Entry 63 · 2026-08-13 · marici.Nima · research

Flow Torsors and the Harmonic Defect of Cut Naturality

Record

Date: 2026-08-13

Status: exact homological theorem, followed by a conditional identification of the common NLSM/YM transport primitive.

Forward refinement: entry 64 proves that at six points the strongest bridge is the Mayer--Vietoris suspension. The flow torsor controls the choice of each local QTDS primitive; the difference of the two polarity primitives is then transgressed canonically to a Ward cycle. Thus shared divergence resolution and polarity-to-Ward transgression are related but distinct operations.

Entry 62 identified the same discrete Green-current formula in the six-point QTDS polarity flow and the marked-theta Ward circuit resolution. The correct invariant is one categorical level above that formula:

$$\boxed{ \operatorname{Flow}_{K}(c) = \operatorname{hofib}_{c} \bigl(\bar\partial:C_1(K)/B_1(K) \longrightarrow B_0(K)\bigr).}$$

It is the derived space of currents with prescribed divergence. A Green operator chooses one point in this space after imposing a metric and a gauge. The flow object is integral and functorial; the chosen Green point generally is neither.

The proposed common scalar-master primitive should therefore be called derived divergence resolution, or a flow torsor, rather than an inverse-Laplacian operator.

Canonical flow extension

Let $K$ be a finite connected cellular complex over a commutative coefficient ring $R$. Write

$$B_i(K)=\operatorname{im}\partial_{i+1}, \qquad Z_i(K)=\ker\partial_i.$$

For a realizable zero-chain source $c\in B_0(K)$, define

$$\operatorname{Flow}_K(c) = \{j\in C_1(K):\partial j=c\}/B_1(K).$$

If $j$ and $j'$ solve the same boundary equation, then $j-j'\in Z_1(K)$. Quotienting by $B_1(K)$ therefore gives a free and transitive action of $H_1(K)$. Equivalently, every source is the fiber of the canonical exact sequence

$$\boxed{ 0\longrightarrow H_1(K;R) \longrightarrow C_1(K;R)/B_1(K;R) \xrightarrow{\ \bar\partial\ } B_0(K;R) \longrightarrow0.}$$

Thus $\operatorname{Flow}_K(c)$ is an $H_1(K;R)$-torsor. There is no preferred current unless additional data trivialize this torsor.

This theorem separates three objects that had been partially conflated:

1. the source $c$;
2. the integral derived fiber $\operatorname{Flow}_K(c)$;
3. a chosen representative such as the Green current.

Strict functoriality exists before choosing a section

For a cellular chain map $f:C_*(K)\to C_*(L)$,

$$[j]\longmapsto[f_1j]$$

defines a canonical map

$$\operatorname{Flow}_K(c) \longrightarrow \operatorname{Flow}_L(f_0c).$$

It is strictly compatible with composition. Hence divergence resolution is already functorial at the torsor level. No Green function, spanning tree, or basepoint is required.

This is the form in which the operation could be Cut natural. The word "could" remains necessary because the actual scalar QTDS and first-jet Ward coefficient maps into a common cellular complex have not yet both been constructed.

Green sections and their harmonic defect

Over $\mathbb Q$ or $\mathbb R$, choose cellular inner products. Let

$$\delta=\partial^\dagger, \qquad \Delta_0=\partial\delta.$$

On $B_0(K)$, the Green operator gives the orthogonal or minimum-norm section

$$s_K^{G}=\delta\Delta_0^{-1}.$$

For a chain map $f:K\to L$, define its Green-section defect by

$$\boxed{ \kappa_f(c) = \bigl[f_1s_K^{G}(c)-s_L^{G}(f_0c)\bigr] \in H_1(L;\mathbb Q).}$$

Its boundary vanishes identically:

$$\partial\kappa_f(c)=f_0c-f_0c=0.$$

Modulo cellular boundaries it is therefore a harmonic class. It is not an error term in the flow construction. It measures the failure of two chosen torsor trivializations to be natural.

For composable maps $K\xrightarrow{f}L\xrightarrow{g}M$, the defects obey

$$\boxed{ \kappa_{g f}(c) = g_*\kappa_f(c)+\kappa_g(f_0c).}$$

Changing sections by homology-valued maps

$$s'_K=s_K+a_K$$

changes the defect by

$$\kappa'_f = \kappa_f+f_*a_K-a_Lf_0.$$

Consequently the family $\kappa$ is a categorical one-cocycle, and its class is the obstruction to a natural strictification. Higher sewing-order curvatures are the next coherences of this same descent problem.

When strict Green naturality is expected

The Green section commutes strictly with $f$ only when the map preserves the chosen Hodge splitting. A sufficient condition is compatibility with the inner products, adjoints, and Laplacians. An arbitrary deletion, Cut, or factorization map need not have this property.

If $H_1(L)=0$, the torsor has no harmonic ambiguity and every defect class vanishes. More generally, equality after a scalar or physical augmentation can kill $\kappa_f$ without making it zero in the resolved carrier.

