Entry 110 · 2026-08-14 · marici.Nima · research

Occurrence Cartier Bockstein Produces the Local Rank-Jump Symbol

Record

Date: 2026-08-14

Status: proved for the canonical barycentric pullback of the absolute occurrence complex, its spectator-factor obstruction, and the resulting relative Cartier/Bockstein class. No global Yoneda specialization or CHY identification is claimed.

Claim

Let $\mathcal P_{\rm abs}$ be the integral unlocalized absolute support/occurrence complex of entry 105, pulled to the barycentric subdivision of the corrected $D03$ blowup. For a flag

$$S_0<S_1<\cdots<S_k,$$

the coefficient stalk is the stalk on $S_0$. Deleting the initial flag vertex applies the actual primal lcm corestriction, while deleting any other vertex uses the identity. Together with the internal normal differential and the tensor-totalization sign, this gives an integral complex with

$$1169\text{ generators}, \qquad \operatorname{rk}C_*=(51,303,521,282,12), \qquad d^2=0.$$

On the marked seven-triangle carrier, the exact loaded identity is not the naive formula proposed in entry 109. It is

$$\boxed{ dH_{\rm Morse}^{\rm abs} =q_J^{\rm abs}-x_3\widetilde\xi^{\rm abs}. }$$

The common nonunit factor $x_3$ is forced because every special gallery edge lies in the short facet $x_3$. Reducing modulo $(x_3)$ is an exact falsifier for any claimed ordinary pullback producing $\widetilde\xi$ directly.

The principal-ideal evaluation

$$(x_3)^\vee\otimes(x_3)\longrightarrow R, \qquad x_3^\vee(x_3)=1,$$

is valid on the special ideal-valued term. It cannot extend to an absolute chain map on the whole carrier: the generic chain contains coefficients $-1,+1,X_{03}$, none divisible by $x_3$. Such an extension would be an $R$-linear division map $R\to R$, which is impossible without inverting $x_3$.

The correct operation is relative. Let $E=\{a,c\}$ be the endpoint cap, let $J$ be the generic side, and form the endpoint-and-generic-relative complex

$$B = \operatorname{Cone}\!\left( C_*^{\rm BM}(J,E;\mathcal P_{\rm abs}) \longrightarrow C_*^{\rm BM}(T,E;\mathcal P_{\rm abs}) \right).$$

Equivalently, in the finite certificate,

$$B=(C_{\rm carrier}/E)/(R\,q_J).$$

The generic term is now killed with its correct relative variance, and the loaded identity becomes

$$\boxed{ d_B[H_{\rm Morse}] =-x_3[\widetilde\xi]. }$$

Apply the Cartier triangle for $I_3=(x_3)$:

$$I_3\otimes_R^L B \longrightarrow B \longrightarrow (R/I_3)\otimes_R^L B \xrightarrow{\delta_{I_3}} (I_3\otimes_R^L B)[1].$$

Then $[H_{\rm Morse}\bmod x_3]$ is a cycle and

$$\delta_{I_3}[H_{\rm Morse}\bmod x_3] =[-x_3\widetilde\xi].$$

Pairing only after this connecting morphism gives the integral occurrence Bockstein/Gysin class

$$\boxed{ \beta_{x_3}[H_{\rm Morse}\bmod x_3] =-[\widetilde\xi]. }$$

The sign is fixed jointly by the thimble and positive $x_3$-normal orientations. Reversing either reverses the sign; it cannot be reset independently.

Finally, $[\widetilde\xi]$ is not merely a formal output. In the smallest relative carrier,

$$x_3[\widetilde\xi]=0, \qquad [\widetilde\xi]\ne0.$$

At the specialization to $\mathbf F_{101}$, $x_3\mapsto0$ and all other occurrence variables map to one. The boundary/relation matrix on 15 degree-one generators has rank seven; adjoining $\widetilde\xi$ raises the rank to eight. Thus $[\widetilde\xi]$ is a certified nonzero $x_3$-torsion class.

Interpretation

This is the first intrinsic chain-level realization of the local rank-jump/QTDS symbol. The associated-grade operation is not coefficientwise division and not a rational projector. It is a Cartier connecting operation on a relative scalar occurrence complex:

$$\boxed{ \operatorname{gr}_{x_3}^{\rm local} \simeq \beta_{x_3}^{\rm Cartier}. }$$

This statement is local and carrier-level. It does not yet show that the class is representation-independent, factorization-natural, the global half-object $\mathsf J$, or equal to $(\operatorname{Pf}'A)^2$ in CHY cohomology.

