Entry 170 · 2026-08-15 · marici.Nima · research

Product-Rees F03 Square and the Forced Long-Normal Completion

Record

Date: 2026-08-15

Status: two scoped theorems proved. The full product-Rees occurrence square constructs the primitive $p_{03}$ and its strict peripheral boundary, and its tensor with the complete line-valued long-normal factor constructs the unique facewise local enhancement. Spatial distribution of the entry-100 Tor packet, extraordinary endpoint/flip counits, three-road normalization gluing, and parity remain unconstructed. No graph admission is claimed.

The full product-Rees occurrence square

Work over the unlocalized polynomial occurrence ring $A$ . Put

I=(x_0,x_1), \qquad J=(x_3,x_4).

The two ordered pairs are disjoint regular sequences. Let

\pi:Y_{03}longrightarrow\operatorname{Spec}A

be the product of the two Rees blowups, and use the convention-free tautological ideal line

\mathcal O_{Y_{03}}(-E_I-E_J).

Its derived pushforward is the product ideal

R\pi_*\mathcal O_{Y_{03}}(-E_I-E_J)=IJ

with no higher contribution in this regular disjoint-pair case. Tensoring the two primitive length-one ideal resolutions gives the exact complex

\boxed{ 0\longrightarrow A \xrightarrow{d_2}A^4 \xrightarrow{d_1}A^4 \longrightarrow IJ \longrightarrow0. }

Its chain ranks are

1\longrightarrow4\longrightarrow4.

Choose the four vertex generators in the order

v_{03}=x_0x_3, \quad v_{04}=x_0x_4, \quad v_{13}=x_1x_3, \quad v_{14}=x_1x_4.

The four middle generators are the oriented sides of the weighted product square $(x_0,x_1)\boxtimes(x_3,x_4)$ , and the top generator is $p_{03}$ . The differentials are the tensor-product Koszul boundaries; with one global product orientation,

d_1d_2=0

symbolically over $A$ .

Exactness, saturation, and primitive top

Each factor resolution has primitive relation, and the two variable pairs are disjoint. Hence

\operatorname{Tor}^{A}_{k>0}(I,J)=0, \qquad I\otimes_AJ\xrightarrow{\sim}IJ.

Both intermediate homology groups vanish. A unit maximal minor of the edge-to-vertex incidence shows that its rank-three boundary lattice is saturated; the top column is primitive as well. Thus the resolution has no integer torsion and

H_0\simeq IJ.

Relative to the short-boundary support $F_B$ , all four edges lie in $F_B$ , while the long-facet top survives. Therefore

\boxed{ [p_{03}]\ne0, \qquad d p_{03}=\partial F_{03}\subset F_B, }

and $[p_{03}]$ is the primitive relative $F03$ class. The product Rees space constructs not just the relative class but its complete strict four-edge peripheral lift.

Exact entry-143 short-H-empty collar

The product-square resolution is exactly the $H=\varnothing$ short collar of the fixed entry-143 target $E=F_K/F_V$ over the long facet $F03$ . The label-preserving identification is

p_{03}\longleftrightarrow[\{D03\},\varnothing],

e_a\longleftrightarrow[\{D03,x_a\},\varnothing], \qquad a\in\{0,1,3,4\},

and

v_{ab}\longleftrightarrow [\{D03,x_a,x_b\},\varnothing], \qquad a\in\{0,1\},\quad b\in\{3,4\}.

These are actual noncrossing faces. The entry-143 radial occurrence differential on this collar is the weighted product-square differential, so the identification is strict before quotienting by $F_B$ . The phrase “short- $H$ -empty” is essential: this theorem does not identify the other circle grades or freely add denominators.

After quotienting the four edges and vertices by $F_B$ , only the primitive $p_{03}$ remains in the local long-facet quotient.

The line-valued long three-state factor

Let $D=D03$ . The long-normal factor is

C_D: \mathbb Z\langle t_D,n_D\rangle \xrightarrow{[y_D,1]} \mathbb Z\langle p_D\rangle,

where

y_D=X_D\otimes u_D^\vee.

This is a line-valued section, not the base-ring fraction $X_D/u_D$ . The reciprocal dual $u_D^\vee$ is evaluated only in the bivariant normal pairing. No global inversion of $u_D$ occurs.

Tensor the entire occurrence square $P_{03}$ of ranks $(1,4,4)$ with $C_D$ , using

d=d_{P_{03}}+(-1)^{\deg P_{03}}d_{C_D}.

The total ranks are

\boxed{2\longrightarrow9\longrightarrow12\longrightarrow4.}

Every peripheral edge and every vertex occurs in both long states. These lower copies are forced by the tensor differential; they are not optional decorations. Direct calculation gives

d^2=0

on every basis generator.

