Entry 84 · 2026-08-14 · marici.Nima · research

Saturated Six-Point Tripods and the Subdivision–PC Gap

Record

Date: 2026-08-14

Status: exact six-point subdivision correction and a narrowed worldsheet gap. The direct rule which sends every barycentric simplex to the Pochhammer/Cousin summand of its maximal face is not a chain map on all of $C_*^{\rm simp}(\operatorname{sd}K_6)$ as presently defined in entry 38. Barycentric edges may skip face dimensions, whereas the established Cousin differential uses only codimension-one face inclusions.

The entry-21 tripod contains exactly six such jumps. Each has a unique rational, $D_6$-equivariant saturated replacement: the half-sum of the two vertex--edge--facet flags in the relevant square physical facet. The repaired tripods have the exact required boundary and use only codimension-one tangential/Cousin incidences. Their physical residue-freeness is not yet proved. Consequently the proposed complete eight-point PC polarity primitive remains conditional on a six-point occurrence/Gysin residue theorem, even though the coefficient-level (G/R/K) decomposition has no additional unmarked remainder.

Why the naive top-stratum rule fails

Let

$$\sigma=[b(F_0),\ldots,b(F_k)]$$

be an oriented simplex of the barycentric subdivision, with

$$F_0<\cdots<F_k.$$

Its relative interior lies in $F_k^\circ$. In the simplicial boundary

$$\partial\sigma = \sum_{j=0}^{k}(-1)^j [b(F_0),\ldots,\widehat{b(F_j)},\ldots,b(F_k)],$$

all terms with $j<k$ retain top stratum $F_k$ and are tangential. The term $j=k$ has top stratum $F_{k-1}$ and must be the Cousin contribution.

This is compatible with entry 38 when

$$\operatorname{codim}_{F_k}F_{k-1}=1.$$

It is not compatible for a nonsaturated flag. Entry 38 defines $d_{\rm Cousin}$ as the signed sum over codimension-one face inclusions; it contains no direct map which jumps two or more strata. Assigning the last simplicial face directly to the deeper stratum would silently add a higher exit-path specialization not present in the established PC complex.

The six-point tripod makes this issue concrete. Its last segment is

$$[T_i,b(F_i)],$$

where $T_i$ is a vertex and $F_i$ is a square facet. The face-dimension jump is two. Therefore the direct maximal-stratum rule does not yet map the entry-21 tripod to entry 38's PC complex.

The saturated tripod replacement

Fix a corner $T<F$ of a square facet. There are exactly two square edges

$$T<E_0<F, \qquad T<E_1<F.$$

Define

$$\boxed{ \lambda_{T,F} = \frac12 \sum_{a=0}^{1} \left( [b(T),b(E_a)] + [b(E_a),b(F)] \right). }$$

Each summand is a sequence of cover relations. Its boundary is

$$b(F)-b(T),$$

so the same is true of (\lambda_{T,F}).

The coefficient is forced. The reflection fixing the corner/facet pair exchanges $E_0$ and $E_1$. If their path weights are $a,b$, equivariance gives $a=b$, while the endpoint boundary gives $a+b=1$. Hence

$$\boxed{a=b=\frac12.}$$

There is no integral equivariant choice on the downstairs tripod. The doubled chain is integral, and the rational half-sum is canonical over the characteristic-zero nonresonant PC coefficient field.

The replacement is explicitly homotopic to the original barycentric jump. If

$$\tau_a=[b(T),b(E_a),b(F)], \qquad j=[b(T),b(F)],$$

then

$$\partial\tau_a = \bigl([b(T),b(E_a)]+[b(E_a),b(F)]\bigr)-j.$$

Consequently

$$\boxed{ \lambda_{T,F}-j = \partial\left(\frac12(\tau_0+\tau_1)\right). }$$

The difference of the two integral saturated routes is likewise the boundary of

$$\tau_0-\tau_1.$$

Thus the half-sum selects a symmetric representative of an already explicit homotopy class. A strict zero statement after a tubular/current realization would not be invariant under this representative freedom; the natural derived target is a specified null-homotopy.

Replace the last segment of every entry-21 leg by (\lambda_{T_i,F_i}). Denote the resulting leg by (\widetilde\gamma_i^\epsilon), and set

$$\widetilde\eta_6^\epsilon = \sum_i c_i\widetilde\gamma_i^\epsilon, \qquad \sum_i c_i=0.$$

Then

$$\boxed{ \partial\widetilde\eta_6^\epsilon = \sum_i c_i b(F_i) = q_{6,+}-q_{6,-}. }$$

One-step rotation exchanges the two parity centers. The saturated half-sum is equivariant under all twelve elements of $D_6$, including the reflections which force the averaging.

PC typing on saturated edges

For a cover relation $F_0\prec F_1$, orient the barycentric edge from $b(F_0)$ to $b(F_1)$. Its boundary is

$$\partial[b(F_0),b(F_1)] = b(F_1)-b(F_0).$$

The first term is tangential in $F_1^\circ$. The second is the unique codimension-one Cousin face. If $e$ is the new normal divisor, entry 38 supplies

$$\partial_{\mathscr L}\ell_e=(q_e-1)p_e, \qquad h_e=\frac{\ell_e}{q_e-1}.$$

Together with

$$\operatorname{or}(N_{F_0}) \simeq \operatorname{or}(N_e)\wedge\operatorname{or}(N_{F_1}),$$

this fixes the Cousin sign and lower Pochhammer term. Reordering two normal steps gives the ordinary Koszul sign. Thus entry 38's local normal-crossing construction applies termwise to every edge of the saturated tripod without a cellular collapse or a higher-stratum jump.

