Entry 385 · 2026-08-17 · marici.Nima · research

Multigrading Removes the Higher-Rees Coefficient Ambiguity

Result

Entry 384 deliberately computed the endpoint-relative target after forgetting the source and its occurrence shifts. In that ungraded shadow,

$$H_0(K_\partial)\simeq(x_0,x_1)$$

and the first $x_3$-Rees symbol does not see $x_3^2(x_0,x_1)$.

That observation does not produce a deformation of the already fixed generic/lower component of $\mathfrak R_{03}$. The established incidence maps are multigraded of degree zero. In the multidegree forced by Entry 378, the chain equation

$$x_3a+\frac{X_{D03}}{u_{D03}}k=0$$

has the homogeneous solutions

$$(k,a)= c\left(x_3,-\frac{X_{D03}}{u_{D03}}\right).$$

Indeed, after multiplying by $u_{D03}$, the equation is

$$x_3(u_{D03}a)+X_{D03}k=0.$$

The occurrence variables $x_3$ and $X_{D03}$ are coprime. Thus $x_3\mid k$ and $X_{D03}\mid u_{D03}a$, giving precisely the common factor $c$ displayed above.

Fixing the primitive realization degree forces $c$ to have multidegree zero. Over the polynomial occurrence base, $c$ is therefore an integer. The positive first Cartier symbol fixes $c=+1$. Consequently

$$\boxed{ k=x_3,\qquad a=-X_{D03}/u_{D03} }$$

is unique in its fixed multidegree.

In particular,

$$\deg(x_3^2x_0)=2\epsilon_3+\epsilon_0, \qquad \deg(x_3^2x_1)=2\epsilon_3+\epsilon_1$$

differs from $\deg(x_3)=\epsilon_3$. Neither higher-Rees witness can be added to the $q_J$-component of a degree-zero realization.

Compatibility with the endpoint kernel

The weighted road shifts also remain homogeneous. For the simultaneous endpoint kernel

$$R\langle e_2\rangle \xrightarrow{(-x_0,x_1)} R\langle e_3,e_4\rangle,$$

the inverse occurrence weights are

$$\deg(e_2)=-\epsilon_4,\quad \deg(e_3)=-\epsilon_0-\epsilon_4,\quad \deg(e_4)=-\epsilon_1-\epsilon_4.$$

Hence

$$\epsilon_0+\deg(e_3) =\epsilon_1+\deg(e_4) =\deg(e_2).$$

This verifies that the ideal found in Entry 384 is a genuinely graded target module. It does not yet compute the degree-zero Hom from the source thimble into that module.

What is now closed

There is no longer a higher-Rees coefficient ambiguity in the fixed generic/lower map. The following data are rigid:

1. $q_J\mapsto x_3q_{03}^Q$;
2. the lower coefficient $-X_{D03}/u_{D03}$;
3. the first Cartier symbol $+1$; and
4. the two endpoint purity restrictions of Entry 381.

The low-cognition delegated audit independently identified the exact degree equation

$$\deg(s)=\deg(\text{coefficient})+\deg(\text{target generator})$$

and correctly refused to promote the conclusion without source shifts. The present result supplies the already established source--target shift for the generic/lower pair, so uniqueness is unconditional for that pair only.

Remaining boundary

This does not prove uniqueness or existence of the endpoint connector. Entry 384's ideal may still contribute through a different source generator. To decide that, one must retain the individual multidegrees of

$$H_{\rm Morse}p-\widetilde\xi h_3,\qquad q_Jp,\qquad d\widetilde\xi\,h_3$$

and calculate the degree-zero complex

$$\underline{\operatorname{Hom}}_{\rm gr} (\mathcal K_{03}^{\rm face},K_\partial).$$

That is now the smallest unresolved computation. If its $H^0$ vanishes, the realization is coefficientwise unique and only a connector homotopy remains. If it does not, its explicit homogeneous generators replace the ungraded ideal as the true deformation space.

Evidence

research/voevodsky/check_d03_rees_multigraded_uniqueness_gate.py verifies the road shifts, the homogeneous chain equation, the common-factor classification in a bounded exponent census, the fixed-degree unit, and the degree mismatch of the two higher-Rees witnesses.

Outcome contract

{
  "claim": "The fixed-degree D03 generic/lower realization has the unique coefficient pair (x3,-XD03/uD03), with positive Cartier symbol +1. The target-only x3^2(x0,x1) sector cannot perturb that component because it has the wrong occurrence multidegree.",
  "status": "proved_scoped_multigraded_uniqueness",
  "closed": [
    "generic coefficient",
    "lower Cech coefficient",
    "first Cartier scalar"
  ],
  "not_closed": [
    "degree-zero source-to-endpoint Hom",
    "endpoint connector homotopy",
    "reflection parity",
    "full mixed-variance realization"
  ],
  "next_experiment": "Extract the source shifts of the three-face thimble cell and compute the degree-zero graded Hom into the endpoint-relative road complex."
}

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