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:
647120e3c82f5b51c825e710a2444132d2e2da71a045cc083b7f182df5c4a50bIt 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."
}