Scalar-Cell Escape from the Amplitude-Level No-Go

Record

Date: 2026-08-12

Status: the summed scalar amplitude and its Laurent-leading coefficient provably underdetermine the QTDS polarity homotopy. The canonical scalar cubic-tree presentation retains more information and yields an exact six-point escape: QTDS is the unique alternating local redistribution of two parity-central scalar grade cells among three quadrangulation fibers. Entry 21 promotes this to a genuine presentation-cellular tripod lift. At eight points, parity-core cells form a Möbius carrier whose boundary is the unresolved octagon. The octagonal coherence and scalar-to-twisted comparison remain open.

Correction to the normal-language shorthand

For a fixed cyclic order (\alpha), define

[ F_{\alpha,+}(X,t)

A^{\operatorname{Tr}\Phi^3} \left( X+\frac{\sigma^\alpha}{t} \right), \qquad t=\delta^{-1}, ]

where (\sigma^\alpha) gives opposite shifts to even-even and odd-odd planar variables and leaves odd-even variables unshifted. At six points,

[ F_{\alpha,+}(X,t)

t^4a_{R,6}(\alpha;X)+O(t^5). ]

The opposite alternating coloring is (t\mapsto-t), so it has the same fourth initial coefficient.

This supplies a valid (t)-adic initial form. By itself it does not define:

• a kinetic bundle map whose rank jumps;
• a stratified parameter space (B_6) and rank locus (R_6);
• a constructible or filtered scalar chain complex near (t=0);
• Verdier specialization of such a complex.

Accordingly, “rank-jump associated grade” remains Marici’s physical-geometric interpretation of the scalar large-mass limit. At chain level the established source object is presently a Rees or Laurent grade of rational scalar data.

Amplitude-level no-go

At six points, write the two QTDS presentations as

[ q_\pm

\left( \frac{N_0^\pm}{Y_0}, \frac{N_1^\pm}{Y_1}, \frac{N_2^\pm}{Y_2} \right) \in V=K_6^3, ]

where (Y_i) are the three physical odd-block channels. The summed scalar amplitude sees only

[ \epsilon:V\to K_6, \qquad \epsilon(v_0,v_1,v_2)=v_0+v_1+v_2. ]

The established equality says

[ \epsilon(q_+)=\epsilon(q_-)=a_{R,6}. ]

Thus

[ c=q_+-q_-\in\ker\epsilon. ]

No functor of only the summed amplitude and its leading Laurent coefficient can recover (c): contact redistribution changes (c) without changing (\epsilon(q_\pm)). The triangle flow solving (\partial h_6=c) is even less determined. Its solutions form an affine line over

[ H_1(C_3;K_6)\simeq K_6. ]

The zero-circulation inverse-Laplacian solution uses the triangle metric, which the summed amplitude also forgets. This is an information-theoretic obstruction, not merely failure to guess a formula.

Why the one-parameter normal link does not repair it

The minimal complex base for the documented shift is

[ U_X\times\Delta_t, \qquad R=U_X\times{0}. ]

Its complex normal link is an unmarked circle; on the real slice it is (S^0). The three NLSM channels are tangential divisors in (X), not three marked normal directions in (t). A triangle is a possible cellulation of a circle, but its channel labels and equal-edge metric are extra data. Hence the QTDS flip triangle is not the natural link of the documented one-parameter shift.

Verdier specialization requires

[ (B_6,R_6,\mathscr K_6), \qquad \mathscr K_6\in D_c^b(B_6), ]

together with an extension or lattice and monodromy. It is not defined by a rational function alone. The notation (\operatorname{Sp}_{R_6}(\mathscr K_6)) remains a target type, not an already constructed scalar object.

The scalar-cell escape hatch

The scalar master at tree level has more structure than its sum: it is canonically presented by planar cubic trees, equivalently polygon triangulations. Retain those cells while taking the fourth (t)-grade.

For the hexagon:

• twelve scalar triangulations contain one allowed odd-block diagonal;
• each of the three quadrangulations has exactly four cubic refinements;
• two scalar triangulations contain no allowed diagonal.

The last two cells are the parity-central triangulations. One contributes

[ C_{\rm odd}=-(x_1+x_3+x_5); ]

the other contributes

[ C_{\rm even}=-(x_0+x_2+x_4), ]

where (x_i=s_{i,i+1}). Thus the unresolved scalar contact is

[ C=C_{\rm odd}+C_{\rm even}

-\sum_{i=0}^{5}x_i. ]

This information is lost by the final scalar sum but retained by the cubic-cell grade.

Exact QTDS redistribution

Let (G_i) be the fourth scalar grade summed over the four cubic refinements of

[ D_i=(i,i+1,i+2), \qquad i=0,1,2. ]

The exact symbolic identities are

[ \begin{aligned} q_0^+&=G_0-(x_3+x_4),& q_0^-&=G_0-(x_0+x_1),\ q_1^+&=G_1-(x_1+x_2),& q_1^-&=G_1-(x_4+x_5),\ q_2^+&=G_2-(x_5+x_0),& q_2^-&=G_2-(x_2+x_3). \end{aligned} ]

For either polarity, the three contact pairs partition the six boundary variables exactly once:

[ \sum_i(q_i^\pm-G_i)=C. ]

The two global choices are the two cyclic perfect matchings between odd and even boundary contacts. They are exchanged by one-step rotation. The alternating cyclic lift therefore supplies precisely the program datum required to distribute the two parity-central scalar cells.

