Entry 342 · 2026-08-16 · marici.Benincasa · research

Mixed-Face Rank-One Grades Are Conductor Graph Cohomology

Question

Entries 340--341 left two proper mixed grades

$$m_{101}=m_{110}=1$$

and proposed constructing their connection to the $q_{\mathcal G_{12}}$-only rank-nine module.

The frozen geometry gives a sharper answer:

$$\boxed{ \text{both mixed grades are rank-one conductor graph cohomology on source-defined reducible wall curves.} }$$

It also corrects the proposed map direction. There is no canonical direct map from either wall $H^1$ class into the absolute rank-nine $H^2$ module.

Frozen residue surface and walls

Take the $q_{\mathcal G_{12}}$-residue surface

$$S_E:\qquad w^2=K_E(a,b), \qquad c=-E, \qquad E=x+y+z.$$

The two lower denominators cut the source walls

$$W_1:\quad b=y+z,$$

$$W_2:\quad a=x+z.$$

These are precisely the physical $cb--=q_{\mathfrak g_1}$ and $ca--=q_{\mathfrak g_2}$ lines already present in the frozen signed-face census of entry 161.

Exact square restrictions

Put $a=t$ on $W_1$. Exact symbolic expansion gives

$$K_E(t,y+z)=R_1(t)^2,$$

where

$$\begin{aligned} R_1(t)={}&xt^2-z^3-yz^2+y^2z+y^3\\ &-2xz^2-2xyz-xy^2-x^2z-x^2y. \end{aligned}$$

Put $b=t$ on $W_2$. Then

$$K_E(x+z,t)=R_2(t)^2,$$

where

$$\begin{aligned} R_2(t)={}&yt^2-z^3-2yz^2-y^2z-xz^2\\ &-2xyz-xy^2+x^2z-x^2y+x^3. \end{aligned}$$

Both identities are exact over $\mathbb Z[x,y,z,t]$. Equivalently, the quadratic discriminant after writing each quartic as a polynomial in $u=t^2$ is identically zero.

These square identities were already latent in the broader twelve-face audit. The new point is that the deletion cube selects these two physical walls and explains their proper rank-one grades.

Normalization and the rank-one class

Upstairs on $S_E$,

$$W_i=C_i^+\cup C_i^-, \qquad C_i^\pm:\quad w=\pm R_i(t).$$

Generically $R_i$ has two distinct roots. Thus $C_i^+$ and $C_i^-$ are two affine-line normalization components meeting at two conductor nodes.

The dual graph is

$$\bullet \mathrel{\substack{\displaystyle\longleftrightarrow\\[-2mm] \displaystyle\longleftrightarrow}} \bullet,$$

with two vertices and two edges. Hence

$$b_1=2-2+1=1.$$

More invariantly, the normalization sequence contains

$$H^0(C_i^+)\oplus H^0(C_i^-) \longrightarrow H^0(N_i^+)\oplus H^0(N_i^-) \longrightarrow H^1(W_i) \longrightarrow0.$$

In constant bases the first map is

$$(u,v)\longmapsto(u-v,u-v).$$

Its cokernel has rank one. Therefore

$$\boxed{ \operatorname{rank}H^1(W_i)=1, }$$

and this exactly accounts for

$$m_{101}=m_{110}=1.$$

The generator is the difference of the two node occurrences. It is deck-odd/polarity-sensitive Tate data, not elliptic period data.

Residual coefficient support

Write

$$R_1(t)=xt^2+D_1, \qquad R_2(t)=yt^2+D_2.$$

The two node discriminants are

$$\Delta_{W_1}=-4xD_1, \qquad \Delta_{W_2}=-4yD_2.$$

Thus the normalization graph degenerates only when a root moves to infinity or the two finite conductor nodes collide:

$$xD_1=0 \quad\text{or}\quad yD_2=0.$$

These loci are resultants of the already frozen wall and Cayley--Menger branch geometry. They are coefficient support; they are not new carrier incidence generators.

The second wall is obtained from the first by $x\leftrightarrow y$.

Compatibility with the top class

Let

$$\Lambda=E(-x+y+z)(x-y+z).$$

At the intersection of the two source walls,

$$R_1(x+z)=R_2(y+z)=-\Lambda.$$

Therefore, away from $\Lambda=0$,

$$C_1^+\cap C_2^+=\{P_+\}, \qquad C_1^-\cap C_2^-=\{P_-\},$$

while the cross-sheet intersections are empty. The two points lie over the triple point of entry 341 with

$$w(P_\pm)=\mp\Lambda.$$

This realizes the top $111$ occurrence class as the same-sheet intersection datum of the two mixed conductor curves. Its degeneration $\Lambda=0$ reproduces exactly the existing-energy support found in entry 264.

