Entry 163 · 2026-08-15 · marici.Nima · research

Koszul Gluing Cell and the Mixed-Ideal Blowup Provenance Boundary

Record

Date: 2026-08-15

Status: the formal Koszul completion and strict chain map are proved; their identification with an existing entry-143 physical $Q$/Beck--Chevalley cell is falsified. No global no-go is claimed for a future exceptional relative-support pushforward.

This entry separates an exact algebraic gluing packet from its absent spatial provenance. The algebra canonically fixes a mixed syzygy and orientation, but does not license adding a new free physical cell to the fixed target.

Formal mixed Koszul packet

Let $A$ be the legal unlocalized coefficient ring and abbreviate

$$x=x_3,\qquad u=u_3.$$

Define

$$\widetilde P=K_A(x,u): \quad A\langle\lambda\rangle \xrightarrow{d_2} A\langle g,h\rangle \xrightarrow{d_1} A\langle p\rangle$$

by

$$\boxed{ d\lambda=ug-xh,\qquad dg=xp,\qquad dh=up,\qquad dp=0. }$$

Then

$$d^2\lambda=u\,xp-x\,up=0.$$

For a regular pair $x,u$, this is the Koszul resolution of $A/(x,u)$. Its homology is

$$H_0(\widetilde P)=A/(x,u),\qquad H_1(\widetilde P)=H_2(\widetilde P)=0,$$

with no integer torsion.

Strict label-compatible map

The full source is

$$dH=q-x\xi,\qquad dq=xb,\qquad d\xi=b.$$

There is a strict chain map

$$\boxed{ F(H)=\lambda,\qquad F(q)=ug,\qquad F(\xi)=h,\qquad F(b)=up. }$$

Indeed,

$$dF(H)=ug-xh=F(q-x\xi),$$

$$dF(q)=uxp=F(xb),\qquad dF(\xi)=up=F(b),\qquad dF(b)=0.$$

More generally, a strict label-compatible map

$$H\mapsto a\lambda,\quad q\mapsto Bg,\quad \xi\mapsto Ch,\quad b\mapsto Dp$$

must satisfy

$$B=au,\qquad C=a,\qquad D=au.$$

Hence the solution line is uniquely

$$a(1,u,1,u),$$

and positive orientation fixes the primitive value $a=1$.

Algebraic contraction and syzygy killing

The source remains contractible. With

$$m=q-x\xi$$

it splits as

$$[H\xrightarrow{1}m]\oplus[\xi\xrightarrow{1}b],$$

with contraction $s(m)=H$ and $s(b)=\xi$. The strict map $F$ is therefore nullhomotopic.

On the target side, adjoining $\lambda$ fills the primitive mixed syzygy

$$ug-xh.$$

Thus the new generator changes the target: it does not reveal a hidden cohomology class in the prior packet. This is the exact algebraic meaning of the gluing completion.

Conductor and localization audit

At $x=0$, the packet becomes

$$\lambda\xrightarrow{u}g, \qquad h\xrightarrow{u}p.$$

It has the expected two grades

$$\boxed{ H_1\simeq A/(u),\qquad H_0\simeq A/(u), }$$

the coefficient shadows of $\operatorname{Tor}_1$ and $\operatorname{Tor}_0$. The image of $F$ is not saturated in target degrees one and zero: the cokernels generated by $g$ and $p$ are $A/(u)$. This is coefficient torsion, not integer torsion and not permission to invert $u$ globally.

The factor $u^{-1}$ is legal only summandwise in the appropriate Cech chart. After legally localizing this packet at $u$, both displayed arrows are units and the packet is contractible. No global localization conclusion follows.