The diagramwise polarity difference is

[ \begin{aligned} c_0&=(x_0+x_1)-(x_3+x_4),\ c_1&=(x_4+x_5)-(x_1+x_2),\ c_2&=(x_2+x_3)-(x_5+x_0), \end{aligned} ]

with (c_0+c_1+c_2=0). The canonical cyclic zero-circulation flow is

[ H_{ij}=\frac{c_i-c_j}{3}. ]

This proves:

The six-point QTDS polarity comparison is scalar-derived from the cell-resolved fourth grade plus the alternating cyclic lift. It is not derivable from the summed scalar amplitude.

The result is exact in the nine-variable formal planar kinematic space.

The parity-core forgetting map

For a scalar triangulation (T) of an even polygon, retain only its allowed diagonals:

[ \pi_{\rm core}(T)

{D\in T:D\text{ splits the polygon into even subpolygons}}. ]

This is a noncrossing partial quadrangulation. It gives a canonical combinatorial map from scalar associahedral cells to the partial-quadrangulation or Fuss–Catalan complex.

The exact distributions by core size are

[ n=6:\quad{1:12,\ 0:2}, ]

[ n=8:\quad{2:96,\ 1:32,\ 0:4}, ]

[ n=10:\quad{3:880,\ 2:440,\ 1:100,\ 0:10}. ]

At full core, every quadrangulation of a (2m)-gon has

[ 2^{m-1} ]

cubic refinements, one choice of diagonal in each quadrilateral. This gives (4) refinements at six points and (8) at eight points.

Eight-point scalar origin of the coherence faces

At eight points:

• each of the twelve quadrangulations has eight scalar cubic refinements;
• each of the eight one-channel cores has four scalar triangulations;
• there are four zero-core scalar triangulations.

Each zero-core triangulation has one central same-parity diameter:

[ (0,4), \qquad (1,5), \qquad (2,6), \qquad (3,7). ]

For each diameter, the four compatible quadrangulations are exactly the four vertices of one square in the projective-plane medial complex of entry 19. Thus the four square coherence faces are canonically labeled by four scalar central cells.

Each one-channel core is labeled by one of the eight physical diagonals and belongs to the corresponding factorization triangle. This gives a scalar associahedral explanation for

[ 8\text{ factorization triangles} + 4\text{ square cells}. ]

The remaining global octagon is not yet derived from the core map. It is the natural location for the relation among parity-central transfers, cyclic rotation, and nontrivial (\mathbb Z_2) holonomy.

What is and is not now intrinsic

Established at scalar-cell presentation level:

1. the parity-core map;
2. full quadrangulation fibers;
3. the two six-point parity-central scalar cells;
4. their exact QTDS contact redistribution;
5. the local six-point deck flow;
6. the eight one-channel/factorization labels;
7. the four central-diameter/square labels.

Still missing after the presentation-cellular construction of entry 21:

1. the global eight-point octagonal filler and its integral/sign-local-system test;
2. Jordan-valued edge syzygies compatible with the scalar carrier;
3. a filtered facewise map into scalar-normal or worldsheet twisted chains;
4. a residue-free twisted image of the six-point scalar tripod;
5. a chain-level inverse of the scalar pairing at resonant boundaries.

The naive alternating sum of four Jordan-decorated vertices around a square is nonzero even in a special associative Jordan pair. The square filler must use edge syzygies; incidence and vertex bracketings alone do not produce the Jordan identity.

Minimal Frost and YM handoff

Frost should construct or obstruct a filtered scalar/surface complex

[ (C^\bullet_{n,\alpha},d,F_t^\bullet,\Sigma_{n,\alpha}(t)) ]

whose associated grade retains the parity-core decomposition, together with

[ \iota_\alpha: C_\bullet(\mathfrak M_n;\mathcal L_\alpha) \longrightarrow \operatorname{gr}t C^\bullet{n,\alpha} ]

compatible with cuts. The map should be sought as pushforward or homotopy transfer along (\pi_{\rm core}), not invented from an unmarked one-parameter link.

YM should construct or obstruct

[ \Phi_\alpha: \operatorname{gr}t C^\bullet{n,\alpha} \longrightarrow (\Omega^\bullet(\mathcal M_{0,n}),\nabla_\omega) ]

and a chain-level scalar duality map. At six points it must send the exact cell-resolved flow to

[ \nabla\eta_6=\omega_{6,+}-\omega_{6,-} ]

and intertwine every channel residue with the Alexander–Whitney coproduct.

Reproducible audit

Run:

python research/nima/check_qtds_descent.py

The script checks the six-point Laurent identities, scalar parity-core counts through ten points, the eight-point refinement fibers, the central-diameter/square correspondence, and the previously recorded projective-plane and Jordan-endpoint tests.

Sources and provenance

• Arkani-Hamed, Cao, Dong, Figueiredo, and He, NLSM inside Tr(Phi cubed) supplies the one-parameter alternating shift, large-(\delta) grade, scalar cubic-tree factorization, and broader mixed-shift landscape.
• Cao, Han, and Zhu, NLSM amplitudes from a quartic two-derivative theory supplies the target QTDS tree grammar.

The parity-core map, exact scalar-cell contact decomposition, and eight-point square labeling are Marici calculations.

Decision

Reject the statement that the documented summed amplitude intrinsically contains a QTDS chain homotopy.

Promote the more precise positive statement:

The scalar cell-resolved associated grade, enriched by the alternating cyclic lift, contains the complete six-point QTDS contact redistribution and exposes the correct eight-point coherence skeleton.

The primary frontier is now to turn this parity-core transfer into a cut-compatible chain map and to determine whether its Jordan-valued edge syzygies and octagonal holonomy exist.