The Six-Point Residue Scalar and the Entry-Counit Gap

Record

Date: 2026-08-14

Epistemic-graph correction event:

ev-000000000023-43c37313-1d66-4da0-9b5f-b203e289a7b5

Status: exact typing and underdetermination theorem for the bounded residue problem left by entry 84. The occurrence-decorated PC residue of the saturated six-point tripod cannot be computed from the presently established maps. The missing datum is one physical-core entry counit on the edge which passes from the zero-core cell to the rank-one channel facet.

Strict residue vanishing is not the invariant target. The saturated half-sum and the original radial barycentric jump differ by an explicit rational two-chain boundary. Entry 38 likewise defines tubular/current realizations only up to filtered chain homotopy. The invariant question is whether the channel residue is null-homotopic, equivalently whether its class in the factorized four-point-by-four-point boundary line vanishes.

The available data reduce that class to one scalar per channel and sheet, related by $D_6$. They do not determine the scalar.

Local carrier at one channel

Fix a physical channel $D$ of the hexagon. Its associahedral facet is

[ F_D\cong K_4\times K_4, ]

a square. Let $T_D^\epsilon$ be the corner reached from the parity center $E_\epsilon$, and let $E_0,E_1$ be the two square edges incident to that corner.

Entry 84 replaces the nonsaturated radial jump

[ j_D^\epsilon

[b(T_D^\epsilon),b(F_D)] ]

[ \Lambda_D^\epsilon

\frac12\sum_{a=0}^{1} \left( [b(T_D^\epsilon),b(E_a)] +[b(E_a),b(F_D)] \right). ]

With

[ \tau_a=[b(T_D^\epsilon),b(E_a),b(F_D)], ]

the exact identity is

[ \boxed{ \Lambda_D^\epsilon-j_D^\epsilon

\partial\left(\frac12(\tau_0+\tau_1)\right). } ]

The two integral route choices differ by

[ \boxed{ p_0-p_1=\partial(\tau_0-\tau_1). } ]

Thus the saturated path is a canonical symmetric representative, but not a strictly distinguished point-set current after passage to PC.

Normal and coefficient typing

Put

[ u_D=q_D-1, \qquad h_D=\frac{\ell_D}{u_D}, \qquad \partial_{\mathscr L}\ell_D=u_Dp_D. ]

The scalar coefficient and the loading remain separate. A term carrying scalar polynomial $c_D(X)$ has local type

[ c_D(X)\otimes h_D\otimes\operatorname{or}(N_D). ]

There is no substitution $X_D\mapsto u_D$. Reversing the normal orientation changes the sign of both $h_D$ and the Gysin contraction.

The saturated tail inside $F_D$ is completely typed by codimension-one Cousin maps. The incoming edge

[ E_\epsilon \longrightarrow T_D^\epsilon ]

is different: it changes the physical core from empty to ${D}$. Its residue requires a natural transformation from the zero-core occurrence module to the $D$-facet module, including the endpoint Cousin term, $h_D$, and the scalar Laurent specialization.

Entries 32 and 37 do not supply this map. Entry 32 defines the physical coaction after a directed physical edge and its source slots are already present. Entry 37 proves base change for an independent scalar-refinement factor. The present entry edge is neither: it is the core-changing incidence which creates the physical channel in the tripod.

Call the missing map

[ \epsilon_D^{\rm entry}. ]

Without it, adding the saturated tail, its Cousin lower terms, and the normal factor does not produce a defined occurrence-decorated residue.

Signed codimension-one ledger for one leg

Let $L_D^\epsilon$ be the physical flip edge joining $E_\epsilon$ to $T_D^\epsilon$. The complete saturated leg is

[ \begin{aligned} \widetilde\gamma_D^\epsilon ={}&[E_\epsilon,b(L_D^\epsilon)] +[b(L_D^\epsilon),T_D^\epsilon]\ &+\frac12\sum_{a=0}^{1} \left( [T_D^\epsilon,b(E_a)] +[b(E_a),b(F_D)] \right). \end{aligned} ]

All six displayed edges are codimension-one incidences. With every edge oriented as written, their signed upper/tangential and lower/Cousin endpoints are

oriented edge	upper/tangential endpoint	lower/Cousin endpoint	physical core
$[E_\epsilon,b(L_D^\epsilon)]$	$+b(L_D^\epsilon)$	$-E_\epsilon$	empty
$[b(L_D^\epsilon),T_D^\epsilon]$	$-b(L_D^\epsilon)$	$+T_D^\epsilon$	empty $\to{D}$
$\frac12[T_D^\epsilon,b(E_a)]$	$+\frac12b(E_a)$	$-\frac12T_D^\epsilon$	${D}$
$\frac12[b(E_a),b(F_D)]$	$+\frac12b(F_D)$	$-\frac12b(E_a)$	${D}$

The two values $a=0,1$ are both present. Consequently $b(L_D^\epsilon)$ cancels, each $b(E_a)$ cancels, and

[ +T_D^\epsilon-\frac12T_D^\epsilon-\frac12T_D^\epsilon=0. ]

This is strict in the undecorated incidence complex. In PC, the last two terms land in the already established $D$-facet occurrence summand and carry

[ c_D(X)\otimes h_D\otimes\operatorname{or}(N_D). ]

The first $T_D^\epsilon$ term is the Cousin lower term of the oppositely oriented entry edge. It lands in that same summand only after applying $\epsilon_D^{\rm entry}$. Thus the displayed three-term cancellation is precisely the unproved occurrence-decorated square; no other codimension-one term is missing from the saturated leg.

