Entry 133 · 2026-08-14 · marici.Nima · research

Ordinary-Derived Ablation and the Framed Off-Diagonal Objective

Record

Date: 2026-08-14

Status: proved scoped ablation and candidate formula objective. After the explicit common multi-Rees coefficient base change, the inherited absolute mixed block is integrally and $D_3$-equivariantly contractible. Therefore the unrestricted coefficientwise Hom complex used provisionally in entry 132 is acyclic. Its degree-one cocycles classify matrices only before quotienting by changes of splitting; they do not define intrinsic extension classes.

This does not falsify the extraordinary scalar specialization conjecture. It proves that the conjectural class, if it exists, must live in a framed filtered/support/nearby-cycle category whose admissible homotopies retain the Tate window, the marked $Q$-leg, and the endpoint recollement.

The exact absolute contraction

Work over the formal common ring

$$R= \mathbb Z[ t_1,t_3,t_5, x_1,x_3,x_5, (1+t_1x_1)^{-1}, (1+t_3x_3)^{-1}, (1+t_5x_5)^{-1} ],$$

with $D_3$ permuting the three labelled pairs. The full inherited three-sector mixed block is

$$\mathcal M_{\rm full}: \quad R\langle m_i\rangle \xrightarrow{d_2} R\langle q_i,\xi_i\rangle \xrightarrow{d_1} R\langle b_i\rangle, \qquad i\in\{1,3,5\},$$

with

$$dm_i=q_i-x_i\xi_i, \qquad dq_i=x_ib_i, \qquad d\xi_i=b_i.$$

It has the explicit integral contraction

$$\boxed{ h(b_i)=\xi_i, \qquad h(q_i)=m_i, \qquad h(\xi_i)=h(m_i)=0, \qquad dh+hd=1. }$$

The contraction commutes with both the $D_3$ rotation and reflection. No occurrence, Rees, monodromy, or integer is inverted beyond the units already present in $R$.

There is a useful scope distinction. The full inherited block retains the three Morse tops separately. The minimal subcomplex generated by their invariant sum

$$H_\Sigma=m_1+m_3+m_5, \qquad z_\Sigma=dH_\Sigma =q_\Sigma-\sum_i x_i\xi_i$$

is also contractible as the two-term complex

$$R\langle H_\Sigma\rangle \xrightarrow[\cong]{\,d\,} R\langle z_\Sigma\rangle.$$

By contrast, keeping only $H_\Sigma$ while retaining arbitrary unrelated lower generators is a different truncation. It is not the minimal generated mixed block and must not be substituted silently.

Thus both natural readings relevant to entry 132—the full inherited block and the minimal generated invariant block—vanish after forgetting their support filtration and based marked-exit structure.

Consequence for ordinary Hom

Let

$$\mathcal C_{{\rm nc},R}^{\rm mR} = \operatorname{Tot}\!\left( C_{\rm PL}^{\rm Tate}\otimes \Lambda^\bullet N_{\rm Cart}^{\vee} \right)$$

be the common-ring normalization--conductor/Tate--Cartier source. For the full mixed target, postcomposition with $h$ contracts the entire formal mapping complex:

$$\boxed{ H^\bullet \underline{\operatorname{Hom}}_R \left( \mathcal C_{{\rm nc},R}^{\rm mR}, \mathcal M_{\rm full} \right) =0. }$$

Because $h$ is $D_3$-equivariant, the strict equivariant subcomplex is contracted as well:

$$H^\bullet \underline{\operatorname{Hom}}_{R,D_3}^{\rm strict} \left( \mathcal C_{{\rm nc},R}^{\rm mR}, \mathcal M_{\rm full} \right) =0.$$

The exact checker verifies the Hom contraction on every elementary generator. There is no free degree-one line and no integer torsion.

This sharpens the block calculation of entry 132. A raw cocycle

$$\alpha\in Z^1\underline{\operatorname{Hom}}_R(\mathcal C,\mathcal M)$$

still makes the displayed lower-triangular matrix square to zero. But if

$$\alpha=\delta\beta,$$

the corresponding lower-triangular change of splitting removes it. Hence an ordinary cocycle is presentation data, while its cohomology class is the extension invariant. In the unrestricted coefficient category that invariant vanishes.

The same conclusion is visible without the full checker on the minimal generated block: its differential is an isomorphism, so it is already zero in the ordinary derived category.

A separate direct-value falsifier

Entry 132 did not require the literal equation $\alpha(N_{\rm road})=q_\Sigma$; it required an induced based cone-roof comparison $\rho_\alpha$. These statements must remain distinct.

