Low-Point Transmutation Counit and Metric-Adjoint Underdetermination

Record

Date: 2026-08-13

Status: exact four- and five-point field-theory audit complete. The augmented lowering map is cyclically reference-independent after quotienting by the annihilator of the canonical Yang–Mills amplitude. It is a transmutation counit/coframe identity. The proposed metric adjoint is not determined by the published data and remains a distinct chain-level problem.

Reproducible certificate:

research/nima/check_low_point_transmutation.rs

Question

Entry 40 isolated three possible outcomes for the low-point relation between scalar scaffolding and the new scaffold differential operators:

  1. a genuine metric adjunction;
  2. a weaker augmented counit/coframe identity;
  3. irreducible dependence on the cyclic scaffold reference.

The proposed adjoint was

[ J^\dagger_{\rm metric}

I_S^{-1}J^{\mathsf T}I_G. ]

The finite audit gives a sharp, but qualified, answer:

Outcome 2 is realized on the canonical tree Yang–Mills amplitude at four and five points. Outcome 3 is absent after augmentation and the canonical amplitude quotient. Outcome 1 is not a matrix that the published constructions presently determine.

This is not merely a failure to finish a large matrix computation. The input needed to define that matrix is absent.

The intrinsic-looking lowering operator

Backus and Figueiredo define

[ \mathcal W_{2i}[F]

\sum_{j\notin{2i,2i\pm1}} \frac{\partial F}{\partial X_{2i,j}}. ]

Let

[ E_n={2,4,\ldots,2n} ]

be the even scaffold labels. Their tree-level, low-energy theorem states that for any two distinct labels (e_1,e_2\in E_n),

[ \left( \prod_{e\in E_n\setminus{e_1,e_2}} \mathcal W_e \right) A_n^{\rm YM}

X_{e_1,e_2}A_n^{\operatorname{Tr}\phi^3}. ]

The scalar amplitude on the right depends only on odd–odd variables. Since (X_{e_1,e_2}) occurs with unit coefficient in both (\mathcal W_{e_1}) and (\mathcal W_{e_2}), either final action removes the prefactor:

[ \mathcal W_{e_1} \prod_{e\notin{e_1,e_2}}\mathcal W_e A_n^{\rm YM}

\mathcal W_{e_2} \prod_{e\notin{e_1,e_2}}\mathcal W_e A_n^{\rm YM}

A_n^{\operatorname{Tr}\phi^3}. ]

It is therefore convenient to define the full transmuter leaving the even label (e_*) unacted:

[ T_{e_*}

\prod_{e\in E_n\setminus{e_}}\mathcal W_e, \qquad T_{e_}A_n^{\rm YM}

A_n^{\operatorname{Tr}\phi^3}. ]

As differential operators in the original scaffold coordinates the (\mathcal W_e) commute. The order dependence discussed in the source concerns the convenient sequence of split/polarization loci, not the final operator identity.

This already supplies a reference-independent augmented lowering operation at the level of the distinguished field-theory amplitude.

Four points

Write ([a,b]=\partial_{X_{a,b}}). Dong–Su–Yang fix the reference derivatives ([2,8][1,4]). Their two graph extractors are

[ D_1=[2,8][1,4][1,6], \qquad D_2=[2,8][1,4][3,6], ]

and obey

[ D_1A_4^{\rm YM}=\frac1{X_{1,5}}, \qquad D_2A_4^{\rm YM}=\frac1{X_{3,7}}. ]

Consequently their cellular augmentation is

[ (D_1+D_2)A_4^{\rm YM}

\frac1{X_{1,5}}+ \frac1{X_{3,7}}

A_4^{\operatorname{Tr}\phi^3}. ]

For the same reference, the Backus–Figueiredo transmuter is

[ T_8=\mathcal W_2\mathcal W_4\mathcal W_6. ]

The three factors contain, respectively,

[ [2,8],\qquad [1,4],\qquad [1,6]\ \text{or}\ [3,6]. ]

Thus (D_1) and (D_2) are literally two monomials in the coordinate expansion of (T_8). But they are not the whole operator. The exact expansion has

[ 5^3=125 ]

raw selections and 124 distinct derivative monomials. Both (D_i) occur with coefficient one. Therefore

[ R_{4,0}

T_8-D_1-D_2 ]

is a nonzero differential operator with 122 distinct monomials and total coefficient weight 123. Nevertheless the two source theorems imply

[ R_{4,0}A_4^{\rm YM}=0. ]

This is already enough to distinguish an operator equality from an equality after evaluation on the canonical amplitude.

