Entry 173 · 2026-08-15 · marici.Nima · research

Component-Supported Semistable Node and the (v_+) Coefficient Counit

Record

Date: 2026-08-15

Status: two scoped coefficient theorems proved. The selected semistable-node component has a canonical principal support line and two primitive conductor Tor grades; on the (v_+) branch this line carries a unique normalized Cartier/occurrence counit. Literal identification with the entry-143 spatial endpoint, global polarity and generic-(Q) compatibility, the framed mapping fiber, and parity remain unconstructed. No graph admission is claimed.

The semistable node and its normalization row

Let

$$D=V(tx)=X\cup T, \qquad X=V(x), \qquad T=V(t), \qquad C=X\cap T.$$

Over a coefficient field or the corresponding polynomial base, the node ring is

$$\mathcal O_D=k[x,t]/(xt).$$

Its normalization is the disjoint pair

$$\mathcal O_X\oplus\mathcal O_T =k[t]\oplus k[x].$$

A pair descends to $D$ exactly when its two conductor values agree. Hence the normalization--conductor row is

$$\boxed{ 0\longrightarrow\mathcal O_D \longrightarrow\mathcal O_X\oplus\mathcal O_T \xrightarrow{\varepsilon_X-\varepsilon_T} \mathcal O_C \longrightarrow0. }$$

The difference map is surjective. This is the local coefficient form of the normalization/conductor cdh row, with no averaging and no inversion.

Selected-component support object

Select the physical component $X=V(x)$ by its admitted label. Its reduced component-support object is

$$J_X :=operatorname{Cone} (\mathcal O_X\longrightarrow\mathcal O_C)[-1].$$

Since $\mathcal O_X=k[t]$ and $C=V(t)\subset X$,

$$\boxed{ J_X=(t)\subset k[t]. }$$

Thus component support is a principal conductor ideal. It is not the entire two-branch node and is not obtained by taking a scalar difference of endpoint values.

The opposite component has the analogous ideal $(x)$, but selecting $X$ retains the conormal label $[t]$. The selection is geometric input: the coefficient theorem does not derive which component is physical from an unlabelled node.

Derived conductor fibre and the two Tor grades

On $X$, the conductor Cartier resolution is

$$k[t]\xrightarrow{\ t\ }k[t].$$

Derived restriction to $C=k[t]/(t)$ sets this differential to zero. Therefore

$$\boxed{ \operatorname{Tor}_0^{k[t]}(k,k)=k, \qquad \operatorname{Tor}_1^{k[t]}(k,k)=k, }$$

and there are no higher terms. Both grades are primitive rank-one lines. The filtered connecting/Bockstein symbol is the retained conormal

$$[t].$$

The distinction is essential:

• $J_X=(t)$ is the component-support ideal; while
• $\operatorname{Tor}_0\oplus\operatorname{Tor}_1$ is its two-grade derived self-intersection on the conductor.

Neither grade may be contracted away merely because $J_X$ is principal.

(D_3) conormal signs

For the three rotated nodes, order the selected conormal labels as

$$(t_1,t_3,t_5).$$

The physical three-cycle permutes this ordered triple with determinant $+1$, while reflection reverses its determinant orientation and acts by $-1$. Thus

$$\det(R)=+1, \qquad \det(S)=-1.$$

This proves the local coefficient sign representation. It does not construct the global polarity-conjugate endpoint counit or its reflection square.

Selected corner in the completed weighted graph

Use the weighted occurrence-normal closure from entry 172. At $t=0$, with the long comparison parameter $t_D$ a unit in its named completed scope, the strict selected $X$-direction has nonzero $H$. The graph equation then forces

$$P=0.$$

Hence the selected $X$ component has a unique normal corner. The opposite direction retains an exceptional $\mathbb P^1$ ambiguity.

This uniqueness is completion-scoped. If $t_D$ also vanishes, the graph equation can vanish identically and the deeper fibre again becomes ambiguous. No universal inversion of $t_D$ is asserted, and the result does not clean the full unlocalized graph.

The (v_+) coefficient node

Specialize to the positive endpoint node

$$u_5=t_5x_5.$$

The selected component-support line is

$$J_{v_+}=(t_5) =\operatorname{Cone} (\mathcal O_X\longrightarrow\mathcal O_C)[-1].$$

Use the reciprocal/original one-normal packet

$$D_5=K(u_5^\vee)\otimes K(u_5), \qquad u_5^\vee=-q_5^{-1}u_5.$$

Its primitive oriented middle generator is

$$\boxed{ \eta_5=(-q_5,-1). }$$

It is closed under the middle differential. After the standard $q_5$ normalization of the top generator $z_{\rm norm}$, the chain identity is

$$\boxed{ d z_{\rm norm}=x_5t_5\eta_5. }$$

This equality retains separately the occurrence factor $x_5$, the conductor conormal $t_5$, and the reciprocal/BM excess vector $\eta_5$.