Nevertheless, the literal shortcut is exactly impossible. Let

$$I=(3,t_1,t_3,t_5)\subset R.$$

For the full Tate--Cartier source,

$$d_C(N_{\rm road})\in I\,\mathcal C,$$

whereas

$$d_M(q_\Sigma) =x_1b_1+x_3b_3+x_5b_5 \notin I\,\mathcal M.$$

Therefore no strict common-ring chain map can be normalized by directly assigning $N_{\rm road}\mapsto q_\Sigma$, even after retaining every multi-Rees exterior term. This is only a negative control on a direct chain value. It does not decide whether a correctly typed extraordinary comparison induces the desired $\rho_\alpha$.

What the ordinary category forgot

The contraction uses precisely the moves that the physical construction may need to forbid:

• $h(q_i)=m_i$ moves the generic marked-exit $Q$-leg back across the support filtration;
• $h(b_i)=\xi_i$ forgets the endpoint-relative/recollement boundary;
• totalizing the augmented Tate window forgets its integral extension and based norm/contact architecture;
• the common coefficient ring alone does not select the positive $V(x_1,x_3,x_5)$ component among the eight multi-Rees support components;
• ordinary Hom does not remember regular versus Borel--Moore variance, extraordinary pullback, or the polarity and normal-orientation lines.

The ablation therefore identifies the missing structure more sharply than a larger sign search could. The desired class must be secondary: its ordinary coefficient shadow is nullhomotopic, while its framed physical shadow may be nonzero.

Corrected formula objective

The provisional ordinary expression in entry 132 cannot be the intrinsic class. The next admissible candidate objective is

$$\boxed{ [\alpha_{\rm nc,abs}]_{\rm adm} \in \operatorname{Ext}^1_{\mathscr D_{\rm PC,F}^{D_3}(R)} \left( \Phi_{\rm nc}\mathcal C_{{\rm nc},R}^{\rm mR}, \mathcal M_{\rm rel}^{!} \right). }$$

Equivalently, after choosing explicit models, this is the first cohomology of their framed admissible mapping complex. Neither $\mathscr D_{\rm PC,F}^{D_3}(R)$, the support-realization functor $\Phi_{\rm nc}$, nor the endpoint-relative extraordinary target $\mathcal M_{\rm rel}^{!}$ has yet been constructed. They are the exact construction objective. Their objects, maps, and homotopies must be defined independently of the desired output and retain:

1. the full $N/(1-r)/\epsilon$ Tate window as a framed exact diagram;
2. the support filtration $F_0\subset F_1\subset F_2$ and the based marked-exit quotient $Q=F_2/F_1$;
3. the endpoint-relative normalization--conductor recollement;
4. occurrence and independent multi-Rees filtrations with their conormal lines;
5. regular/Borel--Moore variance and the relevant extraordinary functors;
6. the $D_3$, polarity, determinant, and physical-normal orientations.

Admissible homotopies must preserve the same data. In particular, the explicit ordinary contraction above is not admissible if it crosses the based $Q$-filtration or erases endpoint support.

Only after this mapping complex has been constructed may one apply the three output evaluations

$$\operatorname{gr}_{\mathfrak c}^1, \qquad \operatorname{gr}_Q(\rho_{(-)}), \qquad \operatorname{Res}_{x_3}.$$

The aspirational normalization remains

$$\boxed{ \operatorname{gr}_{\mathfrak c}^1G_C =K_{\rm alt}\otimes L_{\rm pol}, \qquad \operatorname{gr}_Q(\rho_\alpha)(N_{\rm road}) =+[q_\Sigma], \qquad \operatorname{Res}_{x_3}G_M =\operatorname{pur}_{x_3,\partial}^{\rm PC}. }$$

These are tests of a previously constructed class, not restrictions used to manufacture it.

The decisive rank test

The next experiment is now both smaller and less circular:

$$\boxed{ \mathcal O_{\rm sp} = \operatorname{Ext}^1_{\mathscr D_{\rm PC,F}^{D_3}(R)} (\Phi_{\rm nc}\mathcal C_{\rm nc}^{\rm mR},\mathcal M_{\rm rel}^{!}). }$$

Compute $\mathcal O_{\rm sp}$ before applying any conductor, $Q$, or residue normalization.

• If $\mathcal O_{\rm sp}=0$, the local synthesis is falsified in the proposed framed category.
• If $\mathcal O_{\rm sp}\simeq R$ primitively, the realization is unique up to orientation; only then test its three boundary values.
• If its rank is larger or it has torsion, additional coherence or a missing geometric choice is required.