Five points

For the reference derivatives ([2,10][1,4]), the five pentagon triangulations and their complete extractors are

Cell Scalar diagram Complete differential monomial
(\Gamma_1) (1/(X_{1,5}X_{1,7})) ([2,10][1,4][1,6][1,8])
(\Gamma_2) (1/(X_{3,7}X_{3,9})) ([2,10][1,4][3,6][3,8])
(\Gamma_3) (1/(X_{1,5}X_{5,9})) ([2,10][1,4][1,6][5,8])
(\Gamma_4) (1/(X_{1,7}X_{3,7})) ([2,10][1,4][1,8][3,6])
(\Gamma_5) (1/(X_{3,9}X_{5,9})) ([2,10][1,4][3,8][9,6])

The ray and fan cases are explicit examples or immediate specializations of the Dong–Su–Yang rules; the remaining two follow mechanically from their connected-sequence prescription. Together they give

[ \sum_{a=1}^{5}D_{\Gamma_a}A_5^{\rm YM}

A_5^{\operatorname{Tr}\phi^3}, ]

where

[ \begin{aligned} A_5^{\operatorname{Tr}\phi^3} ={}& \frac1{X_{1,5}X_{1,7}} +\frac1{X_{3,7}X_{3,9}} +\frac1{X_{1,5}X_{5,9}}\ &+\frac1{X_{1,7}X_{3,7}} +\frac1{X_{3,9}X_{5,9}}. \end{aligned} ]

The corresponding full transmuter is

[ T_{10}

\mathcal W_2\mathcal W_4\mathcal W_6\mathcal W_8. ]

Every listed (D_{\Gamma_a}) selects one coordinate derivative from each of these four factors. Hence the full Catalan coframe is again contained monomial-by-monomial in (T_{10}).

The exact expansion now has

[ 7^4=2401 ]

raw selections and 2370 distinct derivative monomials. Each of the five graph extractors occurs with coefficient one. The residual

[ R_{5,0}

T_{10}-\sum_{a=1}^{5}D_{\Gamma_a} ]

has 2365 distinct monomials, total coefficient weight 2396, and satisfies

[ R_{5,0}A_5^{\rm YM}=0. ]

The coframe is therefore a very sparse cellular resolution of the full transmutation operator on this particular amplitude. It is not an equality of free differential operators.

The exact quotient statement

Let (\operatorname{Diff}_n) denote the algebra generated by scaffold coordinate derivatives and define

[ \operatorname{Ann}(A_n^{\rm YM})

{D\in\operatorname{Diff}_n:D A_n^{\rm YM}=0}. ]

At four and five points the low-point result is

[ \boxed{ [T_{e_*}]

\left[\sum_{\Gamma}D_{\Gamma}^{(e_*)}\right] \quad\text{in}\quad \operatorname{Diff}_n/\operatorname{Ann}(A_n^{\rm YM}) } ]

for every cyclic scaffold reference (e_*). Evaluation gives

[ T_{e_*}A_n^{\rm YM}

\sum_\Gamma D_\Gamma^{(e_*)}A_n^{\rm YM}

A_n^{\operatorname{Tr}\phi^3}. ]

This annihilator quotient is a precise low-point quotient. It should not be silently identified with the full gauge-cohomological or PT/KK/BCJ quotient; constructing a comparison to those representation-independent quotients is additional work.

Every cyclic scaffold reference

Let

[ \rho_r(i)=i+2r\pmod{2n} ]

with labels returned to ({1,\ldots,2n}). Rotate the base reference, operators, and cells by (\rho_r), and set

[ e_*^{(r)}=\rho_r(2n). ]

The Rust certificate checks every reference rather than assuming covariance.

At four points:

(r) Rotated fixed even pair (e_*^{(r)}) Cell permutation
0 ((2,8)) 8 ((1)(2))
1 ((2,4)) 2 ((12))
2 ((4,6)) 4 ((1)(2))
3 ((6,8)) 6 ((12))

Every rotated coframe is contained in the corresponding (T_{e_*^{(r)}}) expansion. Each expansion has the same 125/124 raw/distinct count and the same 122-support residual.

At five points:

(r) Rotated fixed even pair (e_*^{(r)}) Cell permutation
0 ((2,10)) 10 (1\to1\to\cdots)
1 ((2,4)) 2 (1\to2\to3\to4\to5\to1)
2 ((4,6)) 4 the square of that 5-cycle
3 ((6,8)) 6 the cube of that 5-cycle
4 ((8,10)) 8 the fourth power of that 5-cycle

Every expansion has the same 2401/2370 raw/distinct count and the same 2365-support residual. Rotation permutes the five scalar diagrams, so their augmentation is unchanged.

