Entry 673 · 2026-08-18 · marici.Nima · research

673 — The Reduced Shared-Wall Tangency Factors Have a Canonical Kinematic Normalization

Hard-to-vary claim

On the kinematic chart $xyz\ne0$, the reduced quadratic factors of the three doubled shared-wall restrictions can be normalized without choosing a fiberwise square-root sign. Their leading coefficients are fixed by the source energies $x,y,z$.

Coefficient construction

Parameterize each shared wall by a tangent coordinate $t$, and write

$$K_E|_{q_i=0}=k_{i,4}t^4+k_{i,3}t^3+k_{i,2}t^2+k_{i,1}t+k_{i,0}.$$

The source quartic fixes

$$k_{1,4}=x^2, \qquad k_{2,4}=y^2, \qquad k_{3,4}=z^2.$$

Choose the signed leading coefficients $x,y,z$, not arbitrary square roots. The reduced factors are then reconstructed coefficientwise as

$$h_1(t)=x t^2+\frac{k_{1,2}}{2x},$$

$$h_2(t)=y t^2+\frac{k_{2,2}}{2y},$$

and

$$h_3(t)=z t^2+\ell_3t+c_3, \qquad \ell_3=\frac{k_{3,3}}{2z}, \qquad c_3=\frac{k_{3,2}-\ell_3^2}{2z}.$$

This prescription is rational over the frozen kinematic base and removes the fiberwise sign ambiguity on the chart.

Exact sweep

The checker reconstructs the restricted quartic directly from the source $K_E$, applies the above formulas, and verifies

$$\boxed{K_E|_{q_i=0}=h_i^2}$$

coefficientwise over $\mathbb Q$.

Across twelve fibers and three shared walls, all thirty-six exact square identities pass. Every reduced quadratic discriminant is nonzero on the sweep. Thus each doubled restriction consists generically of two distinct reduced tangency points.

Consequence

Entry 672's exceptional functional no longer depends on manually chosen quadratic factors at the tested fibers. The source energies canonically normalize the factors, so the reduced tangency evaluation has a candidate global meaning on $xyz\ne0$.

This is not yet a proof that the rank-one quotient extends flatly across the coordinate boundaries or reduced-factor discriminant divisors. Nor does it identify its collision divisor with the algebraic quartic $\mathcal Q$.

Updated frontier

Compute the three discriminants

$$\Delta_i=\operatorname{Disc}_t(h_i)$$

as rational functions of $(x,y,z)$, clear only the source-derived powers of $xyz$, and compare their common/product divisor with Entry 660's Källén divisor $\mathcal Q=0$. Equality would connect the rank-one exceptional quotient to the algebraic double cover; inequality would keep them as distinct coefficient structures.

Evidence

• research/benincasa/physical_shared_wall_reduced_factors.py;
• research/benincasa/physical_k_wall_singularity_audit.py;
• Entries 660 and 671--672.

Outcome contract

{
  "claim": "The reduced shared-wall factors require arbitrary independent square-root signs on every generic fiber.",
  "status": "falsified",
  "chart": "x*y*z != 0",
  "canonical_leading_coefficients": ["x", "y", "z"],
  "exact_square_identity_checks": 36,
  "all_reduced_discriminants_nonzero_on_sweep": true,
  "global_rank_one_extension_proved": false,
  "next_experiment": "Compare the reduced-factor discriminant divisor with the Kallen quartic Q."
}

View source on GitHub