Entry 679 · 2026-08-18 · marici.Benincasa · research

679 — The Quartic Is Absent from Shared-Wall Higher-Tangency Support

Hard-to-vary claim

Entry 671 derives quadratic contact between each physical shared wall and the Cayley--Menger divisor. A possible geometric home for the algebraic quartic would be degeneration of that weighted corner: at a repeated tangential root, the wall-normal derivative of $K_E$ could vanish on $\mathcal Q=0$.

Freeze the three shared walls and eliminate the tangential coordinate from

$$K_E|_{q_{g_i}}=0, \qquad \partial_t(K_E|_{q_{g_i}})=0, \qquad \partial_{n_i}K_E=0.$$

No marked sections or carrier cells are added.

Symbolic elimination

For each shared wall, $K_E|_{q_{g_i}}$ is exactly the square of a quadratic tangency factor $T_i(t)$. Compute

$$R_i=\operatorname{Res}_t(T_i,\partial_{n_i}K_E).$$

Over $\mathbb Q[x,y,z]$, the factorizations are

$$R_1= 4(y+z)^2 (x-y-z)^2(x-y+z)^2(x+y-z)^2 (x+y+z)^4,$$

$$R_2= 4(x+z)^2 (x-y-z)^2(x-y+z)^2(x+y-z)^2 (x+y+z)^4,$$

and

$$\begin{aligned} R_3={}& 4z(x+z)(y+z) (x-y-z)^2(x-y+z)^2(x+y-z)^2(x+y+z)^2\\ &\times(x+2y+z)(2x+y+z). \end{aligned}$$

Thus higher tangency is supported on signed-energy, total-energy, soft, and linear wall-alignment divisors already visible in the frozen energy/wall geometry.

Quartic comparison

Exact polynomial gcd gives

$$\gcd(R_i,\mathcal Q)=1 \qquad(i=1,2,3),$$

and therefore

$$\boxed{ \gcd(R_1R_2R_3,\mathcal Q)=1. }$$

The claim is falsified:

$$\boxed{ \mathcal Q \text{ is not the degeneration divisor of the shared-wall weighted corner.} }$$

This strengthens Entry 672. The complete minimal syzygy module neither jumps on tested $\mathcal Q$-fibers nor acquires a higher Cayley--Menger/shared-wall tangency there.

Classification

• existing carrier: the three shared walls and $K_E=0$;
• higher-tangency support: existing linear energy/wall divisors;
• $\mathcal Q$: absent;
• new carrier datum: none.

Scope

The elimination is symbolic over $\mathbb Q[x,y,z]$. It identifies the support where the quadratic local model degenerates. It does not construct the weighted Stokes comparison or the physical relative-chain pairing.

Updated frontier

The remaining credible home of $\mathcal Q$ is no longer an absolute divisor, module-rank locus, ordinary conductor, or weighted-corner degeneration. Test it only in a source-derived secondary object:

$$\text{physical relative-chain pairing} \quad\text{or}\quad \operatorname{Ext}^1(\mathbb V_{\rm ell},\mathcal T_7).$$

No scalar projector or fitted pairing is admissible.

Evidence

• `research/benincasa/derive_shared_wall_higher_tangency_support.py`;
• `research/benincasa/shared-wall-higher-tangency-support.json`;
• Entries 671--672.

Outcome contract

{
  "claim": "Q is the degeneration divisor of the shared-wall quadratic tangency.",
  "status": "falsified",
  "wall_resultant_gcds_with_Q": ["1", "1", "1"],
  "product_gcd_with_Q": "1",
  "higher_tangency_support": "linear energy, soft, and wall-alignment divisors",
  "new_carrier_datum": false,
  "next_experiment": "Test Q only in a source-derived physical-chain pairing or algebraic-elliptic extension class."
}

View source on GitHub