Incidence Projection No-Go and Pre-Quotient Correspondence Gate
Result
Entry 352 makes the target of (q^!) type-correct, but the tempting incidence-graph projection from the two-route (D03) diamond does not define the required central-flip correspondence.
The source contains the pre-quotient generic and lower generators
[ dq_J=x_3r_J, ]
while the ringed target contains the radial pair
[ dn_D=(X_D/u_D)p_D. ]
For an incidence-only degree-zero pairing, put
[ k=langle q_J,p_D angle,\qquad a=langle r_J,n_D angle. ]
The ringed chain-map condition is
[ x_3a mathbin{pm} (X_D/u_D)k=0. ]
Reducing modulo the conductor (x_3=0) proves that (k=pm1) is impossible. More generally, (x_3mid k), and the least monomial solution is
[ k=x_3,\qquad a=mp X_D/u_D. ]
Thus the incidence relation supplies only an orientation shadow. It cannot be promoted to the primitive generic projection needed by the central-flip span. Inverting (x_3) is not a repair: it removes the conductor support on which the extraordinary map must be nonzero.
Consequence for a category-of-elements span
Let (Z_{ m inc}) be the category of elements of the visible relation between the source diamond and the target state poset. Its two forgetful order maps exist as maps of finite spaces. Equipping a point with the target localization ring also makes the target projection ringed.
This still does not construct the required span. A ringed source projection must intertwine the source differential (x_3) with the target radial coefficient (X_D/u_D). The equation above shows that the unit incidence map fails that condition. Pulling back the target module along the forgetful relation merely repackages the target coefficient diagram; it does not supply the missing mixed-variance source map.
Exact replacement for the projection block
A viable correspondence must be constructed before taking endpoint or generic quotients. It must retain, in one ringed object:
- the Morse top (H), the generic term (q_J), and lower term (r_J);
- the principal ideal line ((x_3)) without inverting (x_3);
- both occurrence and repeated-normal Tor grades;
- every target Cech lower and overlap term;
- the shifted Cartier/Gysin arrow on the conductor; and
- a Beck--Chevalley homotopy whose generic and special restrictions are induced by the same correspondence.
Accordingly the former single obligation d_central_flip_projections
splits into:
d_central_flip_prequotient_correspondence: construct the full ideal-valued ringed/log object (Z);d_central_flip_projection_beck_chevalley: prove compatibility of its generic localization and special extraordinary restrictions; andd_central_flip_projections_assembled: only then record the actual morphisms (p,q) with the hypotheses needed for (q^!) and pushforward.
The labelled double-Rees relative cap of entry 176 remains necessary local input, but its universal closure has extra special-fibre components and does not itself choose this pre-quotient correspondence.
Scope
This is a sharp no-go for the naive incidence projection, not a global nonexistence theorem. It leaves open an ideal-valued DNC/nearby-cycle correspondence with all lower terms and its shifted extraordinary Beck--Chevalley cell.
Evidence is the exact coefficient theorem and checker of entry 177,
research/voevodsky/check_d03_generic_incidence_pairing_obstruction.rs,
together with entries 120, 160, 174, 176, 348, and 352. Delegated runs
run-3557dcd4cb374e1186c3ae43038c06e5,
run-365bae6be8c84b38817e53dad63c35a6, and
run-5b6a82a7b0e14bc08f878a4ac28ca783 all failed without results and are
not used as evidence.