Why a top-only attachment fails

Suppose one attaches only $t_D$ or $n_D$ to the nonclosed top $p_{03}$ , omitting the corresponding peripheral copies. Then

d^2(t_D\otimes p_{03}) =y_D\,d_{P_{03}}p_{03}\ne0,

and

d^2(n_D\otimes p_{03}) =d_{P_{03}}p_{03}\ne0.

Both failures have four nonzero peripheral components. Therefore the naive top-only attachment is not a chain complex. The full facewise tensor is the minimal strict completion: its signed cross terms cancel precisely because all lower copies are present.

Local Q row after the full completion

Only after constructing the complete tensor may one quotient every peripheral $P_1/P_0$ copy. The remaining local long-facet row is

\boxed{ [,y_D\mid1,] = \left[\frac{X_D}{u_D}\ \middle|\ 1\right] }

in the legal target Cech realization. The displayed fraction is the image of the line-valued section in the summand where $D\in S\setminus H$ ; it is not a base localization. This is the entry-143 local $Q$ radial/normal row.

Thus the quotient matrix is a consequence of the full collar complex. It cannot be used to define a top-only antecedent, because that antecedent has nonzero $d^2$ before quotienting.

Entry-100 excess packet: algebraic preservation and spatial gate

The entry-100 reciprocal/BM excess packet is an external tensor factor. Its repeated- $u_3$ grades

\operatorname{Tor}_0, \qquad \operatorname{Tor}_1

are algebraically preserved by tensoring with the complete local enhancement; no naive contraction or integer torsion appears.

This algebraic preservation is not yet a spatial distribution theorem. In entry 143, Cech states are indexed by varying circle sets $H\subseteq S$ . The two Tor grades and their lower terms must be placed across those actual face/circle summands with the reciprocal source/original-BM target variance. The current checkers do not construct:

the extraordinary endpoint counits on the two marked half-corridors;
the first- and second-flip spatial incidence maps;
the distribution of the repeated- $u_3$ excess through all lower $H\subseteq S$ terms; or
the comparison identifying the resulting overlap class with the literal entry-143 states.

Consequently the local tensor proves coefficient compatibility and forces the correct lower ranks, but it does not complete the entry-100 spatial mixed-variance attachment.

Global gluing boundary

The theorem is confined to one $F03$ branch. It does not identify the three branchwise generic tops with one global top, construct their $D_3$ -compatible overlaps, or build the normalization-sheet source map. In particular, it does not construct

dH=q_J-x_3\widetilde\xi_{03}

as a comparison into the fixed endpoint/ $Q$ target. The peripheral collar lies in $F_B$ ; its strict existence does not supply the retained nonzero $q_J$ leg.

The outstanding global data are:

extraordinary endpoint/flip counits distributing the Tor grades;
a support-typed source-normalization to local- $Q$ specialization;
three-road $D_3$ -equivariant generic-top gluing;
the entry-160 logarithmic Beck--Chevalley cells; and
both endpoint comparison cells.

Until these exist, the endpoint-fixed mapping fiber is uninstantiated and reflection parity is undefined.

Anti-circularity controls

Do not retain $p_{03}$ while discarding its strict four-edge boundary before forming the chain complex.
Do not attach only the long top states; the resulting differential does not square to zero.
Do not interpret $y_D=X_D\otimes u_D^\vee$ as global inversion of $u_D$ .
Do not infer the full varying- $H$ Cech distribution from the $H=\varnothing$ collar.
Do not identify algebraically preserved Tor grades with spatially attached endpoint residues.
Do not infer three-road gluing, source normalization, a generic $Q$ leg, Beck--Chevalley, parity, or graph admission from the local tensor theorem.

Falsifiers and scope

The product-Rees theorem would be falsified by nonzero intermediate homology, torsion or nonsaturation, failure of $d_1d_2=0$ , a nonprimitive top, or a peripheral edge outside $F_B$ .

The local enhancement theorem would be falsified by failure of its rank census, nonzero $d^2$ in the full tensor, a lower copy not forced by the Leibniz differential, or failure of the peripheral quotient to give $[y_D,1]$ . Conversely, a top-only complex with $d^2=0$ would falsify its scoped negative control.

The spatial boundary would be crossed by explicit extraordinary counits and a face/circle-indexed distribution of both entry-100 Tor grades, followed by source normalization and three-road gluing retaining $q_J$ . No global nonexistence theorem is asserted.

Provenance and exact certificates

The exact checkers are

research/voevodsky/check_d03_product_rees_occurrence_square.rs; and
research/voevodsky/check_d03_local_long_normal_enhancement.rs.

Their SHA-256 hashes are, respectively,

616d86db3c4398cb2bc3701447e2ba830df1e6e150f56a4c4e9d0f7e4fe46e9d; and
520060a6873ad59fde16c300d29d6943fda6555208033c0411edd66495c0bd62.

The first verifies the exact saturated $1\to4\to4$ resolution, primitive top, strict peripheral boundary, vanishing positive Tor, and absence of torsion. The second verifies the $2\to9\to12\to4$ rank census, signed total differential, forced lower copies, local $[y_D,1]$ quotient, and both top-only $d^2$ failures.

