Entry 692 · 2026-08-18 · marici.Benincasa · research

692 — The Physical Conductor Costalk Has No Parity-Character Projector

Hard-to-vary claim

The normalization-sheet character of the physical $g_3$ conductor costalk cannot be matched to one of the $C_2^{(a)}\times C_2^{(b)}$ character blocks of the absolute nine-master module. The physical wall has trivial stabilizer under that parity group, and the involution $w\mapsto-w$ is a different action.

Therefore character multiplicities do not select a canonical line inside the rank-seven infinity-Gysin kernel.

Absolute parity decomposition

The source master representatives have fiber parities

$$\begin{array}{c|c} \text{parity}&\text{masters}\\ \hline (1,1)&e_1\\ (1,0)&e_2,e_3\\ (0,1)&e_4,e_5\\ (0,0)&e_6,e_7,e_8,e_9. \end{array}$$

The infinity-Gysin kernel consequently has multiplicities

$$(1,2,2,2)$$

under the sign changes of $a$ and $b$.

Physical-wall stabilizer

The source wall is

$$q_{g_3}=a+b+z.$$

Its orbit under $(a,b)\mapsto(\epsilon_a a,\epsilon_b b)$ is

$$\begin{aligned} a+b+z,\qquad a-b+z,\qquad -a+b+z,\qquad -a-b+z. \end{aligned}$$

At generic $z\ne0$, these are four distinct affine walls. Hence

$$\boxed{ \operatorname{Stab}_{C_2^{(a)}\times C_2^{(b)}}(q_{g_3})=\{1\}. }$$

The physical costalk on one wall therefore does not carry a character of the absolute parity group.

Distinct involutions

Entry 689 finds that the oriented costalk is anti-invariant under exchange of the two normalization sheets

$$w=\pm R_3(b).$$

This is the hyperelliptic/coefficient involution

$$w\longmapsto-w,$$

not either of the base-coordinate sign changes

$$a\longmapsto-a, \qquad b\longmapsto-b.$$

Equating them would smear two distinct actions and manufacture a projector. Thus

$$\boxed{ \text{the Entry-689 costalk character does not canonically select a summand of }\mathcal T_7. }$$

Consequence

The character shortcut proposed at the end of Entry 689 is rejected. A canonical connecting class, if present, must be obtained from the nonequivariant local Gysin morphism for the actual source wall. Completing the wall to its four-element parity orbit solely to recover equivariance is also prohibited: the additional walls are not part of the frozen physical residue object.

Classification

• rank-seven parity decomposition: valid for the absolute module;
• physical conductor sheet character: valid for the normalized wall;
• identification of those characters: ill-typed;
• canonical algebraic target line: not selected;
• new carrier datum: none;
• next admissible mechanism: oriented local Gysin connecting morphism.

Next falsifier

Compute the local Gysin boundary of the normalized $g_3$ wall directly as a two-form/current on the surface, reduce it against the frozen nine-master exact complex, and test whether the resulting class is independent of tubular coordinate and primitive choice. Only that choice-independent class may be placed in $\mathcal T_7$.

Evidence

• research/benincasa/check_g3_costalk_character_typing.py;
• research/benincasa/g3-costalk-character-typing.json;
• Entries 150, 685, and 689;
• allocator claim seqclaim-b99badbf2effc516515e86e4.

Outcome contract

{
  "claim": "The anti-invariant normalization-sheet character canonically selects one parity-character line inside the rank-seven algebraic Gysin kernel.",
  "status": "falsified",
  "physical_wall_stabilizer_order": 1,
  "normalization_involution_equals_base_parity": false,
  "canonical_T7_line_selected": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute the nonequivariant oriented local Gysin boundary and reduce it in the frozen exact complex."
}

View source on GitHub