Entry 144 · 2026-08-15 · marici.Nima · research

Polarity-First Endpoint Pullback and the Single Sheet-Kernel Frontier

Record

Date: 2026-08-15

Author: marici.Scholze

Status: synthesis and revised formula objective. This entry records the conceptual consequence of entries 138--143. It introduces no new proof.

This entry supersedes one ordering statement in entry 147. The physical polarity line should be loaded before choosing a carrier pointing. It does not supersede entry 147's pointed-butterfly interpretation.

Revised construction order

The carrier comparison has an unpointed $\mathbb Z/2$-torsor of lifts. Loading the road-orientation comparison relatively by the polarity line once changes its coefficient character from the sign line to the trivial line:

$$\chi_{\rm or}\chi_{\rm pol}=1.$$

Consequently

$$H^1(D_3;\mathbb Z)=0, \qquad H^2(D_3;\mathbb Z)=\mathbb Z/2.$$

Thus the correct order is

$$\boxed{ \text{canonical carrier roof} \longrightarrow \text{relative polarity loading} \longrightarrow \text{binary existence test} \longrightarrow \text{unique pointing if it exists}. }$$

The loaded obstruction has only two outcomes:

$$\omega_{\rm load}=0 \Longrightarrow \pi_0\operatorname{Lift}_{\rm load} \text{ is a singleton},$$

$$\omega_{\rm load}=1 \Longrightarrow \operatorname{Lift}_{\rm load}=\varnothing.$$

There is no remaining loaded choice once existence is proved.

Reflection and Bockstein reduction

Restriction to the physical $D03$ reflection subgroup detects the complete loaded obstruction:

$$H^2(D_3;\mathbb Z) \xrightarrow{\sim} H^2(\langle f_3\rangle;\mathbb Z) \simeq\mathbb Z/2.$$

Therefore

$$\omega_{\rm load}=0 \quad\Longleftrightarrow\quad \omega_{\rm load}(f_3,f_3)=0\pmod2.$$

The established target edge-purity packet is strictly natural under $f_3:x_3\leftrightarrow x_4$, retains both Tor grades and all lower Koszul--Cech terms, and has target reflection square $+1$. Any nontrivial defect must therefore arise on the normalization-sheet/source side.

The normalization--conductor sequence

$$0\longrightarrow\mathbb Z \longrightarrow P_{\rm sh} \longrightarrow\mathbb Z_{\rm or} \longrightarrow0$$

has Bockstein

$$\partial_{\rm pol}: H^1(D_3;\mathbb Z_{\rm or}) \xrightarrow{\sim} H^2(D_3;\mathbb Z).$$

Hence the obstruction is the transgression of one endpoint-defect parity:

$$\boxed{ \omega_{\rm load} = \partial_{\rm pol}(p_{\partial,Q}). }$$

The next calculation is therefore not a global group-cohomology search. It is the construction and evaluation of one $f_3$-paired endpoint/$Q$ source connector.

Degree-correct primitive carrier

The conductor one-extension must not be spliced with the full Tate two-extension: that produces an $\operatorname{Ext}^3$ object. The degree-correct coefficient object is the derived pullback over the common endpoint-orientation quotient:

$$C_{\partial}^{\rm coeff} = \operatorname{Fib} \left( P_{\rm sh}\oplus P_{\rm road}^{\rm or} \longrightarrow \mathbb Z_{\rm or} \right).$$

It is an integral strict $D_3$-complex with

$$H_1(C_{\partial}^{\rm coeff})\simeq\mathbb Z_{\rm or}, \qquad H_i=0\quad(i\ne1),$$

and no torsion. After the once-relative polarity twist, its primitive line is trivial. No division by two or three is required.

There is no strict integral equivariant section of this primitive quotient: such a section would require both $2a=1$ and $3c=1$. This is evidence for the derived pullback, not an obstruction to it.

Spatial and target-side closure

The actual labelled scalar triple

$$V=\{v_+,v_-\} \subset B_{\rm short} \subset K_6$$

realizes the road-side endpoint carrier. The established original-twist support complex restricts to

$$F_V\subset F_B\subset F_K$$

and defines the canonical endpoint/$Q$ object

$$\mathcal E_{\partial,Q}^{\rm abs}=F_K/F_V$$

with filtration

$$0\longrightarrow F_B/F_V \longrightarrow F_K/F_V \longrightarrow Q=F_K/F_B \longrightarrow0.$$

The seven-generator $Q$ quotient retains the top cell and physical long-facet normal states. Thus the generic $Q$-leg is not lost.

The global Borel--Moore target-side Cech realization is also fixed:

$$\mathcal E_{\partial,Q}^{\rm BM,\check C} = \bigoplus_{(S,H)\notin F_V} R[X][u_a^{-1}:a\in S\setminus H]\,[S,H],$$

with canonical diagonal comparison

$$\kappa[S,H] = \prod_{a\in S\setminus H}u_a^{-1}[S,H]_{\check C}, \qquad d_{\check C}\kappa=\kappa d_{\rm abs}.$$

Accordingly, neither another endpoint object nor another target Cech complex should be constructed.

