Direct Mixed-Cell Generator and Fourteen-Point Rust Stress Test

Record

Date: 2026-08-13

Status: the mixed (K_2\times I) carrier is now generated directly from the rooted dependency chains, without enumerating the ambient associahedron. An independent optimized Rust audit reproduces every aggregate Python result at ten and twelve points and extends the complete decorated test to fourteen points.

No asymmetric cut support or nonzero occurrence-level curvature was found.

Local generator

Let (c_r) be a rooted dependency chain with at least two unperformed flips and let (c_i) be an independent active chain. At a common prefix state (x), define

[ z

\operatorname{flip}_{c_r[p+1]}x, ]

using the second upcoming flip of the repeated chain, and

[ u

\operatorname{flip}_{c_i[q]}x. ]

Independence of the chains gives

[ v

\operatorname{flip}_{c_i[q]}z. ]

Then

[ x\longleftrightarrow z ]

is the lower scalar-refinement edge,

[ u\longleftrightarrow v ]

is its upper scalar-refinement edge, and the common replacement of the independent flip is the physical cut (e).

This constructs the scalar fixed-diagonal facet of the pentagonal prism directly as the missing (K_2\times I) product square. It requires no rank-three face index and no global triangulation census.

The generator reproduces exactly:

20 distinct mixed squares from 40 marked occurrences at (n=10);
336 distinct mixed squares from 720 marked occurrences at (n=12);
3,920 distinct mixed squares from 8,820 marked occurrences at (n=14).

The fourteen-point carriers occur at base-core degrees zero, one, and two.

Independent Rust audit

The executable

research/nima/check_mixed_prism_all_arity.rs

reimplements from scalar polygon data:

zero-core scalar sources;
marked rooted dependency chains;
the direct (K_2\times I) mixed-cell generator;
regional contact-mark choices;
regional marked Catalan forward transfer;
general directed edges and componentwise forest sinks;
physical cut support and two-slot data;
regional inverse Catalan descent;
reconstruction of both upper scalar edges;
exact one-step deck covariance of the full decorated atlas.

It does not call the Python implementation and does not import a QTDS table.

Compile and run:

rustc --edition=2021 -O research/nima/check_mixed_prism_all_arity.rs -o "$env:TEMP\marici-mixed-prism.exe"
& "$env:TEMP\marici-mixed-prism.exe"

Regression agreement

At ten points Rust reproduces the Python totals exactly:

[ 120

50_{\rm supported} + 70_{\rm common\ zero}. ]

At twelve points it reproduces:

[ 2568

1092_{\rm supported} + 1476_{\rm common\ zero}. ]

The Rust profile retains the cyclic order of component-region sizes, whereas the Python report sorts those sizes. After aggregation, the totals and supported/absent partition agree exactly.

Fourteen-point result

The optimized run completes the full fourteen-point decorated atlas in approximately (1.3) seconds on the current host.

It finds:

[ 84 ]

zero-core scalar sources,

[ 3920 ]

distinct mixed squares from

[ 8820 ]

rooted-chain occurrences, and

[ 39228

16898_{\rm supported} + 22330_{\rm common\ zero} ]

decorated transports per polarity.

For every decorated transport:

cut support agrees at the two lower endpoints;
if absent, both routes vanish;
if present, the cut source quadrilateral and both scalar slots agree;
each slot reconstructs a genuine scalar edge at the enlarged core;
exactly one reconstructed edge is the forced prism edge;
exactly one is the parallel slot edge;
the coefficient-level cellular curvature has empty support;
one-step rotation gives the corresponding opposite-polarity record exactly.

This includes all base-core-degree-two spectator profiles available at fourteen points.

Structural interpretation

The direct construction explains why the mixed relation is local.

The scalar refinement belongs to the (K_2) factor generated by two consecutive dependent flips. The physical cut belongs to the independent (I) factor. Their carrier-level interchange is therefore a product square.

The coefficient statement has additional content: the marked Catalan transfer must preserve support and the source quadrilateral of the (I)-factor flip. The audits show that it does so through fourteen points, even though the target quadrilateral may slide.

The two-slot formula remains

[ G_e(h)

-\frac{X_{d_e^0}}{X_e}h_e^0

\frac{X_{d_e^1}}{X_e}h_e^1, ]

with

[ \partial G_e(h)=G_e(\partial h). ]

Epistemic status

Established computationally and independently through fourteen points:

the local rooted-chain generator;
all decorated support tests;
all upper-edge reconstructions;
spectator stability through base degree two;
exact deck covariance;
zero occurrence-level mixed curvature.

Strongly indicated, but not yet written as a formal all-arity proof:

Independent rooted dependency-chain factors remain independent under regional marked Catalan transfer; hence physical cut support, source slots, and the two reconstructed upper edges are natural at arbitrary even arity and arbitrary spectator core.

The proof should be a local rooted-dual-tree argument, not an induction over enumerated triangulations.

Remaining frontier

The evidence no longer points toward a scalar-cellular obstruction. The next serious tests are:

prove the rooted-tree naturality lemma at all arity;
construct the finite-nonresonant-(\alpha’) loaded Pochhammer/Cousin image of the cellular coaction;
prove that the filtered comparison commutes with physical specialization;
identify its derived worldsheet class with ((\operatorname{Pf}’A)^2);
test pairing compatibility before cohomology.

Decision

Promote:

The universal mixed prism is generated directly by one dependent-chain scalar flip and one independent-chain physical flip. Its decorated Beck–Chevalley curvature vanishes in exhaustive, independently implemented tests through fourteen points, including all spectator cores of degree at most two.

Do not yet promote:

An all-arity twisted-chain half-object has been constructed.

That statement still requires the rooted-tree proof and the filtered worldsheet comparison.