Zero-Section Trace No-Go and the Principal-Dual-Line Gate

Record

Date: 2026-08-15

Status: scoped falsification of ordinary normalization-zero-section assembly. No no-go is claimed for a marked extraordinary Gysin/nearby-cycle correspondence, and physical parity remains undefined.

Entry 154 retyped the missing kernel as the primal trace

[ \operatorname{Tr}^{\rm biv}{\rm sh,\partial,Q}: \mathcal S{\rm sh}^{\rm norm,reg}\otimes^L \mathcal E_{\partial,Q}^{\rm BM,\check C} \longrightarrow\mathbf1_{\chi_N}. ]

There is a smallest exact test of whether the known normalization zero-sections can define this trace. It fails before any residue, Alexander–Tate comparison, or reflection-parity computation.

Maximal formal branch object

For (I_+={1,3,5}), (I_-={0,2,4}), and

[ A_\sigma=B_\sigma[t_i,(1+t_ix_i)^{-1}:i\in I_\sigma], \qquad u_i=t_ix_i, ]

the established mixed block is

[ M_{\sigma,2}=A_\sigma\langle m_i\rangle, \quad M_{\sigma,1}=A_\sigma\langle\mathbf q_i,\xi_i\rangle, \quad M_{\sigma,0}=A_\sigma\langle b_i\rangle ]

with

[ dm_i=\mathbf q_i-x_i\xi_i, \qquad d\mathbf q_i=x_ib_i, \qquad d\xi_i=b_i. ]

Tensoring with the independent reciprocal-regular multi-Rees packets gives a bounded square-zero branch totalization and retains both Tor grades on the conductor. Transported labels also close a formal polarity action. This does not construct the normalization-sheet source: only the coefficient row

[ 0\to B\to B_+\oplus B_-\to C\to0 ]

has canonical normalization provenance.

Primitive trace equation

For each long target facet (D), the constructed endpoint/(Q) quotient contains (n_D,p_D) with

[ d_E n_D=\epsilon_Dp_D, \qquad \epsilon_D\in{+1,-1}. ]

If an ordinary zero-section trace (T) were a chain map, its value on (\mathbf q_i\otimes n_{D(i)}) would obey

[ \boxed{ x_iT(b_i\otimes n_{D(i)}) =\epsilon_{D(i)}T(\mathbf q_i\otimes p_{D(i)}). } ]

Primitive (Q)-framing makes the right side a signed unit. The left side belongs to the proper ideal ((x_i)), so the equation has no solution over the unlocalized occurrence ring. On the conductor it becomes

[ 0=\pm1. ]

The contradiction is sectorwise, hence neither the three-road sum nor (D_3) covariance can cancel it. Globally inverting (x_i) would erase the support being specialized and is inadmissible.

Why the existing gallery quotients do not repair it

The absolute block remains valid:

[ dH_\Sigma=q_\Sigma- \sum_{i=1,3,5}x_i\widetilde\xi_i, \qquad d^2=0. ]

But each proved local Cartier gallery is already endpoint-and-generic relative and has killed its (q_i). A homotopy colimit using only those objects cannot reconstruct (q_\Sigma). The two natural alternatives also fail:

• quotienting by the short boundary makes (q_\Sigma) bound only by deleting all three special galleries;
• adjoining (dc=q_\Sigma) fails absolutely because (d^2c=x_1b_1+x_3b_3+x_5b_5\ne0).

Thus the required source must be constructed before the endpoint and generic relative quotients.

Minimal admissible repair

The equation identifies the missing coefficient type. An extraordinary correspondence may carry the principal dual line

[ (x_i)^\vee\otimes(x_i)\longrightarrow\mathbb Z, \qquad (x_i)^\vee(x_i)=1, ]

without making (x_i) a unit. This is not ordinary restriction and must be earned geometrically.

The minimal new object is one (D_3)-equivariant, two-sheet-compatible, ringed normalization/nearby-cycle correspondence constructed before the relative quotients. It must carry simultaneously:

1. the full ((m_i,\mathbf q_i,\xi_i,b_i)) column;
2. the principal occurrence dual line;
3. independent multi-Rees lines and both Tor grades;
4. the nonzero generic (q_\Sigma) leg;
5. reciprocal-regular source and BM–Cech target variance;
6. both endpoint comparison cells and the polarity conjugate.

Only then is the endpoint-fixed mapping fibre defined. Consequently

[ p_{\partial,Q}\in H^1(D_3;\mathbb Z_{\rm or}) ]

remains undefined. The even reflection of the formal coefficient cone is inadmissible because that cone fails the primitive trace equation.

Evidence

• entry 93: normalization–conductor coefficient row and first symbol;
• entry 113: absolute mixed block and subquotient/filler controls;
• entry 143: primal endpoint/(Q) BM–Cech target;
• entry 154: primal trace retyping and mandatory ablations;
• research/voevodsky/check_primal_zero_section_trace_obstruction.rs.

Outcome contract

{
  "claim": "Ordinary normalization zero-section gluing cannot support a primitive primal endpoint/Q trace: its chain equation requires x_i times a coefficient to equal a signed unit and becomes 0=+/-1 on the conductor. This falsifies only zero-section assembly, not an extraordinary principal-dual-line Gysin correspondence.",
  "status": "falsified",
  "assumptions": [
    "The occurrence ring remains unlocalized.",
    "The established mixed and endpoint/Q differentials are retained.",
    "Primitive Q framing has signed-unit normalization.",
    "No desired residue or principal-dual evaluation is inserted."
  ],
  "evidence_refs": [
    "research/voevodsky/check_primal_zero_section_trace_obstruction.rs",
    "src/ledger/20260814-93 Alternating Fusion Normalization-Conductor Square.md",
    "src/ledger/20260814-113 Marked-Exit Tate Detector and the Mixed Boundary-Crossing Block.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-154 Primal Bivariant Trace Retyping and the Double Object Gate.md"
  ],
  "factorization_test": {
    "branch_total_d_squared": "passed",
    "primitive_zero_section_trace": "falsified",
    "conductor_specialization": "0=+/-1",
    "three_road_cancellation": "impossible sectorwise",
    "generic_Q_leg": "retained in the mixed block",
    "extraordinary_principal_dual_repair": "unconstructed and not falsified",
    "endpoint_Q_parity": "undefined"
  },
  "counterevidence": [
    "Global occurrence localization would erase the tested support.",
    "Existing relative quotients lose the primitive Q leg or all special galleries.",
    "The formal polarity-even cone fails the primitive trace equation."
  ],
  "next_experiment": "Construct a marked ringed occurrence-Gysin/nearby-cycle correspondence carrying the principal dual occurrence line, both Tor grades, q_Sigma, and both endpoint cells; then form the endpoint-fixed mapping fibre and compute reflection parity."
}