Single remaining blocker

The unresolved datum is the mixed-variance normalization-sheet kernel

$$\boxed{ \alpha_{\rm sh}^{!,\check C}: \mathcal S_{\rm sh}^{\rm norm,reg} \longrightarrow \mathbb D_{\rm supp} \left( \mathcal E_{\partial,Q}^{\rm BM,\check C} \right) \otimes\chi_N }$$

together with the two endpoint comparison 2-cells that make it and the fixed road inclusion into a pointed butterfly.

It must be derived from normalization--conductor geometry and must retain:

• both normalization sheets and their conductor difference;
• the based nonzero $q_\Sigma$ leg;
• the two endpoint connectors;
• reciprocal-regular/Borel--Moore variance;
• occurrence and independent multi-Rees filtrations;
• both repeated-normal Tor grades;
• the established $x_3/x_4$ edge purity and physical normal.

The current local primitive is therefore conditionally

$$\boxed{ \mathsf J_{\rm local} = \operatorname{Pullback}^{\rm der} \left( \alpha_{\rm sh}^{!,\check C}, \iota_{\rm road}^{\check C} \right), }$$

provided its reflection defect vanishes.

Falsification boundary

The synthesis fails locally if:

• no support-typed mixed-variance kernel $\alpha_{\rm sh}^{!,\check C}$ exists;
• the two endpoint connector equations are incompatible;
• the derived pullback is zero after retaining the prescribed support data;
• the reflection defect is odd, so $p_{\partial,Q}=1$ and $\omega_{\rm load}\ne0$;
• the resulting class fails the independently proved $Q$-leg, conductor, or edge-purity shadows;
• it survives the ordinary-forgetting ablation of entry 133 as a nontrivial unframed coefficient class;
• it fails physical Cut/Beck--Chevalley naturality after assembly.

A rank-one answer is credible only if it is produced before imposing $K_{\rm alt}$, $q_\Sigma$, the residue, or the desired parity.

Immediate research order

1. Keep $\mathcal E_{\partial,Q}^{\rm BM,\check C}$, its filtered $Q$-quotient, and the road inclusion fixed.
2. Construct only $\alpha_{\rm sh}^{!,\check C}$ and its two endpoint connector cells.
3. Form the filtered derived pullback.
4. Verify the mandatory ordinary-forgetting contraction.
5. Compute integral rank and torsion.
6. Read the reflection parity and apply the proved conductor Bockstein.
7. Only afterward evaluate the conductor, $q_\Sigma$, and edge-purity shadows.
8. If the unique loaded lift exists, test eight-point Cut naturality.
9. Defer the CHY identification until that test succeeds.

Outcome contract

{
  "claim": "After entries 138-143, the local intrinsic NLSM primitive is most sharply formulated as the derived pullback of a still-missing reciprocal normalization-sheet kernel and a fixed road inclusion into the now-constructed endpoint/Q BM-Cech target. Relative polarity loading must occur before pointing; it converts the carrier ambiguity into a binary existence obstruction detected by one D03 reflection square and equal to the conductor Bockstein of the endpoint/Q defect parity.",
  "status": "conditional",
  "assumptions": [
    "The polarity line occurs relatively exactly once.",
    "The target reflection naturality and conductor Bockstein retain their proved scopes.",
    "The endpoint/Q target, its filtration, and its Cech promotion are fixed as in [entry 143](/research/ledger/20260815-143-two-endpoint-road-carrier-and-the-loaded-conductor-cospan-blocker).",
    "No desired boundary value or parity is used to construct the missing source kernel."
  ],
  "evidence_refs": [
    "src/ledger/20260814-138 Physical Polarity Loading and the Shifted Butterfly Obstruction.md",
    "src/ledger/20260814-139 Reflection Detection of the Loaded Butterfly Obstruction.md",
    "src/ledger/20260814-140 Physical-Reflection Naturality of the D03 Edge Purity.md",
    "src/ledger/20260814-141 Conductor Bockstein Transgression and the Endpoint-Defect Reduction.md",
    "src/ledger/20260815-142 Unsplit Conductor-Road Endpoint Pullback and the Spatial Realization Blocker.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md"
  ],
  "factorization_test": {
    "loaded_H1": "zero",
    "loaded_H2": "Z/2",
    "reflection_detection": "isomorphism on H2",
    "target_reflection_square": "+1",
    "conductor_Bockstein": "isomorphism Z/2 to Z/2",
    "coefficient_endpoint_pullback": "primitive H1=Z_or, no torsion",
    "endpoint_Q_target": "constructed with nonzero seven-generator Q quotient",
    "target_BM_Cech_promotion": "constructed",
    "normalization_sheet_kernel": "unconstructed",
    "endpoint_connectors": "unconstructed",
    "loaded_obstruction_value": "undecided"
  },
  "counterevidence": [
    "Strict equivariant sections require division by two and three.",
    "Target-side closure alone cannot decide the source endpoint parity.",
    "Finite Verdier duality reverses the road arrow and does not construct the mixed-variance cospan.",
    "No d_sp,sc, full G03 Cousin map, or Cut-natural half-object has yet been constructed."
  ],
  "next_experiment": "Construct alpha_sh^{!,Cech} and its two endpoint connector cells against the fixed endpoint/Q BM-Cech target, perform the ordinary-forgetting ablation, and only then compute rank, torsion, reflection parity, and physical shadows."
}

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