2993 — D03 Supplies a Formal Support-Sensitive Boolean Derived Calculus
Scope
This audit concerns only the fixed D03 occurrence-level Cox/Koszul--Cech and coefficient/Cousin models implemented by
research/voevodsky/check_d03_toric_cox_cousin_trace.rs;research/voevodsky/check_d03_x3_loaded_pc_endpoint_boundary.rs.
It does not identify those models with the actual ringed PC extraordinary costalk, physical Cut sewing, or a G03 source map.
Hard-to-vary claim
The D03 packet is a genuine positive model of
[ ext{labelled Boolean incidence} + ext{support-sensitive cubical maps} + ext{source-defined coefficient transport} ]
at the fixed formal coefficient/Cousin level.
Unlike the five-site row audited in Entry 2992, its decisive maps are not inserted as a predeclared nonzero row.
Construction audit
The first checker derives the oriented road square from principal monomial labels, verifies both weighted chain identities, and shows that the normalized complex resolves the Cartier ideal
[ (M),\qquad M=x_0x_1x_3x_4. ]
On the (x_3) edge it constructs the one- and two-normal Koszul-to-Cech maps explicitly. For endpoint normal (x_i), the degree-one comparison is
[ g_i^0(r)=(r/x_i,0), \qquad g_i^1(t)=t/x_i, ]
and the checker verifies
[ d,g_i^0=g_i^1d. ]
The resulting top class is (1/(x_ix_3)), with both (v_{00}) and (v_{10}) residues fixed by the retained occurrence orientations.
The second checker preserves this construction after loading the frozen D03 coefficient packet. It verifies:
- the endpoint quotient chain maps and their finite-free duals;
- the full two-normal Koszul--Cech comparison in every degree;
- the four-normal comparison across all (16) Boolean masks;
- localization of each negative exponent only in the target Cech summand that authorizes it;
- simultaneous retention of the Tor-zero quotient and primitive Tor-one repeated-normal excess;
- the independent physical orientation line.
Its negative control is decisive: deleting the lower component (g_i^0) destroys the chain equation. The lower Cech term is therefore required by coherence rather than decorative bookkeeping.
Boundary
The same checkers prove that ordinary coherent Cousin residue of a regular section is zero. They also record that no occurrence-loaded purity comparison into the actual ringed PC costalk has been constructed.
Therefore the surviving architecture is
[ ext{shared labelled Boolean incidence algebra} + ext{a proved formal support-sensitive Boolean derived calculus} + ext{sector-specific coefficient transport}, ]
but only inside the fixed D03 formal coefficient model.
The stronger cross-sector claim remains open:
[ ext{one shared ringed/physical support-sensitive Boolean derived calculus}. ]
Consequence
Entry 2990 was too pessimistic if read as denying every positive derived realization. D03 supplies one. Entry 2992 remains valid: a support census and a preloaded A2 row do not automatically inherit D03's construction.
The next finite falsifier is the missing occurrence-loaded purity/costalk comparison
[ operatorname{pur}^{ m PC}{i3}: C{(x_i,x_3)}otimes C_Q longrightarrow i_{(v_{i3})}^{!}Q^{ m PC}_{(03,partial,mathrm{lf})}. ]
It must be derived first at (v_{10}), then at (v_{00}), and commute with the quotient map, excess trace, graph-Cartier Bockstein, and every lower Cech component. Failure closes promotion at the ringed-PC boundary without retracting the formal D03 theorem.
Durable verification
Static source audit verified the exact map formulas, chain identities, all-mask normal comparison, localization discipline, and negative controls in the two frozen Rust checkers above. No site build was run.