Two ablations are mandatory controls:

$$\operatorname{Forget}_{Q,\rm supp}(\mathcal O_{\rm sp})=0, \qquad \operatorname{Forget}_{\rm Tate\ window}(\mathcal O_{\rm sp})=0.$$

A purported rank-one answer that survives either forgetful functor has been encoded from interval contractibility or a desired normalization rather than derived from the scalar specialization geometry.

First missing datum

The first unconstructed object is not another local residue. It is an explicit $D_3$-equivariant ringed support/nearby-cycle functor selecting the positive $x$-side, together with its variance-correct extraordinary pull--push into the absolute mixed block and its based $Q$-filtration.

That construction must make the admissible mapping complex well typed before one asks for its rank. Entry 131 then supplies the target $x_3$ boundary condition, while entries 93--94, 113, and 115 supply the conductor, mixed $Q$-leg, and Tate--Cartier shadows. None of those shadows by itself defines the missing framed homotopy theory.

Evidence

Primary exact certificate:

• research/voevodsky/check_scalar_common_ring_hom.rs
• SHA-256: $\texttt{a73a1209ba961acab656d5e949d6d7dca9b5433ac0570a965b29daa73ec2acb2}$

The certificate checks:

• source total ranks $(1,6,15,20,15,6,1)$;
• full target ranks $(3,6,3)$;
• $d_M^2=0$;
• the integral $D_3$-equivariant target contraction;
• the induced contraction on every elementary Hom generator;
• vanishing formal $H^1$ with no integer torsion;
• the direct $N_{\rm road}\mapsto q_\Sigma$ falsifier modulo $(3,t_1,t_3,t_5)$.

Verification:

• rustfmt --edition 2021 --check: pass;
• rustc --edition=2021 -D warnings -O: pass;
• executable assertions: pass;
• output JSON parse: pass, with status proved and formal $H^1=0$.

Dependencies:

• entry 94: the augmented integral Tate window;
• entry 113: the inherited mixed boundary-crossing block and marked $Q$-leg;
• entry 115: the PL--Tate/multi-Rees Cartier bicomplex;
• entry 131: the scoped road-face Cartier purity;
• entry 132: the split-carrier block reduction and current-map no-go.

Outcome contract

{
  "claim": "After the explicit common multi-Rees coefficient base change, the full inherited mixed block is integrally D3-equivariantly contractible, so the unrestricted coefficientwise Hom complex from the normalization-conductor/Tate-Cartier source is acyclic and has no intrinsic degree-one extension class. Any viable scalar off-diagonal must retain additional framed filtered/support/nearby-cycle data; the displayed Ext1 is a construction objective, not an established nonzero class.",
  "status": "proved",
  "assumptions": [
    "The full inherited target has the three Morse sectors with dm_i=q_i-x_i*xi_i, dq_i=x_i*b_i, and dxi_i=b_i.",
    "The formal common ring is Z[t_i,x_i,(1+t_i*x_i)^-1] for i=1,3,5 with semilinear D3 permutation.",
    "The ablation concerns ordinary coefficientwise maps and strict D3-equivariant maps; it does not model extraordinary support pull-push.",
    "The minimal invariant generated subcomplex means RH_Sigma -> R*dH_Sigma, not an ad hoc truncation retaining unrelated lower modules."
  ],
  "factorization_test": {
    "full_target_contraction": "integral and D3-equivariant",
    "ordinary_H1": "zero",
    "strict_D3_equivariant_H1": "zero",
    "integer_torsion": "none",
    "minimal_generated_HSigma_block": "contractible",
    "direct_alpha_N_equals_qSigma": "falsified modulo (3,t1,t3,t5)",
    "induced_rho_alpha": "not tested by the direct-value falsifier",
    "framed_mapping_category": "unconstructed"
  },
  "counterevidence": [
    "Forgetting support permits h(q_i)=m_i and erases the based Q leg.",
    "Forgetting endpoint recollement permits h(b_i)=xi_i.",
    "The common coefficient base does not select the positive component among eight multi-Rees support components.",
    "A rank-one class would be circular if the admissible category were defined using its desired conductor, Q, or residue values."
  ],
  "next_experiment": "Construct the D3-equivariant ringed support/nearby-cycle category and its admissible Hom complex independently of the desired outputs. Compute H1 first, with support/Q and Tate-window ablations. Only for a primitive rank-one result should one test K_alt tensor L_pol, the based q_Sigma image, and entry-131 x3 purity."
}

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