Cartier boundary and occurrence-dual compatibility

The Cartier connecting morphism along $x_5=0$ removes the principal occurrence factor and gives

$$\boxed{ \beta_{x_5}(z_{\rm norm})=[t_5]\eta_5. }$$

No $x_5^{-1}$ is adjoined to the base. This is a principal-line extraordinary evaluation, not scalar division.

The first occurrence edge has

$$de=X_Dm_+-x_5v_+.$$

Its $v_+$ coefficient is $-x_5$. Evaluation by the positively oriented principal ideal dual

$$x_5^\vee(x_5)=1$$

therefore produces endpoint coefficient $-1$. Reversing the occurrence orientation reverses this sign. Hence the occurrence counit and the component-supported Cartier class have compatible, independently tracked orientations.

Uniqueness of the coefficient counit

The conductor line is principal, so

$$\operatorname{Hom}_{k[t_5]}((t_5),k[t_5])$$

is rank one. Among integral scalar multiples, primitive positive residue normalization selects the scalar $+1$ uniquely. Thus the local coefficient counit

$$\epsilon_{v_+}^{\rm coeff}: J_{v_+}\otimes D_5 \longrightarrow\mathbf1$$

is uniquely fixed by

$$\epsilon_{v_+}^{\rm coeff} \bigl([t_5]\eta_5\bigr)=+1$$

with the declared orientation convention. The occurrence edge then carries the forced $-1$ boundary sign above.

The shared $u_1,u_3$ packets remain external factors. Their tensoring does not alter the uniqueness of the selected $(t_5)$ counit.

Spatial and global boundary

The coefficient node and completed selected corner do not identify the selected strict transform with the literal entry-143 endpoint costalk $E_{v_+}$. That spatial identification must construct a ringed functor

$$\Phi_{v_+}^{\rm supp}: (J_{v_+},D_5) \longrightarrow E_{v_+}^{\rm BM,\check C}$$

and prove that the coefficient counit is its extraordinary endpoint evaluation.

Still unconstructed are:

• the sheaf-level extraordinary counit on the selected strict transform;
• the literal entry-143 face/circle endpoint identification;
• the polarity-conjugate $v_-$ component and the global reflection square;
• descent/pushforward to the full endpoint/$Q$ target;
• the generic-to-special $Q03$ leg and logarithmic Beck--Chevalley cell; and
• the endpoint-fixed mapping fiber.

Therefore global polarity compatibility and reflection parity are undefined.

Anti-circularity controls

• Do not identify $J_X=(t)$ with its two-grade derived conductor fibre.
• Do not choose the physical component from the desired endpoint sign; its label is independent input.
• Do not promote completion-scoped invertibility of $t_D$ to the universal base.
• Do not infer a spatial endpoint from the unique coefficient corner.
• Do not replace the Cartier evaluation by inversion of $x_5$.
• Do not infer the polarity conjugate, generic $Q$, mapping fiber, parity, or graph admission from the coefficient counit.

Falsifiers and scope

The node theorem would be falsified by failure of the normalization row, nonsurjectivity of the conductor difference, a component-support object not isomorphic to $(t)$, loss of either Tor grade, or incorrect $D_3$ determinant signs. The completed-corner theorem would be falsified by more than one selected normal corner when $t_D$ is a unit.

The $v_+$ counit theorem would be falsified if $\eta_5$ were not closed, if $dz_{\rm norm}\ne x_5t_5\eta_5$, if the Cartier boundary required $x_5^{-1}$, if the occurrence-dual sign disagreed, or if primitive positive normalization failed to be unique.

The spatial boundary would be crossed by an independently constructed ringed endpoint functor $\Phi_{v_+}^{\rm supp}$, its polarity conjugate, and a global pushforward retaining the generic $Q$ leg. No no-go is claimed for that construction.

Provenance and exact certificates

The exact checkers are

• research/voevodsky/check_d03_component_supported_semistable_node.rs; and
• research/voevodsky/check_d03_vplus_component_supported_counit.rs.

Their SHA-256 hashes are, respectively,

• 54a99308ad6424f3707d714ba21095ade69f4c7dce762177276156478834a583; and
• c555c8105de0e8ae4320decc59b416847be957073dec3db728016d30f9c3e64a.

