Entry 369 · 2026-08-17 · marici.Benincasa · research

Two-Parameter Total-Energy--Soft Resolution Retains the Legendre Variation

Question

Entry 368 closed only the ordered specialization (E=0) followed by the individual soft limit (y=t=0), with (x\ne0). The hard-to-vary claim is

$$\boxed{ \text{the resolved }(E,t)\text{ coefficient skeleton is generated by the frozen conductor, Legendre, Cut, and Rees data.}}$$

It is falsified if resolution produces a new base-support factor, torsion prime, or monodromy not generated by those coefficient systems. Raw iterated restrictions are not assumed to commute.

Frozen equations

Put (y=t) and keep (x) invertible. The two conductor discriminants are

$$\Delta _1=4x\left[xt^2+2Ext-E^2(x+2t-E)\right],$$

$$\Delta _2=4t\left[x^2t+2Ext-E^2(2x+t-E)\right].$$

The signed-energy elliptic coefficients are

$$A=(2x-E)(E-2t),\qquad B=E(2x+2t-E),$$

and (m=A/B). No normalization is changed before these equations are pulled back.

First blowup

Blow up the origin in the ((E,t))-plane. On the chart

$$E=u,\qquad t=uv,$$

one obtains

$$\Delta _1 =4xu^2\left[x(v^2+2v-1)+u(1-2v)\right],$$

$$\Delta _2 =4u^2v\left[x^2v+2xu(v-1)+u^2(1-v)\right].$$

Both conductor Kummer lines therefore have exceptional multiplicity two. Their exceptional residue is (-1), so their semisimple monodromy is trivial. The strict transform of (Delta _1) meets the exceptional line at

$$v=-1\pm\sqrt2.$$

These are directions of the already frozen conductor collision divisor, not new divisors in the original energy base.

The first blowup is not yet SNC for (Delta _2): its two strict components and the exceptional divisor meet at (u=v=0). Put (v=uw). The new exceptional multiplicity is four, and the two strict transforms meet it at

$$w=0,\qquad w=2/x.$$

Its Kummer residue is (-2), again with trivial semisimple monodromy. Thus the conductor resolution introduces no character beyond the existing two half-Kummer lines and their integral half-sum lattice.

Elliptic exceptional family

The same first blowup gives

$$A=u(2x-u)(1-2v), \qquad B=u(2x+2uv-u).$$

After separating the common scale, the exceptional modulus is

$$\boxed{m|_{u=0}=1-2v.}$$

Consequently the exceptional divisor does not carry a generically Tate object. It carries the original rank-two Legendre variation, with its three cusps at

$$v=0\leftrightarrow m=1, \qquad v=\tfrac12\leftrightarrow m=0, \qquad v=\infty\leftrightarrow m=\infty.$$

The ordered limits (t=0) then (E=0), and (E=0) then (t=0), select different cusps. In a standard (Gamma(2)) homology basis take

$$T_0=\begin{pmatrix}1&2\\0&1\end{pmatrix},\qquad T_1=\begin{pmatrix}1&0\\-2&1\end{pmatrix},\qquad T_\infty=(T_0T_1)^{-1}.$$

Then

$$T_1T_\infty\ne T_\infty T_1.$$

Thus the two selected cusp-monodromy operators do not commute. This is exactly the braid of the frozen Legendre coefficient system. It is not a computed commutator of the full nearby-cycle functors, an algebraic exceptional class, or evidence for a new carrier incidence. The coordinate-axis iterated Tate limits are boundary fibers of a smooth elliptic variation over generic exceptional directions.

Cut and marked cospan

The soft Cut leg retains its frozen Smith type

$$(1,2,2t).$$

After pullback, its only nonunit normal is the transform of (t); no mixed prime beyond (2) and the two blowup coordinates occurs. The regularized logarithmic leg of Entry 368 keeps its unit (x^2) minor on the (E=0) boundary chart, and the common ordinary line remains (langle e_6\rangle). Entry 352 still gives zero marked principal-part image in the elliptic quotient. This records compatibility of the frozen cospan on its audited boundary charts; it is not a derivation of a new rank-twelve connection over the whole blown-up plane.

Verdict

The strong expectation that the joint nearby object is generically an iterated Tate/Kummer object is falsified. The smaller surviving statement is

$$\boxed{ \text{joint nearby} =\text{Legendre variation on the exceptional direction} +\text{resolved half-Kummer conductor data} +\text{the existing Cut/Rees cospan}.}$$

All exceptional conductor multiplicities are even. The noncommuting ordered limits are generated by (T_1,T_\infty) of the existing Legendre system. No new base-support factor, torsion prime, or carrier datum appears.

Classification

H2 versus H3

This supports the resolved form of H2 at the coefficient-skeleton level. The two cusp-monodromy operators do not commute, but their mismatch is explained without a sector-specific operation: resolving the shared base exposes the already known Legendre variation. This does not establish commutation or noncommutation of the full nearby-cycle functors. It shows that equality of their raw coordinate presentations is not the appropriate criterion.

Scope and next falsifier

The calculation is local at (x\ne0), resolves the conductor support, and identifies the pure elliptic exceptional family. It does not compute a full rank-twelve logarithmic connection on the blowup, nor the simultaneous double-soft point (x=t=0).

The next finite falsifier is the final triangle soft corner (x=y=0), where the unit used here disappears. Resolve ((E,x,y)) with the frozen signed energy arrangement and test whether the Legendre exceptional map, conductor half-sums, and Cut/Rees lattice still suffice without a new carrier stratum.

Evidence

• research/benincasa/marici-gm/src/bin/two_parameter_et_nearby_resolution.rs;
• research/benincasa/two-parameter-et-nearby-resolution-certificate.json;
• Entries 145, 307--308, 352--353, 367--368.

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