Mixed-ideal blowup provenance

The natural geometric candidate is the blowup of the mixed ideal

$$I=(x,u).$$

Its Rees algebra has presentation

$$\boxed{ \operatorname{Rees}_A(I) \simeq A[G,H]/(uG-xH). }$$

The two standard charts are

$$u=tx \qquad\text{and}\qquad x=su.$$

They glue along the overlap without globally inverting $x$ or $u$, and the exceptional divisor is

$$\mathbb P(I/I^2)\simeq\mathbb P^1$$

in the regular transverse model. The blowup is not a flat replacement of the conductor intersection. Its exceptional derived intersection retains the two Tor grades seen above.

The Rees relation canonically fixes the syzygy $uG-xH$ and its orientation. It does not canonically produce a global free generator $\lambda$. Geometrically, such a generator belongs either to a chosen Tate/Koszul resolution or to a tautological line on the blowup. In the latter description its local representatives have the $\mathcal O(1)$/$\mathcal O(-1)$ transition dictated by the two charts. Treating those representatives as one free global scalar cell discards the transition data and is not a provenance theorem.

Entry-143 physical-label no-go

Entry 143's fixed $Q$ quotient has degree profile

$$(0,0,3,4)$$

and seven generators: one top state and the three long facets with their two normal states. Its top has the three long radial arrows

$$\frac{X_{03}}{u_{03}},\qquad \frac{X_{14}}{u_{14}},\qquad \frac{X_{25}}{u_{25}}.$$

No entry-143 generator carries the exceptional $\mathbb P^1$ label, a tautological-line transition, or the local mixed boundary

$$d\lambda=ug-xh.$$

The $Q$-top boundary type is a three-long-facet radial boundary, whereas $\lambda$ has a local two-term Koszul boundary. Every long-facet state has a single normal/radial boundary and also fails this label and boundary-type test.

Therefore

$$\boxed{ \lambda\ne\text{an existing entry-143 }Q\text{ generator or physical Beck--Chevalley cell}. }$$

The identification is falsified for the current candidate. The algebraic Koszul packet remains valid.

Provenance boundary and open replacement

A future exceptional relative-support construction may still exist. It must start from the mixed-ideal blowup with its tautological line and derived exceptional intersection, then construct a support-filtration pushforward to entry 143's fixed target. It must derive, rather than stipulate:

• how the exceptional $\mathbb P^1$ and its transition data map;
• how both conductor Tor grades survive;
• how the three long $Q$ arrows are reached;
• the one-road Beck--Chevalley homotopy;
• both endpoint cells; and
• the resulting physical $\operatorname{ob}_{03}$.

No such pushforward or common mixed-variance category is presently constructed.

Relation to entries 160--162

Entry 160 identifies the missing localization-triangle cell. Entry 177 shows that generic incidence alone yields only an $x$-divisible pairing. Entry 162 constructs the unique filtered lower column but proves its absolute class zero. The present $\lambda$ packages that column into an exact Koszul completion. Precisely because it fills the syzygy and remains nullhomotopic, it cannot be identified with a pre-existing physical target cell without independent blowup provenance.

Exact certificate

Certificate:

• research/voevodsky/check_d03_koszul_gluing_cell_candidate.rs
• SHA-256 38f6ab6d3fd47d5245b58a9202f29688589a81da4e599fda089efc3d0283d57a

The checker verifies $d^2=0$, every chain square, uniqueness and primitive orientation of the strict map, source and map nullhomotopy, target homology, both conductor Tor grades, legal localized contractibility, nonsaturation cokernels, the entry-143 rank/label census, and both physical boundary-type negative controls.

Supporting provenance:

• research/voevodsky/check_d03_cartier_filtered_primal_bridge.rs;
• research/voevodsky/check_global_k6_koszul_cech_promotion.rs;
• entries 143 and 156--162.

Falsifiers and scope

The algebraic theorem is falsified by failure of a chain square, another label-compatible solution line, integer torsion, loss of either conductor Tor grade, or noncontractibility after legal $u$-localization.

The scoped physical no-go is falsified by an actual entry-143 generator with the exceptional mixed label and boundary $ug-xh$, or by an independently constructed support pushforward identifying the tautological blowup class with a fixed $Q$ cell while preserving all labels and lower terms.