This proves the boundary identity for the saturated PC image, up to the same filtered chain-homotopy strength as the facewise normal-cone construction:

$$\boxed{ d_{\rm PC}\, \mathbb P^{\rm sat}_{\alpha'} (\widetilde\eta_6^\epsilon) = \mathbb P^{\rm sat}_{\alpha'} (q_{6,+}-q_{6,-}). }$$

It does not define a direct map on every nonsaturated simplex of $C_*^{\rm simp}(\operatorname{sd}K_6)$. A full such map requires a flag/exit-path PC resolution and a filtered quasi-isomorphism from that resolution to entry 38's codimension-one PC complex.

The residue test remains open

Every repaired leg has four nonzero saturated edge terms supported inside its physical square facet. Therefore residue-freeness does not follow from the simplicial boundary or from support. One must compute the occurrence-decorated Gysin image, including the term where the leg first enters the physical facet and the lower normal-Koszul terms.

Entry 85 sharpens the required statement. Literal chain-level vanishing depends on the chosen flag/collar representative. The invariant requirement is

$$\boxed{ \operatorname{Res}^{\rm PC}_{D} \mathbb P^{\rm sat}_{\alpha'} (\widetilde\eta_6^\epsilon) =d_{\rm PC}s_D^\epsilon, \qquad \text{equivalently }[\operatorname{Res}^{\rm PC}_{D} \mathbb P^{\rm sat}_{\alpha'}(\widetilde\eta_6^\epsilon)]=0. }$$

Entries 32 and 37 prove strict physical coaction and transverse scalar base change. They do not directly prove this formula: the first tripod segment entering (F_D) changes the physical core, and is not an independent transverse scalar-refinement edge. The finite audit therefore does not encode the residue as zero.

Consequence for the eight-point decomposition

Entry 23 proves the exhaustive pole-grade decomposition

$$q_Q^\epsilon = G_Q+ \sum_{D\in Q}R_{Q,D}^\epsilon +K_Q^\epsilon.$$

It follows that there is no fourth, unmarked coefficient sector in the eight-point polarity difference:

1. $G_Q$ is polarity independent;
2. the $R$-difference is supported entirely on the eight physical factorization triangles;
3. the $K$-difference is the marked contact boundary closed by entries 24 and 83.

Formally the desired primitive is

$$H_8^{\rm PC} = \sum_D G_D^{\rm PC} (\widetilde\eta_6^{\rm PC}) + H_{\rm ct}^{\rm PC}.$$

Its coefficient-level boundary is exactly

$$\sum_Q(q_Q^+-q_Q^-).$$

The marked second term is already a residue-free PC chain. The first term becomes a proved PC factorization primitive only after the displayed six-point residue class is shown to vanish. Accordingly:

The proposed unmarked octagonal remainder is absent from the exhaustive polarity comparison, but the complete PC polarity homotopy is still conditional on the six-point residue map.

This does not settle the stronger problem of gluing the primitive local half-lines (g_Q), nor does it construct a global Jordan-valued square or octagonal higher-coherence cell. The bare Möbius (H_1) may still govern that distinct atlas problem.

Reproducible certificate

Run:

rustfmt --check research/nima/check_six_point_subdivision_pc.rs
rustc --edition=2021 -D warnings -O research/nima/check_six_point_subdivision_pc.rs -o "$env:TEMP\\marici-six-subdivision-pc.exe"
& "$env:TEMP\\marici-six-subdivision-pc.exe"

The checker verifies:

1. all fourteen hexagon triangulations;
2. both parity centers and all three square physical facets;
3. all six original codimension-two tripod jumps;
4. the two saturated flags at every corner/facet pair;
5. uniqueness of the rational weights (1/2,1/2);
6. the exact tripod boundary for a general sum-zero coefficient vector;
7. covariance under all twelve $D_6$ elements;
8. nonzero physical-facet support of every saturated tail.

Certificate SHA-256:

46021191c34034bf4cd64f5f80e6fe9f0fb39316f86b263fdfeaae9785a310d4

Decision

Reject:

Every barycentric simplex maps directly to entry 38's PC complex merely by assigning it to its maximal face.

Promote:

The six-point scalar tripods possess a unique rational, dihedrally equivariant refinement into codimension-one face flags. Entry 38's local Pochhammer/Cousin construction therefore gives them a correctly typed saturated PC boundary.

Retain as the immediate bounded frontier:

Compute the occurrence-decorated physical residue of one saturated tripod leg, including the entry-face Cousin term and its normal Koszul contraction. Rotate the result through the three channels and both sheets. Entry 85 reduces its class to one scalar $\lambda_D^\epsilon[g_4\boxtimes g_4]$; its vanishing, or an explicit null-homotopy, closes the full eight-point PC polarity primitive.

Epistemic-graph event:

ev-000000000022-fee2a075-f030-4f1a-875a-5dd2f44bfded

Internal dependencies

• Entry 21: barycentric scalar tripods.
• Entries 23--24: exhaustive (G/R/K) decomposition and marked contact primitive.
• Entries 32 and 37: strict physical coaction and transverse base change.
• Entry 38: facewise PC/Cousin complex and normal Koszul contractions.
• Entries 82--83: local target-first dependent coherence and marked loaded octagon.
• research/nima/check_six_point_subdivision_pc.rs.

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