Integral Ambient Log-Blowup Invariance and the Persistent Bivariant Q-Leg Gap
Record
Date: 2026-08-14
Status: proved for the integral loaded stellar subdivision, its filtered strong deformation retract, literal invariance of the relative road quotient, and transport of the global Yoneda extension. Falsified as a construction of the missing marked Beck–Chevalley map.
The ambient modification
Let
[ p:\widetilde K_6=\operatorname{Bl}_{C}K_6\longrightarrow K_6, \qquad C={D03,x_1}. ]
The center is an actual codimension-two face of the labelled hexagon associahedron. It is contained in the short-diagonal boundary (B_{\rm short}) and is disjoint from the positive central vertex
[ v_+={x_1,x_3,x_5}. ]
Use the full inverse-image filtration
[ \widetilde F_0=F_0, \qquad \widetilde F_1=PC_{\rm supp}(p^{-1}B_{\rm short}), \qquad \widetilde F_2=PC_{\rm supp}(\widetilde K_6). ]
The ordinary and blown-up face censuses are
[ (1,9,21,14) \quad\longrightarrow\quad (1,10,24,16), ]
and the loaded degree ranks are
[ (14,63,93,45) \quad\longrightarrow\quad (16,72,106,51). ]
Thus the absolute loaded complex grows from 215 to 245 generators, with
[ \operatorname{rk}(\widetilde F_0,\widetilde F_1,\widetilde F_2) =(8,238,245). ]
Occurrence and monodromy are different layers
The blowup exposed a necessary correction. The exceptional monodromy character is multiplicative:
[ q_E=q_{03}q_1, \qquad u_E=u_{03}+q_{03}u_1 =u_{03}+u_1+u_{03}u_1. ]
There is no corresponding additive occurrence variable (X_E=X_{03}+x_1). Occurrence coefficients come from the lcm-labelled cellular resolution.
Over the marked point
[ b={D03,x_1,x_3}, ]
the exceptional interval separates vertices (b_1,b_D). The occurrence boundaries are
[ de_c’=X_{03}b_1-x_5a, \qquad dh_E=b_D-b_1, \qquad de_r’=x_0c-x_1b_D. ]
The exceptional edge therefore has unit cellular boundary. The unique primitive expanded chain is
[ \boxed{ \widetilde\xi =x_1e_c’+X_{03}x_1h_E+X_{03}e_r’ } ]
and it obeys
[ \boxed{ d\widetilde\xi=X_{03}x_0c-x_1x_5a. } ]
The blowdown and section are
[ \begin{aligned} r(b_1)=r(b_D)&=b,& r(h_E)&=0,& r(e_c’)&=e_c,&r(e_r’)&=e_r,\ s(b)&=b_1,&s(e_c)&=e_c’,& s(e_r)&=e_r’+x_1h_E. \end{aligned} ]
They satisfy (rs=1), carry (\xi\leftrightarrow\widetilde\xi), and the unit exceptional edge gives an integral chain homotopy between (sr) and the identity.
The independent normal subdivision is the saturated complex
[ L_0=R\langle p\rangle, \qquad L_1=R\langle h_{03},h_E,h_1\rangle, \qquad L_2=R\langle A,B\rangle, ]
with
[ dA=h_E-h_{03}-q_{03}h_1, \qquad dB=u_1h_{03}-u_{03}h_1. ]
It retracts integrally to (K(u_{03},u_1)) by
[ r(h_E)=h_{03}+q_{03}h_1, \qquad r(A)=0, \qquad r(B)=h_{03}\wedge h_1, \qquad H(h_E)=A. ]
Tensoring this local contraction with every spectator packet preserves it: the two totalization cross terms cancel with opposite Koszul signs. No Rees parameter, normal variable, occurrence coefficient, or integer is inverted.
Filtered invariance and the literal road quotient
The occurrence and normal contractions are supported entirely over the center, hence entirely inside (\widetilde F_1). Extending them by the identity gives an integral filtered strong deformation retract
[ (\widetilde F_0\subset\widetilde F_1\subset\widetilde F_2) \simeq (F_0\subset F_1\subset F_2) ]
which is the identity on (F_0) and on
[ Q=F_2/F_1. ]
More strongly, the blown-up quotient is literally the same seven-generator chain complex:
[ \widetilde Q=\widetilde F_2/\widetilde F_1=Q, \qquad (\operatorname{rk}Q_i)_i=(0,0,3,4). ]
Its three long-facet occurrence attachments and three normal-circle boundaries retain their original coefficients and signs.