The collar identification also uses entry 143's fixed face/circle labels; the entry-100 Tor statement uses its independently proved reciprocal/BM coefficient packet. Neither checker claims the missing global spatial attachment.

Next experiment

Construct the extraordinary endpoint/flip counits on the complete $2\to9\to12\to4$ local complex and distribute the entry-100 $\operatorname{Tor}_0/\operatorname{Tor}_1$ packet across every required entry-143 state $[S,H]$ . Verify all lower Cech squares before quotienting. Then rotate the construction to $F14$ and $F25$ , glue the three generic tops through a normalization-provenanced source retaining $q_J$ , and test the logarithmic Beck--Chevalley and endpoint cells. Only afterward instantiate the framed mapping fiber or compute parity.

Outcome contract

{
  "claim": "The full F03 product-Rees occurrence square is an exact saturated 1-to-4-to-4 resolution with primitive p03 and strict four-edge boundary in F_B; tensoring the entire square with the line-valued long three-state factor gives the canonical 2-to-9-to-12-to-4 local enhancement with d squared zero, forced lower copies, and local Q row [X_D/u_D|1], while every top-only attachment fails d squared.",
  "status": "proved",
  "scope": "one F03 product-Rees occurrence square, its complete line-valued long-normal tensor, and relative local-Q quotient only; no graph admission, global three-road gluing, source normalization, or parity",
  "assumptions": [
    "A is the unlocalized polynomial occurrence ring and (x0,x1),(x3,x4) are disjoint regular pairs.",
    "One global product orientation fixes the occurrence-square signs.",
    "y_D means X_D tensor u_D^vee before its legal target-Cech realization.",
    "Entry-143 face/circle labels and entry-100 reciprocal/BM Tor packet remain fixed."
  ],
  "factorization": {
    "product_Rees_ideals": ["I=(x0,x1)", "J=(x3,x4)"],
    "occurrence_resolution_ranks": [1, 4, 4],
    "occurrence_d_squared": "zero",
    "occurrence_intermediate_homology": [0, 0],
    "occurrence_saturation": "unit maximal minor",
    "occurrence_positive_Tor": "zero",
    "relative_top": "primitive p03",
    "peripheral_boundary": "all four edges in F_B",
    "entry143_short_H_empty_collar": "p03, four D03-short edges, and four D03-short-short vertices identified strictly",
    "long_factor": "[t_D,n_D] -> p_D with row [y_D,1]",
    "long_line_typing": "y_D=X_D tensor u_D^vee",
    "enhanced_total_ranks": [2, 9, 12, 4],
    "enhanced_d_squared": "zero",
    "forced_lower_copies": "all four edges and four vertices in both long states",
    "naive_t_top_only": "falsified: d2=y_D*d(p03) is nonzero",
    "naive_n_top_only": "falsified: d2=d(p03) is nonzero",
    "local_Q_row": ["X_D/u_D", "1"],
    "entry100_Tor0_Tor1": "algebraically preserved",
    "varying_H_spatial_distribution": "unconstructed",
    "extraordinary_endpoint_flip_counits": "unconstructed",
    "three_road_source_normalization_gluing": "unconstructed",
    "mapping_fiber": "uninstantiated",
    "parity": "undefined"
  },
  "evidence_refs": [
    "research/voevodsky/check_d03_product_rees_occurrence_square.rs",
    "research/voevodsky/check_d03_local_long_normal_enhancement.rs",
    "src/ledger/20260814-100 Support-Directed Can-Var Packet and Three Local Cousin Traces.md",
    "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/20260815-166 Two-Support MV Excess and the Missing D03 Chart-Generic Leg.md",
    "src/ledger/20260815-168 Full Rees First-Flip Occurrence Kernel and the External Normal Gate.md"
  ],
  "checker_sha256": {
    "product_rees_occurrence_square": "616d86db3c4398cb2bc3701447e2ba830df1e6e150f56a4c4e9d0f7e4fe46e9d",
    "local_long_normal_enhancement": "520060a6873ad59fde16c300d29d6943fda6555208033c0411edd66495c0bd62"
  },
  "counterevidence": [
    "A top-only long attachment has nonzero second differential on all four peripheral components.",
    "The H-empty collar does not determine the varying-H Cech distribution of the entry-100 Tor packet.",
    "The strict peripheral lift lies in F_B and does not itself supply the retained generic q_J leg.",
    "No extraordinary endpoint/flip counits or three-road normalization gluing have been constructed."
  ],
  "next_experiment": "Construct extraordinary endpoint/flip counits on the full 2-to-9-to-12-to-4 complex, distribute both entry-100 Tor grades over every required [S,H] state, then rotate and glue the three roads through a normalization-provenanced source retaining q_J before testing Beck-Chevalley, the framed mapping fiber, or parity."
}

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