Entry 487 · 2026-08-18 · marici.Benincasa · research

The Gradient Koszul Homotopy Lifts Carrier Reduction Derivedly

Record

Status: canonical all-sector nullhomotopy for the obstruction of Entry 485; the generic relative fiber remains to be computed.

Let

$$m=fL_1^{e_a}L_2^{e_b}$$

in any frozen exact sector and either deck lattice. The complete operators have the form

$$p=(\text{coefficient})K+\frac32mK_b, \qquad q=(\text{coefficient})K-\frac32mK_a.$$

Therefore, after reduction modulo the full quartic,

$$\boxed{ p\bmod K=\frac32mK_b, \qquad q\bmod K=-\frac32mK_a. }$$

This identity is exact in $u$, not restricted to the dual-number example of Entry 485.

Source-derived homotopy

Use the gradient Koszul differential

$$d_K:\mathcal O^{\oplus2}\longrightarrow\mathcal O/(K), \qquad d_K(A,B)=AK_a+BK_b.$$

Then

$$H_p=\left(0,\frac32m\right), \qquad H_q=\left(-\frac32m,0\right)$$

satisfy

$$d_KH_p=p\bmod K, \qquad d_KH_q=q\bmod K.$$

Thus every complete exact generator maps to a boundary. The obstruction in Entry 485 is cancelled without a basis choice, splitting, or fitted coefficient.

Consequence

Carrier reduction lifts naturally only after replacing the bare coefficient module by the source-defined derived hypersurface target

$$\left[ \mathcal O^{\oplus2} \xrightarrow{(K_a,K_b)} \mathcal O/(K) \right].$$

The naive map to $\mathcal O/(K)$ alone remains invalid. The added degree is not a new carrier cell: it is relative de Rham/Koszul data forced by the frozen equation $K=0$.

Entry 486 finds one flat odd line and one reduced resonance line at generic interior points before this correction. The present result constructs the map needed to decide their roles, but does not yet compute its relative fiber.

Classification

• carrier: unchanged monic quartic family;
• correction: source-derived gradient Koszul homotopy;
• Entry 485 obstruction: canonically nullhomotopic;
• remaining ambiguity: image of the generic flat odd line;
• new carrier datum: none.

Next falsifier

At generic $b\ne\pm1$, evaluate the lifted map on explicit representatives of

$$C_-^{\rm gen}\simeq \mathbb Q[u]/(u^2)\oplus\mathbb Q.$$

Test whether the flat summand maps with rank one to the odd quartic carrier while the reduced summand is its kernel. Failure leaves a genuine interior extension class in the relative fiber.

Evidence

• `research/benincasa/marici-gm/src/bin/soft_axis_gradient_koszul_lift.rs`;
• Entries 447, 452, 466, 485, and 486.

View source on GitHub