For $g\in D_6$, the finite incidence chain obeys

[ g\widetilde\gamma_D^\epsilon

\widetilde\gamma_{gD}^{g\epsilon}. ]

A one-step rotation has $g\epsilon=-\epsilon$ and cycles the three channels. After choosing ordered normal lines, the loaded residue transforms by the normal-orientation character $\chi_N(g)$:

[ g[r_D^\epsilon]

\chi_N(g)[r_{gD}^{g\epsilon}]. ]

There is no additional deck sign in the incidence chain. The only residual sign is $\chi_N(g)$; assigning it a numerical value without fixing the ordered normal convention would be spurious.

What is nevertheless fixed

Let

[ \eta_6^{\epsilon,{\rm PC}} ]

denote any PC realization of the saturated tripod extending the established codimension-one terms. Since the two QTDS presentations have identical physical residue on every channel,

[ \operatorname{Res}^{\rm PC}D d{\rm PC}\eta_6^{\epsilon,{\rm PC}}=0. ]

Therefore

[ r_D^\epsilon := \operatorname{Res}^{\rm PC}_D \eta_6^{\epsilon,{\rm PC}} ]

is closed whenever $\epsilon_D^{\rm entry}$ makes residue a chain map.

Entry 77 identifies the induced maximally factorized boundary object with the primitive line

[ \mathcal J_4\boxtimes\mathcal J_4

\mathbf k_{\rm nr} \langle g_4\boxtimes g_4\rangle. ]

Hence there is a scalar

[ \boxed{ [r_D^\epsilon]

\lambda_D^\epsilon [g_4\boxtimes g_4]. } ]

One-step rotation sends $D$ to the next channel and exchanges the two parity sheets. Reflection fixes the symmetric saturated tail. Thus the six numbers $\lambda_D^\epsilon$ belong to one $D_6$-orbit, with only the normal-orientation character changing signs. Computing one representative determines all six.

Boundary, deck covariance, and the local Pochhammer identity do not determine its value. If a closed lift with nonzero primitive boundary residue is available, the replacement

[ \eta_6^{\epsilon,{\rm PC}} \longmapsto \eta_6^{\epsilon,{\rm PC}}+\kappa_D, \qquad d_{\rm PC}\kappa_D=0, ]

changes $\lambda_D^\epsilon$ while preserving the endpoint boundary. The established axioms do not exclude such a lift and do not provide a period that fixes its coefficient. Scalar provenance must select the value through $\epsilon_D^{\rm entry}$.

Strict zero versus null-homotopy

Because

[ \Lambda_D^\epsilon-j_D^\epsilon

\partial\left(\frac12(\tau_0+\tau_1)\right), ]

changing between the radial and saturated representatives adds an explicit boundary. A literal chain equality

[ r_D^\epsilon=0 ]

is therefore not invariant under the allowed filtered chain-homotopy freedom.

The correct condition is

[ \boxed{ [r_D^\epsilon]=0, } ]

or, with a chosen representative, an explicit homotopy

[ r_D^\epsilon=d_{\rm PC}s_D^\epsilon. ]

If $\lambda_D^\epsilon=0$, the residue is null-homotopic and the six-point contact primitive belongs to the derived residue-free fiber. If it is nonzero, the primitive has a genuine lower factorized obstruction. A point-set strict zero can be imposed only after choosing the null-homotopy; it is not primary data.

Consequence for eight points

The formal pole-grade primitive remains

[ H_8^{\rm PC}

\sum_D G_D^{\rm PC} (\eta_6^{\rm PC}) +H_{\rm ct}^{\rm PC}. ]

Its coefficient boundary is exactly

[ \sum_Q(q_Q^+-q_Q^-) ]

because entry 23 exhausts the full symbol by $G$, $R$, and $K$. The marked $K$ part is closed by entry 83. The $R$ part is factorization-natural in the derived PC category if and only if

[ \lambda_D^\epsilon=0. ]

There is still no extra unmarked coefficient remainder. The remaining obstruction is the single six-point entry-counit scalar and, separately, the stronger global problem of gluing local quadrangulation half-lines with Jordan higher coherence.

Exact finite content

The finite carrier identities are certified by

research/nima/check_six_point_subdivision_pc.rs

with SHA-256

46021191c34034bf4cd64f5f80e6fe9f0fb39316f86b263fdfeaae9785a310d4

The checker proves the saturated paths, their $D_6$ covariance, and the explicit rational and integral two-chain fillers. It deliberately does not assign a value to $\epsilon_D^{\rm entry}$ or $\lambda_D^\epsilon$.

Decision

Reject:

The established PC map proves that the six-point tripod residue is strictly zero.

Also reject:

Nonzero physical-facet support by itself proves a nonzero residue class.

Promote:

The invariant six-point obstruction is one scalar multiplying the primitive $K_4\times K_4$ boundary line. Strict zero is representative dependent; null-homotopy is the correct condition.

Retain as the next bounded experiment:

Construct $\epsilon_D^{\rm entry}$ from the occurrence-resolved scalar Laurent grade for one center-to-channel edge and pair its residue with the dual $g_4^\vee\boxtimes g_4^\vee$. This computes $\lambda_D^\epsilon$. Rotation and reflection then determine every channel and sheet.

Internal dependencies

Entries 20–21: scalar contact redistribution and tripods.
Entry 23: exhaustive eight-point pole-grade decomposition.
Entries 32 and 37: domains of the established physical coaction and transverse base change.
Entry 38: PC normal factors and filtered representative strength.
Entry 77: primitive factorized boundary half-line.
Entries 83–84: marked octagon and saturated tripod correction.
research/nima/check_six_point_subdivision_pc.rs.