Exact QTDS and Jordan Lift Audit

Record

Date: 2026-08-12

Status: all implemented exact checks pass. They verify the order-relative tree lift through eight points, complete six-point PT-basis reconstruction, six/eight-point cut products, one nested eight-point residue, and the special rectangular Jordan identity. They do not construct the missing twisted half-chain augmentation.

Reproducible artifact

Run:

python research/nima/check_qtds_lift.py

The script uses only Python’s standard library and exact rational arithmetic. Its Mandelstam data are generic points of the formal Gram-free massless space

[ s_{ij}=s_{ji}, \qquad s_{ii}=0, \qquad \sum_j s_{ij}=0. ]

It imports the independently audited scalar associated-grade implementation from check_j_reconstruction.py.

Quartic recursion

Rooting a planar quartic tree on the last external leg turns it into an ordered ternary tree. Each off-shell current contains an odd consecutive block, and every nontrivial current is assembled from three odd blocks. The recursion uses only:

  • the alternating (+,-,+,-) assignment;
  • the propagator (1/K^2);
  • the vertex (-2K_1\cdot K_3), with the momentum-crossing sign handled when the root slot has positive polarity.

The resulting tree counts are

[ N_4=1, \quad N_6=3, \quad N_8=12, \quad N_{10}=55, \quad N_{12}=273, ]

in agreement with the quartic-tree sequence quoted in the QTDS source.

Period comparison

For both global polarity assignments, the script verifies

[ A_n^{\rm QTDS}(\alpha,\varepsilon)

A_n^{\rm QTDS}(\alpha,-\varepsilon)

(-1)^{n/2-1}a_{R,n}(\alpha) ]

at:

Multiplicity Exact ordering samples Result
4 1 pass
6 2 pass
8 2 pass

Each of the six- and eight-point sets includes a noncanonical label ordering. The alternating sign is only the relative convention between the paper’s quartic vertex and the scalar-shift grade; it can be absorbed into the quartic coupling convention.

At six points the recursion also reproduces equation (6) of the QTDS paper exactly. The three individual tree contributions are not invariant under the global polarity flip, while their exact sum is invariant. This is direct evidence that polarity flip is an equivalence of the evaluated tree presentation, not a diagram-by-diagram redundancy.

Missing-choice audit

For a bare set of (n) labels, the number of unordered balanced bipartitions is

[ \frac12\binom{n}{n/2}. ]

The script records

[ 3, \quad 10, \quad 35 ]

at (n=4,6,8). None is a permutation-invariant distinguished choice. QTDS does not use an arbitrary balanced bipartition: a cyclic order selects its alternating double cover. The count is included only to expose how much coloring data the bare class has forgotten.

Rectangular Jordan audit

For exact rational matrices

[ x\in\operatorname{Mat}{2\times3}, \qquad y,z\in\operatorname{Mat}{3\times2}, ]

the script implements

[ Q_x(y)=xyx ]

and checks the typed fundamental formula

[ Q_{Q_x y}(z)

Q_x\bigl(Q_y(Q_x(z))\bigr) ]

entry by entry. The equality is exact.

Interpretation

The checks establish five useful facts:

  1. one finite quartic grammar evaluates to the scalar-derived ordered period at low multiplicity;
  2. the two fibers of the alternating cyclic cover have the same tree evaluation but different internal decompositions;
  3. the matrix QTDS realization obeys the Jordan coherence identity expected of its target algebra.
  4. both six-point polarity families reconstruct the same half-class from all six independent Parke–Taylor periods;
  5. audited cut residues split as products of lower QTDS periods, including an eight-point codimension-two corner.

They do not establish:

  • a canonical choice of order, polarity, or Jordan pair from (\mathsf J);
  • a chain map into twisted cohomology before PT pairing;
  • a sewing-stable quotient of presentation differences;
  • the all-Jordan-pair strictifiability classification;
  • an exact arbitrary-topology representative rather than a cut-equivalence class.

Next executable extension

The complete-basis and presentation-level residue tests are now implemented here. Entry 19 and check_qtds_descent.py construct the six-point local flip flow and the exact eight-point coherence complex. The remaining executable target is the generic Jordan-valued solution of its 24 local edge equations and the four square curvatures, followed by an augmentation into a scalar-normal twisted-chain model.