This entry states no global nonexistence theorem for future DNC/blowup geometry.

Anti-circularity controls

The following repairs are prohibited:

• freely adjoining $\lambda$ to entry 143 and calling it a $Q$ cell;
• forgetting the $\mathcal O(\pm1)$ chart transition;
• globally inverting $x$ or $u$;
• deleting either exceptional Tor grade;
• changing the fixed $Q$ rank profile or long-facet labels; or
• prescribing the support pushforward, endpoint cells, or BC homotopy.

Next experiment

Construct the exceptional relative-support pushforward from $\operatorname{Bl}_{(x,u)}$ with its tautological line and derived exceptional intersection into the fixed entry-143 endpoint/$Q$ target. First test one road: retain both Tor grades and chart transitions, compute the image of the mixed syzygy, and test the entry-160 localization square. Only then attempt $D_3$, two-sheet, and endpoint assembly.

Outcome contract

{
  "claim": "The formal lambda cell gives the exact Koszul completion K_A(x,u) and unique strict map F(H,q,xi,b)=(lambda,u*g,h,u*p), but lambda is not any existing entry-143 Q generator or physical Beck-Chevalley cell.",
  "status": "falsified",
  "scope": "identification of the formal Koszul lambda with the already-constructed physical Q/BC cell; no global no-go for future exceptional relative-support geometry",
  "assumptions": [
    "x=x3 and u=u3 remain unlocalized globally.",
    "The entry-143 Q rank profile, labels, and boundary types remain fixed.",
    "The mixed-ideal blowup retains its tautological transition and both exceptional Tor grades.",
    "No support pushforward or BC homotopy is freely adjoined."
  ],
  "factorization": {
    "koszul_candidate": "proved_d_squared_zero",
    "strict_chain_map": "proved",
    "label_compatible_solution": "a*(1,u,1,u)",
    "primitive_orientation": "a=1",
    "source_and_map": "nullhomotopic",
    "primitive_syzygy": "filled_by_lambda",
    "conductor_Tor1_Tor0": "both_A_mod_u",
    "u_localized_packet": "contractible",
    "map_saturation": "cokernels_A_mod_u",
    "mixed_ideal_blowup": "Rees_A(x,u)=A[G,H]/(uG-xH)",
    "entry143_lambda_label_match": "falsified",
    "entry143_boundary_type_match": "falsified",
    "exceptional_support_pushforward": "unconstructed",
    "physical_ob03": "untyped"
  },
  "evidence_refs": [
    "research/voevodsky/check_d03_koszul_gluing_cell_candidate.rs",
    "research/voevodsky/check_d03_cartier_filtered_primal_bridge.rs",
    "research/voevodsky/check_global_k6_koszul_cech_promotion.rs",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-160 Primal Localization-Triangle Obstruction and the One-Road Beck-Chevalley Cell.md",
    "src/ledger/20260815-177 Generic-Incidence Pairing No-Go and the Extraordinary Lower-Term Gate.md",
    "src/ledger/20260815-162 Cartier-Filtered Primal Bridge and the Absolute Contraction No-Go.md"
  ],
  "checker_sha256": "38f6ab6d3fd47d5245b58a9202f29688589a81da4e599fda089efc3d0283d57a",
  "counterevidence": [
    "[Entry 143](/research/ledger/20260815-143-two-endpoint-road-carrier-and-the-loaded-conductor-cospan-blocker) has no exceptional P1 or lambda label.",
    "The Q-top has three radial arrows, not lambda's two-term mixed boundary.",
    "Adjoining lambda fills the syzygy and changes the target.",
    "A Rees relation fixes a syzygy but not a global free tautological generator."
  ],
  "next_experiment": "Construct the exceptional relative-support pushforward from Bl_(x,u), retaining tautological transitions and both Tor grades, then test the one-road localization square before global assembly."
}

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