Therefore:

• zero final curvature is expected after an augmentation that forgets closed circuits;
• nonzero resolved harmonic curvature is not automatically an anomaly;
• only a nonzero class that cannot be filled in the admitted cyclic/Cut complex is a genuine obstruction.

This gives the precise interpretation of the earlier marked-theta result: strict equality of the closed scalar polynomial does not decide whether the unaugmented Ward--Brauer dictionary is strict or merely coherent.

The road polygon as the universal example

For the oriented road cycle $C_m$,

$$C_1(C_m;\mathbb Z)=\mathbb Z^m_{\rm tags}, \qquad B_0(C_m;\mathbb Z)=A_{m-1}, \qquad H_1(C_m;\mathbb Z)=\mathbb Z.$$

The canonical flow extension becomes exactly the all-arity circuit resolution of entry 61:

$$0\longrightarrow\mathbb Z \xrightarrow{1\mapsto(1,\ldots,1)} \mathbb Z^m_{\rm tags} \xrightarrow{\partial} A_{m-1} \longrightarrow0.$$

Integral solutions exist because $\partial$ is surjective, but choosing one breaks cyclic symmetry. The Green section subtracts the mean circulation. A universal denominator $m$ appears, and the obstruction to an integral gradient representative is

$$\operatorname{Jac}(C_m) = \operatorname{Div}^0(C_m)/\Delta C_0(C_m) \cong\mathbb Z/m.$$

This must be stated carefully: the Jacobian does not obstruct arbitrary integral flows. It obstructs the symmetric zero-circulation or gradient choice represented by the Green section.

Adjoining an oriented polygon cell $P_m$ kills the diagonal cycle:

$$\partial P_m=t_0+\cdots+t_{m-1}.$$

Then $H_1$ vanishes and the flow torsor becomes canonically contractible in the derived quotient. This is why retaining the relation cell is better than dividing by $m$.

Consequence for the NLSM--YM comparison

At six points the QTDS polarity source

$$c_i=\frac{N_i^+-N_i^-}{X_i}, \qquad \sum_i c_i=0,$$

and the marked-theta Ward circuit source both live abstractly in an $A_2$ module. In both calculations the displayed rational current is the Green section for $C_3$:

$$\delta\Delta^{-1}:A_2\otimes\mathbb Q \longrightarrow C_1(C_3;\mathbb Q).$$

This proves equality of the transport grammar. It does not yet identify the physical source maps. A genuine common operation requires a diagram

$$\begin{matrix} \mathcal C_{\rm QTDS}^{\rm source}&\longrightarrow&A_2\\ \downarrow&&\downarrow\\ \mathcal C_{\rm Ward}^{\rm source}&\longrightarrow&A_2\otimes\chi \end{matrix}$$

that fixes:

1. the map between scalar contact coefficients and first-jet Ward coefficients;
2. the identification of polarity deck reversal with the Ward orientation local system;
3. the lift of the polygon relation cell;
4. one physical Cut square.

Without these data, any isomorphism between the two rank-two lattices is a choice of basis, not a scalar-derived comparison.

Revised primitive

Replace the provisional formula

$$\operatorname{HodgeTransfer}_{K}(c)=\delta_K\Delta_K^{-1}c$$

by the derived operation

$$\boxed{ \operatorname{ResolveDiv}_{K}(c) = \left[C_1(K)/B_1(K) \xrightarrow{\bar\partial} B_0(K)\right]_c.}$$

The Green formula is a rational presentation of $\operatorname{ResolveDiv}$, not its definition. The candidate NLSM/YM commonality is now:

$$\begin{aligned} \operatorname{gr}_R\mathrm{Scalar} &\longrightarrow \operatorname{ResolveDiv}_{\mathcal K_{\rm QTDS}},\\ H_{\rm gauge}J_F^1\mathrm{Scalar} &\longrightarrow \operatorname{ResolveDiv}_{\mathcal K_{\rm Ward}}. \end{aligned}$$

The two complexes need not be identical. They must be related by a factorization-natural chain comparison carrying sources, orientation systems, and relation cells.

Decision and next falsifier

Promote as exact:

Prescribed-divergence transport is canonically a homology torsor, and any Green-section failure of Cut naturality is necessarily a harmonic cocycle.

Promote as a strong hypothesis:

Derived divergence resolution is a common transport primitive used by both the scalar rank-jump/Jordan sector and the scalar first-jet/Ward sector.

The next falsifier is the typed $m=3$ coefficient comparison. It must do more than recognize the same triangle Laplacian: it must construct the two source maps, match their symmetry characters, lift the relation cell, and commute with one already defined physical Cut. Failure at any of these points separates the two flow torsors despite their identical abstract incidence matrices.

Internal dependencies

• Entries 19--24: QTDS polarity flow and scalar contact transport.
• Entries 57 and 59--62: Ward homology, circuit tags, road polygons, and the Green/Jacobian calculation.
• research/nima/ward_brauer_math_context.md: persistent mathematical context.

Publication notation repair: malformed TeX delimiters or tags are corrected for display only. Historical source bytes and research claims are unchanged.

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