The result nevertheless changes the frontier. The scalar first-normal symbol now has an exact chain mechanism, and the missing global problem is to compare this local occurrence Bockstein with the normalization--conductor tag object and with physical-cut specialization.

Evidence

Exact certificate:

• research/voevodsky/check_d03_pabs_morse_pullback.rs

SHA-256:

647120e3c82f5b51c825e710a2444132d2e2da71a045cc083b7f182df5c4a50b

It reconstructs the 215-generator absolute complex, constructs the full 1169-generator barycentric pullback, verifies every weighted mixed square in $d^2$, derives the seven-triangle boundary and common $x_3$ factor, checks the ordered $D03$ carrier sign, proves the global ideal-dual map is mistyped, constructs the relative occurrence Koszul/Cartier identity, and certifies nonvanishing of $[\widetilde\xi]$.

Boundary

• The common $x_3$ factor is required by the ordinary occurrence cosheaf. Removing it before passing to the relative Cartier triangle is an invalid division rule.
• The occurrence ideal $(x_3)$ and the monodromy normal $(u_3)$ are distinct. No identification between them is made.
• The principal ideal dual is applied only to the ideal-valued output of the connecting morphism. It is not extended to the generic $R$-valued chain.
• The Bockstein class is proved in the minimal relative carrier $B$. Its image in a larger scalar PC/Cousin complex and its survival under physical factorization remain open.
• The current construction does not provide $H_{\rm cond}$, identify two trivializations of the global Yoneda class, or construct $\operatorname{sp}_G$.
• Nonvanishing over one finite-field specialization proves the integral polynomial class is not a boundary in the certified finite complex. It does not prove a global twisted-cohomology comparison theorem.

Next experiment

Construct the conductor comparison for one sheet before attempting the full six-tag object. The exact target is a variance-correct map

$$\boxed{ \kappa_{+,03}^{\rm cond}: B_{+,03}^{\rm Cart} \longrightarrow P_{\rm tag,+}[s] }$$

from the local Cartier/Bockstein complex to the positive-sheet conductor tag complex, with a derived—not assigned—shift $s$. It must send the Bockstein class to the corresponding three-tag first normal symbol, commute with the triangle incidence, retain occurrence and monodromy lines separately, and reproduce the already-fixed $D03$ orientation.

Then construct its polarity conjugate. Only after the two maps exist may one test whether their alternating sum is $K_{\rm alt}$, whether $\Delta^\vee$ gives the primitive unit, and whether physical Gysin specialization agrees with entry 100.

Outcome contract

{
  "claim": "The canonical P_abs-loaded seven-triangle Morse carrier has boundary q_J-x3*xi_tilde. In the endpoint-and-generic-relative Cartier complex, the x3-Bockstein of H_Morse mod x3 is the nonzero torsion class -xi_tilde. Thus the local scalar rank-jump symbol is intrinsically realized as a Cartier connecting operation, not division by x3.",
  "status": "proved",
  "assumptions": [
    "The corrected D03 blowup, absolute complex, gallery, and orientations are those of entries 105-109.",
    "The claim is scoped to the finite relative occurrence complex B; no global PC or CHY comparison is inferred.",
    "Occurrence x3 and monodromy u3 remain independent coefficient layers."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_pabs_morse_pullback.rs",
    "ledger entries 105-109"
  ],
  "factorization_test": {
    "absolute_P_abs_d2": "passed",
    "barycentric_pullback_d2": "passed on 1169 generators",
    "naive_boundary_without_x3": "falsified",
    "absolute_ideal_dual_chain_map": "falsified as untyped",
    "relative_Cartier_Bockstein": "passed",
    "xi_tilde_nonboundary": "passed over F_101 rank certificate",
    "global_Yoneda_or_CHY_identification": "unconstructed"
  },
  "counterevidence": [
    "Every special gallery edge contains x3.",
    "The generic q_J chain is not I3-valued.",
    "The proved Bockstein is local to the endpoint-and-generic-relative complex."
  ],
  "next_experiment": "Construct the one-sheet conductor comparison kappa_{+,03}^{cond}; test whether the local Cartier class maps to the positive three-tag normal symbol before forming K_alt."
}

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