Consequently the global Yoneda two-extension
[ e_F= [0\to F_0\to F_1\to F_2/F_0\to Q\to0] \in\operatorname{Ext}^2(Q,F_0) ]
transports canonically through the filtered subdivision equivalence. This is an actual invariance theorem, not merely equality of associated grades.
The surviving obstruction
The same support calculation prevents an overclaim. Since (C\subset B_{\rm short}), the complete exceptional divisor and every cell of the expanded gallery lie in (\widetilde F_1). Therefore
[ q_{\widetilde Q}\circ\widetilde\gamma=0. ]
The literal marked restriction of the transported Yoneda class remains the entry-105 zero:
[ \operatorname{pb}_{03}^{\rm lit}(p^*e_F)=0. ]
The nonzero chain (\widetilde\xi) is a canonical secondary trivialization inside (\widetilde F_1). It is not itself a representative of a map from (Q) to (F_0[2]). Blowup invariance transports the already known gallery class; it does not create the missing extraordinary-pullback leg.
The first unconstructed arrow is therefore
[ \boxed{ \operatorname{BC}^{\log}{+;03}: R!\operatorname{Hom}(\widetilde Q,\widetilde F_0[2]) \longrightarrow \mathcal H^{\rm loc}{+;03}, } ]
where (\mathcal H^{\rm loc}_{+;03}) is the normalized local bivariant Hom complex of entries 97 and 100. It must be constructed independently and satisfy
[ \operatorname{BC}^{\log}{+;03}(p^*e_F)=\Theta{03}^{\rm loc}. ]
Its associated-grade carrier must be (\widetilde\xi); its excess component must be the labelled two-copy class (\eta_{3,\rm mix}); and its endpoint, physical-normal, determinant, and support variances must all be visible. Entry 97 proves uniqueness only after such a closed, correctly typed, unit-normalized cocycle exists. It cannot supply existence.
Rejected shortcut
If one incorrectly introduces an additive occurrence coefficient (X_E=X_{03}+x_1), the subdivision quotient contains (K(X_E)), with
[ H_0=R/(X_E), ]
and a blowdown contraction would require (X_E^{-1}). This is a useful negative control: it detects precisely the forbidden conflation of lcm occurrence weights with Kummer-character multiplication.
Evidence
Exact certificate:
research/voevodsky/check_d03_global_log_blowup_relative_q.rs
SHA-256:
07b2fb9eae2390c779140ce1e37542ebc83c475cb7747c5af9c31a152869da2dIt verifies the full face and loaded-generator censuses, exceptional square, occurrence contraction, saturated normal contraction, spectator signs, filtered ranks, literal relative quotient, transported extension, and zero exceptional (Q)-image. The entry-105 and entry-106 certificates were also rerun.
Consequence
The ambient log blowup is now understood exactly:
[ \boxed{ \text{canonical integral resolution and invariance theorem,} \quad \text{not the missing bivariant comparison.} } ]
The scalar master already contains the local secondary carrier and the global two-extension, and the blowup relates each to its own transform. The remaining mathematics is the extraordinary operation relating those two categorical degrees.
Outcome contract
{
"claim": "The ambient toroidal blowup along C={D03,x1} gives an integral filtered stellar-subdivision equivalence, preserves the global Yoneda two-extension, and canonically expands the marked gallery, but its exceptional/gallery support has zero relative-Q image and therefore does not construct the missing Beck--Chevalley map.",
"status": "proved",
"assumptions": [
"The blowup filtration uses the full inverse image of the short-diagonal boundary.",
"Occurrence coefficients use the lcm-labelled cellular resolution and remain independent of monodromy parameters.",
"The ordered center normals are (D03,x1), fixing q_E=q_D03 q_1 and the positive exceptional orientation."
],
"evidence_refs": [
"research/voevodsky/check_d03_global_log_blowup_relative_q.rs",
"ledger entries 97, 100, 105, and 106"
],
"factorization_test": {
"absolute_filtered_SDR": "passed integrally",
"relative_Q": "literal seven-generator identity",
"global_Yoneda_transport": "passed",
"expanded_gallery": "passed",
"ordinary_exceptional_Q_leg": "falsified; identically zero",
"global_to_local_Beck_Chevalley": "unconstructed"
},
"counterevidence": [
"The exceptional divisor and expanded gallery lie wholly in F1.",
"The local secondary trivialization and global Ext2 class inhabit different functorial types.",
"Rank-one local uniqueness does not construct a bivariant source map.",
"An additive exceptional occurrence variable produces a noncontractible K(X_E) quotient and is rejected."
],
"next_experiment": "Construct BC_log_{+;03} as an independently typed extraordinary pull--push kernel with a nonzero Q leg, then test its carrier, repeated-normal excess class, endpoints, and physical normal before invoking local uniqueness."
}