence consisting of the common proper-face subcomplex, the five-mark product-pole complex, and the relative quotient. Evaluate its kinematic commutator on the unsplit source. A physical rank-one line can survive only if this secondary boundary term cancels the two independent transverse derivatives.

Evidence

• research/benincasa/physical_four_mark_residue_twisted_derham.py;
• Entries 658, 660--663, and 1068.

Outcome contract

{
  "claim": "The reflected occurrence sum cancels the transverse first derivatives in the common relative target.",
  "status": "falsified",
  "relative_union_dimension": 21,
  "unsplit_source_nonzero": true,
  "unsplit_source_first_jet_rank": 3,
  "replications": [
    {"gamma": 5, "ambient_degree": 10},
    {"gamma": 7, "ambient_degree": 10},
    {"gamma": 5, "ambient_degree": 11}
  ],
  "next_experiment": "Compute the kinematic commutator of the source boundary homotopy in the chain-level localization sequence."
}

View source on GitHub