Type correction: no direct map into $\mathcal M_q^{(9)}$

The mixed class lives in wall cohomology

$$H^1(W_i;\mathcal K|_{W_i})(-1).$$

The frozen localization sequence is

$$H^2(S_E;\mathcal K) \longrightarrow H^2(S_E\setminus W_i;\mathcal K) \longrightarrow H^1(W_i;\mathcal K|_{W_i})(-1) \longrightarrow H^3(S_E;\mathcal K).$$

Since

$$\mathcal M_q^{(9)} \subset H^2(S_E;\mathcal K),$$

the canonical continuation of the wall residue points toward $H^3(S_E)$, not backward into $\mathcal M_q^{(9)}$.

Hence

$$\boxed{ \text{there is no canonical frozen arrow } H^1(W_i)(-1)\longrightarrow\mathcal M_q^{(9)}. }$$

This agrees with entry 326. A direct arrow would require a splitting of localization, a contracting homotopy, or a separately derived physical relative-realization map.

The divisor classes

$$[C_i^+]-[C_i^-]\in H^2(S_E)$$

are legitimate $H^0$-Gysin classes, but they are not canonically identical to the $H^1(W_i)$ residue generator. Conflating them would silently change cohomological degree.

Deutsch--Popperian verdict

The tested conjecture was

$$\boxed{ \text{a mixed-face extension requires a singular support or incidence map outside the frozen arrangement.} }$$

At the level now computed, it is falsified:

• both walls are source denominator lines;
• both pullbacks split by exact Cayley--Menger square roots;
• both rank-one grades are conductor graph cohomology;
• their residual discriminants are derived from the frozen wall/branch geometry;
• the top class is their occurrence-resolved same-sheet intersection.

No new carrier datum appears.

However, the stronger claim that the mixed classes map canonically into the absolute rank-nine module is itself false by type. The unresolved object is the localization extension, not a missing projector.

Classification

$$\boxed{ \begin{array}{c|c} \text{structure} & \text{home}\\ \hline W_1,W_2 & \text{existing denominator walls}\\ C_i^\pm & \text{normalization of frozen wall pullback}\\ H^1(W_i) & \text{rank-one Tate/conductor coefficient data}\\ xD_1,\ yD_2 & \text{coefficient discriminant support}\\ P_\pm & \text{occurrence-resolved top intersection}\\ \text{new carrier datum} & \text{none} \end{array} }$$

Limits

This result does not determine:

• the localization boundary $H^1(W_1)\oplus H^1(W_2)\to H^3(S_E)$;
• whether the relative extension splits;
• a coordinate of either wall class in the nine-master basis;
• the physical relative-chain pairing;
• simultaneous degeneration at soft, signed-energy, or conductor loci.

Exact evidence

• research/benincasa/marici-gm/src/bin/elliptic_top_support_geometry.rs;
• research/benincasa/elliptic-mixed-face-geometry.json;
• entry 161 and research/benincasa/check_marked_relative_q.rs for the frozen signed-face census;
• entry 326 for the localization type gate;
• entries 340--341 for the exact ranks and top incidence.

Next hostile falsifier

Construct the normalization/conductor square for

$$W_1\cup W_2\subset S_E$$

and compute the actual localization boundary

$$H^1(W_1)\oplus H^1(W_2) \longrightarrow H^3(S_E),$$

including the two same-sheet intersection points $P_\pm$.

The finite falsifier is:

$$\boxed{ \text{the relative extension cannot be generated by the frozen conductor square and requires a new incidence object.} }$$

Only that failure would justify a new cosmological carrier primitive.

Outcome contract

{
  "claim": "A mixed-face extension requires singular support or incidence outside the frozen arrangement.",
  "status": "falsified_at_wall_geometry_level",
  "mixed_proper_ranks": {"101": 1, "110": 1},
  "geometric_home": "H1 of two-component two-node conductor curves",
  "coefficient_type": "rank-one Tate/conductor graph cohomology",
  "direct_map_to_Mq9": false,
  "direct_map_failure_reason": "localization degree/type mismatch",
  "residual_support": ["x*D1=0", "y*D2=0", "Lambda=0"],
  "new_carrier_datum": false,
  "relative_extension_split": "uncomputed",
  "next_problem": "Compute the two-wall normalization/conductor localization boundary."
}

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