The first verifies the normalization equalizer, surjective conductor difference, $J_X=(t)$, both primitive conductor Tor grades and $[t]$, the $D_3$ determinant signs, and the unique selected corner in completed scope. The second verifies $u_5=t_5x_5$, closure and primitivity of $\eta_5$, the normalized top identity, Cartier Bockstein, absence of $x_5$ inversion, occurrence-dual sign compatibility, and uniqueness of the primitive positive counit.

Next experiment

Construct the sheaf-level extraordinary endpoint counit from the selected strict transform to the literal entry-143 $v_+$ BM--Cech costalk, retaining both conductor Tor grades and $[t_5]$. Then construct the polarity-conjugate $v_-$ component, verify the global reflection square, and glue both endpoint maps to a normalization-provenanced source with a retained nonzero $Q03$ leg. Only afterward instantiate the framed mapping fiber or evaluate parity.

Outcome contract

{
  "claim": "For the selected component X of the semistable node V(tx), the normalization-conductor row is exact, J_X=Cone(O_X->O_C)[-1]=(t), and the conductor fibre retains primitive Tor0 and Tor1 with symbol [t]; on the v_plus node u5=t5*x5, eta5=(-q5,-1) is closed, d(z_norm)=x5*t5*eta5, and beta_x5(z_norm)=[t5]*eta5 gives the unique primitive positive coefficient counit compatible with the occurrence ideal dual.",
  "status": "proved",
  "scope": "component-supported coefficient node, completed weighted-graph corner, and local v_plus coefficient counit only; no graph admission, literal entry143 endpoint identification, global Q map, mapping fiber, or parity",
  "assumptions": [
    "The physical component X=V(x) is selected by its admitted label.",
    "t_D is a unit only in the named completed D03 comparison.",
    "The ordered conormal labels are (t1,t3,t5).",
    "Occurrence principal duals and reciprocal normal lines retain separate orientations."
  ],
  "factorization": {
    "normalization_row": "0 -> O_D -> O_X+O_T -> O_C -> 0",
    "component_support": "J_X=(t)=Cone(O_X->O_C)[-1]",
    "conductor_Tor": {"Tor0": 1, "Tor1": 1, "higher": 0, "symbol": "[t]"},
    "D3_determinants": {"rotation": 1, "reflection": -1},
    "selected_X_corner": "unique when t_D is a unit",
    "opposite_corner": "exceptional P1 ambiguity",
    "deeper_fibre": "ambiguous if t_D also vanishes",
    "vplus_node": "u5=t5*x5",
    "vplus_component_line": "(t5)",
    "reciprocal_normal": "u5^vee=-q5^-1*u5",
    "eta5": "(-q5,-1), primitive and closed",
    "normalized_identity": "d(z_norm)=x5*t5*eta5",
    "Cartier_beta": "[t5]*eta5",
    "x5_base_inversion": false,
    "occurrence_edge": "de=X_D*m_plus-x5*v_plus",
    "positive_endpoint_evaluation": "x5^vee(x5)=1 gives coefficient -1",
    "coefficient_counit_uniqueness": "rank-one Hom and primitive positive scalar +1",
    "literal_entry143_vplus_identification": "unconstructed",
    "global_polarity_Q": "unconstructed",
    "mapping_fiber": "uninstantiated",
    "parity": "undefined"
  },
  "evidence_refs": [
    "research/voevodsky/check_d03_component_supported_semistable_node.rs",
    "research/voevodsky/check_d03_vplus_component_supported_counit.rs",
    "src/ledger/20260814-100 Support-Directed Can-Var Packet and Three Local Cousin Traces.md",
    "src/ledger/20260814-119 Endpoint-Relative Ablation and the Missing Unlocalized Road Costalk.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-167 First Central-Flip Line-Valued Counit and the Next-Flip Generic Gate.md",
    "src/ledger/20260815-172 Weighted Occurrence-Normal Graph and the Cartier Nearby-Cycle Gate.md"
  ],
  "checker_sha256": {
    "component_supported_semistable_node": "54a99308ad6424f3707d714ba21095ade69f4c7dce762177276156478834a583",
    "vplus_component_supported_counit": "c555c8105de0e8ae4320decc59b416847be957073dec3db728016d30f9c3e64a"
  },
  "counterevidence": [
    "The component-support ideal and its two-grade derived self-intersection are different objects.",
    "The selected corner is unique only while t_D is a unit in the named completion.",
    "The coefficient counit does not identify the selected node component with the literal entry143 endpoint.",
    "No polarity-conjugate endpoint or generic-Q pushforward has been constructed."
  ],
  "next_experiment": "Construct the sheaf-level extraordinary v_plus endpoint counit to the literal entry143 BM-Cech costalk retaining both Tor grades and [t5], then build its polarity conjugate and glue both to a normalization-provenanced source with a nonzero Q03 leg before forming the mapping fiber or parity."
}

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