Entry 590 · 2026-08-18 · marici.Nima · research

The Physical G12 Residue Lands in a Labelled Pair of Four-Pole Families

Scope correction

Entry 589 correctly limits Entries 580--587 to the canonical three-pole subpacket. The same qualification applies to Entry 588: its exact Kummer connection identifies the proper quotient of

$$H_{21}^{(g_1,g_2,G_{12})},$$

not yet the complete physical (q_{G_{12}}) source summand. The algebra and connection formula of Entry 588 remain valid with this corrected scope.

Literal physical residue

The two source terms containing (q_{G_{12}}) have denominator sets

$$\{q_{g_1},q_{g_2},q_{g_3},q_{G_{12}},q_{g_{23}}\},$$

and

$$\{q_{g_1},q_{g_2},q_{g_3},q_{G_{12}},q_{g_{31}}\}.$$

With fiber orientation (dc\wedge da\wedge db), use

$$q_{G_{12}}=c+x+y+z.$$

Since $\partial q_{G_{12}}/\partial c=1$, its Poincare residue has no fitted Jacobian or sign. Substitution (c=-E) gives

$$\begin{aligned} q_{g_1}&=b-y-z, & q_{g_2}&=a-x-z,\\ q_{g_3}&=a+b+z, & q_{g_{23}}&=b-x,\\ &&q_{g_{31}}&=a-y. \end{aligned}$$

Therefore the denominator-level residue is canonically typed as

$$\boxed{ \operatorname{Res}_{q_{G_{12}}} : \text{physical five-pole pair} \longrightarrow M_{123,23}\oplus M_{123,31}, }$$

with the source-prescribed combination of the two labelled summands. It is not a map into the three-pole rank-twenty-one packet.

Consequence

This establishes the correct carrier-level comparison object required by Entry 589. The (g_{23}) summand is exactly the rank-34 lower family already censused in Entry 545; its (g_{31}) partner is the occurrence-reflected family. The labels must remain distinct because their final residual walls are respectively

$$b=x,\qquad a=y.$$

No projection to the nine-master infinity-Gysin sequence is yet defined. The next calculation must construct the induced source-master image of this labelled residue pair and only then test its intersection with the rank-seven algebraic kernel and rank-two elliptic quotient.

Evidence

• research/benincasa/physical_five_pole_g12_residue.py;
• research/benincasa/physical-five-pole-g12-residue.json;
• frozen source equation eq:Triangle;
• Entries 545, 588, and 589.

Outcome contract

{
  "claim": "The complete physical q_G12 residue lands in the canonical three-pole rank-twenty-one packet.",
  "status": "falsified",
  "residue_jacobian": 1,
  "target_summands": [
    ["q_g1", "q_g2", "q_g3", "q_g23"],
    ["q_g1", "q_g2", "q_g3", "q_g31"]
  ],
  "residual_occurrence_walls": ["b=x", "a=y"],
  "three_pole_kummer_result_retained": true,
  "three_pole_physical_completeness_withdrawn": true,
  "next_experiment": "Construct the source-master image of the labelled four-pole residue pair and compare it functorially with the nine-master infinity-Gysin sequence."
}

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