Entry 384 · 2026-08-17 · marici.Nima · research

Endpoint Faces Leave a Torsion-Free Road Ideal

Result

The Sommerfeld-style minimal calculation can be completed at the exact coefficient-shadow level, but it also exposes why the physical hidden-extension group cannot yet be assigned a rank.

There are two distinct statements.

First, the unrestricted one-road mixed block

R\langle m\rangle \xrightarrow{(1,-x_3)} R\langle q,\xi\rangle \xrightarrow{(x_3,1)} R\langle b\rangle

has the integral contraction

h(b)=\xi, \qquad h(q)=m.

Thus its ordinary mapping complex is acyclic, in agreement with Entry 133. The desired realization cannot be an ordinary coefficientwise extension.

Second, use the literal one-road target and its two endpoint quotients from Entries 130 and 380. Their simultaneous degreewise kernel is

K_{\partial,2}=0, \qquad K_{\partial,1}=R\langle e_2\rangle, \qquad K_{\partial,0}=R\langle e_3,e_4\rangle,

with differential

d e_2=-x_0e_3+x_1e_4.

Consequently

\boxed{ H_1(K_\partial)=0, \qquad H_0(K_\partial)\simeq (x_0,x_1) }

up to the evident grading shift. The isomorphism is induced by

(a,b)\longmapsto x_1a+x_0b,

whose syzygy is generated by $(-x_0,x_1)$ . This module is rank one and torsion-free, but it is not a free line at the joint endpoint ideal. Hence the two endpoint restrictions leave a genuine mixed road-interior receptacle; they do not leave a new free top cell.

Strong versus weak generic restriction

The endpoint-relative module $(x_0,x_1)$ has no $x_3$ -torsion. Therefore the localization map

(x_0,x_1)\longrightarrow (x_0,x_1)[x_3^{-1}]

is injective. If “the generic face is fixed” means that the entire deformation becomes zero after generic $x_3$ -localization, then

\boxed{ \ker(\operatorname{res}_{v_{00}}, \operatorname{res}_{v_{10}}, \operatorname{res}_{x_3\ne0})=0 }

in this coefficient shadow.

That strong statement is not what Entries 378--381 currently construct. They fix the component

q_J\longmapsto x_3q_{03}^{Q}

and its first Cartier symbol. A first-symbol condition alone does not control higher filtration. In particular

x_3^2(x_0,x_1)\ne0

but has zero first $x_3$ -Rees symbol. Thus componentwise agreement on the listed faces is weaker than a restriction map on the entire deformation complex.

This is the precise well-posedness gate revealed by the hidden-extension audit: before computing

H^\bullet\mathcal D_{03}^{\rm rel},

one must define the generic, Cartier, lower-Cech, and endpoint restrictions as morphisms out of one common admissible mapping complex. Otherwise the answer changes depending on whether “generic face” means one checked coefficient or full localization of every deformation.

Reconciliation with the earlier lift spaces

The three previously observed phenomena concern different quotients and do not conflict:

Entry 136's affine rank-nine family consists of strict full-cone representatives. The canonical AW roof is common to every member, so those nine parameters are not nine intrinsic extension classes.
Entry 133's ordinary Hom complex is acyclic because its contraction is allowed to cross the based (Q)-filtration and erase endpoint support.
Entries 141 and 143 isolate a possible endpoint-fixed $\mathbb Z/2$ reflection defect only after admissible connector data are supplied. That mapping fibre is still uninstantiated, so its parity is not presently a computed class.

Benincasa's Entry 313 remains the correct warning: compatible slices do not by themselves determine an interior. The present calculation sharpens that warning for (D03): there is no hidden coefficient class under the strong full-generic interpretation, while the weaker face list does not define a unique restriction kernel at all.

Exact next experiment

Do not search another free coefficient or endpoint normalization. Construct the actual restriction diagram

\underline{\operatorname{Hom}}_{\rm adm} (\mathcal K_{03}^{\rm pre},\mathcal E_{03}^{\rm PC,\check C}) \longrightarrow \mathcal M_{\rm gen}\times^h \mathcal M_{\rm Cart}\times^h \mathcal M_{\rm lower}\times^h \mathcal M_{v_{00}}\times^h \mathcal M_{v_{10}}

and compute its homotopy fibre. In the existing carrier language this is equivalent to constructing the normal--Cech enhanced AW collar and both endpoint columns on the literal Entry-143 stalk diagram, then solving

d_{\operatorname{Hom}}h =i_{\rm road}a\pi-\delta_E\Phi

without changing the already fixed coefficient map $\Phi$ .

Only this construction can decide whether the physical fibre is empty, contractible, or has the residual $\mathbb Z/2$ component.

Evidence

research/voevodsky/check_d03_face_restriction_strength_gate.py verifies:

the explicit one-road contraction;
the two endpoint quotient chain maps and their simultaneous kernel;
the restricted differential $(-x_0,x_1)$ ;
the Hilbert-function identity for the shifted ideal $(x_0,x_1)$ ;
absence of $x_3$ -torsion in the coefficient model; and
the nonzero higher-Rees sector missed by a first-symbol test.

The result also uses Entries 133, 136, 141, 143, 313, and 378--381. It does not construct $\mathfrak R_{03}$ , the physical admissible mapping complex, or the endpoint connector parity.

Outcome contract

{
  "claim": "The exact two-endpoint road coefficient kernel is the two-term complex R -> R^2 with differential (-x0,x1), so its only homology is the torsion-free rank-one ideal (x0,x1). Full generic x3-localization would kill every endpoint-invisible coefficient class, but the currently proved qJ value and first Cartier symbol do not define that full restriction on the unconstructed admissible mapping complex.",
  "status": "proved_scoped_well_posedness_gate",
  "factorization_test": {
    "ordinary_one_road_Hom": "acyclic",
    "endpoint_relative_H1": "zero",
    "endpoint_relative_H0": "(x0,x1), grading shifted",
    "integer_torsion": "none",
    "x3_torsion": "none",
    "strong_endpoint_plus_generic_kernel": "zero at coefficient level",
    "first_Rees_symbol_only": "insufficient; misses x3^2(x0,x1)",
    "physical_admissible_fibre": "uninstantiated"
  },
  "counterevidence": [
    "The affine rank-nine strict lift lattice is presentation freedom before admissible pointing.",
    "Ordinary Hom acyclicity uses homotopies forbidden by the physical support and Q framing.",
    "The known qJ coefficient is not a full generic restriction functor on arbitrary deformations."
  ],
  "next_experiment": "Construct the five restriction maps from one common filtered admissible Hom complex, or equivalently type the normal-Cech AW collar and both endpoint columns on the literal Entry-143 stalk diagram, then compute the homotopy fibre and its reflection parity."
}

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