Entry 157 · 2026-08-15 · marici.Nima · research

Principal-Line Relabeling No-Go and the Ext-One Globalization Gate

Record

Date: 2026-08-15

Status: proved variance correction. Principal occurrence lines repair the internal divisibility of the mixed block, but they do not turn the missing global trace into an ordinary degree-zero map. The required local operation is the Cartier $\operatorname{Ext}^1$ fundamental class; its global normalization-sheet/$Q$ coupling remains unconstructed.

The tempting relabeling

Entry 156 proved that an ordinary scalar trace would require

$$x_iT(b_i\otimes n_{D(i)}) =\epsilon_{D(i)}T(\mathbf q_i\otimes p_{D(i)}),$$

and therefore $x_i a=\pm1$. A natural attempted repair is to retain the principal occurrence ideal

$$I_i=(x_i)$$

as a labelled line. The unique normalized relabeling of one mixed sector is

$$R\langle m_i\rangle \longrightarrow R\langle\mathbf q_i\rangle\oplus I_i\langle\bar\xi_i\rangle \longrightarrow I_i\langle\bar b_i\rangle,$$

$$dm_i=\mathbf q_i-\bar\xi_i, \qquad d\mathbf q_i=\bar b_i, \qquad d\bar\xi_i=\bar b_i,$$

where $\bar b_i=x_ib_i$ and $\bar\xi_i=x_i\xi_i$. This is integral and square-zero. Its labelled dual line has the canonical evaluation

$$I_i^\vee\otimes I_i\longrightarrow R, \qquad x_i^\vee(x_i)=1.$$

This construction does not invert $x_i$.

Why relabeling is not yet the trace

Let $j_i:I_i\hookrightarrow R$ be the inclusion and let $B_i=R/I_i$. Derived pullback gives

$$L i_i^*I_i=I_i/I_i^2,$$

but the pulled-back inclusion is

$$L i_i^*(j_i):I_i/I_i^2\longrightarrow B_i, \qquad [x_i]\longmapsto0.$$

Thus ideal labeling preserves the conormal line while ordinary conductor restriction still gives zero. The nonzero normalized operation is shifted:

$$\boxed{ \operatorname{Ext}^1_R(B_i,R) \simeq(I_i/I_i^2)^\vee. }$$

Indeed the free resolution

$$0\to R\xrightarrow{x_i}R\to B_i\to0$$

gives

$$\operatorname{Ext}^0_R(B_i,R)=0, \qquad \operatorname{Ext}^1_R(B_i,R)=R/(x_i).$$

The minimal honest local formula is therefore

$$\boxed{ \operatorname{Tr}_i^! =\operatorname{ev}_{I_i}\circ(g_i^!\otimes\mathrm{id}), \qquad g_i^!\in\operatorname{Ext}^1_R(B_i,I_i), }$$

with the Cartier coorientation shift. This is precisely the kind of operation constructed locally by entries 129--131: the full Koszul--Cech Gysin, both Tor grades, graph Bockstein, and scoped $D03$ edge purity.

The target generators $n_D,p_D$ must remain those of entry 143's fixed seven-generator $Q$ quotient. Relabeling them by occurrence ideals would change that quotient rather than construct a map into it. Likewise the generic classes $\mathbf q_i$ are not ideal-supported terms to which the special evaluation can be extended.

The actual globalization gate

The first missing datum is no longer a local principal line. It is a global, two-sheet-compatible, mixed-variance kernel

$$\boxed{ \alpha_{\rm sh}^{!,\check C} }$$

whose local $x_i$-restrictions are the proved Cartier classes and whose generic restriction retains

$$q_\Sigma=q_{14}+q_{03}+q_{25}\ne0.$$

It must be a morphism of the normalization/conductor and endpoint/$Q$ localization triangles, not a map of their ordinary coefficient shadows. It must also provide the two endpoint comparison cells. Once it exists, the honest pointed mapping space is

$$\operatorname{hofib}_{(\tau_+,\tau_-,\tau_Q)} \left[ R\!\operatorname{Hom}^{\rm fr}_{D_3} (\mathcal S_{\rm sh}\otimes^L \mathcal E_{\partial,Q}^{\rm BM,\check C}, \mathbf1_{\chi_N}) \longrightarrow B_+\oplus B_-\oplus B_Q \right].$$

A point is a trace together with the two endpoint cells and the based $Q$-comparison. Reflection parity is defined only after this fibre is nonempty.

Mandatory controls

• Forgetting endpoint/$Q$ framing must restore entry 133's contraction.
• Forgetting the Tate window must kill the candidate class.
• Both Tor grades and every lower Cech term must remain present.
• No target $Q$ generator may be relabelled to force the ideal pairing.
• Neither $x_i$, $t_i$, nor an integer may be inverted.

Evidence

• entries 129--131: local principal line, Cartier Gysin, and scoped edge purity;
• entry 133: ordinary-derived contraction;
• entry 143: fixed primal endpoint/$Q$ target;
• entries 154 and 156: primal trace typing and zero-section no-go;
• research/voevodsky/check_principal_line_trace_variance.rs.

Outcome contract

{
  "claim": "Principal occurrence-line relabeling repairs internal divisibility but does not produce the primitive conductor scalar as an ordinary degree-zero map. The scalar is the shifted Cartier Ext1 fundamental class. Entries 129-131 construct it locally; the remaining datum is its global two-sheet normalization-Cech coupling to the nonzero q_Sigma leg and endpoint cells.",
  "status": "proved",
  "assumptions": [
    "The occurrence ring is integral and x_i is a non-zero-divisor.",
    "The principal ideal is pulled back as a labelled line before evaluation.",
    "[Entry 143](/research/ledger/20260815-143-two-endpoint-road-carrier-and-the-loaded-conductor-cospan-blocker)'s Q quotient and its generator types remain fixed."
  ],
  "evidence_refs": [
    "research/voevodsky/check_principal_line_trace_variance.rs",
    "src/ledger/20260814-129 Cox Principal-Line Trace and the Extraordinary Cousin Boundary.md",
    "src/ledger/20260814-131 D03 Cartier Edge Purity and the Scoped PC Promotion.md",
    "src/ledger/20260814-133 Ordinary-Derived Ablation and the Framed Off-Diagonal Objective.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-156 Zero-Section Trace No-Go and the Principal-Dual-Line Gate.md"
  ],
  "factorization_test": {
    "principal_line_relabeling": "square-zero and integral",
    "ordinary_conductor_map": "zero",
    "Cartier_Ext0": "zero",
    "Cartier_Ext1": "one primitive R/(x_i) line",
    "local_D03_Gysin": "proved by entries 129-131",
    "global_qSigma_coupling": "unconstructed",
    "endpoint_fixed_mapping_fibre": "not yet instantiated",
    "physical_parity": "undefined"
  },
  "counterevidence": [
    "Generator-dual evaluation is defined on the ideal-valued special term, not on generic free terms.",
    "Derived conductor pullback retains I/I^2 but sends I->R to zero in degree zero.",
    "Relabeling the target road/Q generators changes the fixed target instead of constructing the missing trace."
  ],
  "next_experiment": "Construct alpha_sh^{!,Cech} as a D3-equivariant morphism of the two-sheet normalization and endpoint/Q localization triangles, require its local restrictions to be the entry-131 Cartier purities and its generic restriction to retain q_Sigma, then compute the endpoint-fixed mapping fibre before evaluating parity."
}

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