The stronger omitted-pair audit also passes:

  • at four points there are 6 omitted pairs, 12 labelled choices of final (\mathcal W), and 4 distinct full transmuters (T_{e_*});
  • at five points there are 10 omitted pairs, 20 labelled choices of final (\mathcal W), and 5 distinct full transmuters.

Every (T_{e_*}) appears (n-1) times among the pair/final-action choices and has the same scalar output. Thus raw coordinates remain reference-dependent, but no reference defect survives the augmentation/annihilator quotient at these arities.

Why the metric adjoint cannot be computed

For a genuine adjunction one needs an actual linear map

[ J^+:S_{2n}^+ \longrightarrow G_n^+\otimes L_{\mathfrak f} ]

on specified paired spaces, not only a rule sending one distinguished scalar master amplitude to one Yang–Mills amplitude.

In the most favorable generic twisted-cohomology model,

[ \dim S_{2n}=(2n-3)!, \qquad \dim G_n=(n-3)! ]

before including the external gauge-state fiber. The low-point counts are:

(n) (\dim S_{2n}) (\dim G_n) Entries of full (J) Free after one master-section value Optimistic free count after (n) independent cyclic values
4 120 1 120 119 at least 116
5 5040 2 10080 10078 at least 10070

The last column is deliberately overgenerous: the cyclic values are related by relabeling, so they need not supply (n) independent constraints. Even under that favorable fiction the matrix is overwhelmingly undetermined.

There are four independent missing pieces:

  1. the published fusion residue gives the image of the distinguished scalar master section, not the action of (J) on a complete source basis;
  2. no chain-level physical gauge pairing (I_G), including transverse-state coevaluation, is supplied;
  3. (J^\dagger) returns to a (2n)-point normal object carrying (L_{\mathfrak f}^\vee), whereas (T_{e_*}) is an arity-preserving (n)-point transmutation;
  4. the diagram derivatives are specified through their values on the canonical amplitude, not as a map on an arbitrary gauge cohomology class.

If a smaller fusion-normal source is intended instead of the global twisted-cohomology space, that source and its perfect pairing must first be defined. Merely restricting both constructions to their distinguished one-dimensional lines makes an adjoint tautological and normalization-dependent; it does not compute the claimed intrinsic adjoint.

Therefore

[ I_S^{-1}J^{\mathsf T}I_G ]

is presently undefined as an explicit low-point matrix. Invertible choices of (I_S) and (I_G) would transport, not remove, the large ambiguity in (J).

Three-way verdict

1. Genuine adjunction

Not established and not presently evaluable. The literal identification with the published (D_\Gamma) or (T_{e_*}) is type-incompatible. A future normal-line-corrected Gysin/Thom comparison could still produce a genuine adjunction theorem, but it would be new structure.

2. Weaker counit/coframe identity

Established at tree field theory for (n=4,5) on the canonical amplitude:

[ T_{e_} \equiv \sum_\Gamma D_\Gamma^{(e_)} \pmod{\operatorname{Ann}(A_n^{\rm YM})}, ]

and both sides evaluate to the complete planar scalar amplitude. The (D_\Gamma) are sparse Catalan coordinates of the transmutation counit.

3. Irreducible reference dependence

Falsified at the augmented canonical-amplitude level through five points. All cyclic references give the same scalar output and differ only by a permutation of the cellular presentation plus an amplitude-annihilating operator. Raw operator representatives remain reference-dependent.

Conceptual update

The correct emerging algebra is not yet a metric raising/lowering algebra. What is actually visible is

[ \text{scalar master} \xrightarrow{\text{fusion normal residue}} \text{YM distinguished section} \xrightarrow{\text{transmutation counit}} \text{scalar amplitude}, ]

with the second arrow admitting a Catalan cellular coframe. The composite may eventually be recognized as a counit, trace, or Frobenius-type contraction, but calling it an adjoint now would erase the arity, normal-line, and pairing data that remain missing.

The next non-tautological task is consequently not another low-point derivative expansion. It is to construct a full chain-level fusion map and physical gauge pairing, then ask whether its Verdier/Gysin adjoint descends to the already identified transmutation class.

Reproduction

rustc --edition=2021 -O research/nima/check_low_point_transmutation.rs `
  -o "$env:TEMP\marici-low-point-transmutation.exe"
& "$env:TEMP\marici-low-point-transmutation.exe"

The certificate checks:

  • both four-point and all five five-point graph/operator pairs;
  • membership with coefficient one in the appropriate full (W) expansion;
  • all four and five cyclic scaffold references;
  • exact graph permutations and augmentation invariance;
  • raw, distinct, and residual operator-support counts;
  • every omitted-even-pair/final-action choice;
  • the low-point matrix-identifiability bounds.

Sources