Part II

Polylogarithms and Iterated Propagation: Transcendental Weight as Loop Order

Matthew Longhep-th20 pagesHaskell + Lean 4

Abstract

The second rung introduces the graded family of special functions defined on the domain of observable ratios: the classical polylogarithm, the multiple polylogarithms, multiple zeta values, and Chen's iterated path integrals, organized by a weight and depth bigrading. The emergent structural property of this rung is a precise slogan: transcendental weight equals loop order. We prove Chen's path-independence theorem, show that the iterated reduced coproduct of a weight-n polylogarithm terminates at weight-one logarithmic leaves, and give a complete three-form treatment of the five-term relation for the dilogarithm.

1 Introduction

1.1 The governing perspective

The organizing idea of this library is a single sentence, which we ask the reader to keep in view throughout:

A Goncharov-style Lie coalgebra is not merely a formal algebraic object; it is a bookkeeping system for how physical quantities break apart into irreducible informational constituents.

A physical observable — a scattering amplitude, a Feynman integral, a period, an entropy — is treated as the numerical realization of a deeper, pre-numerical, structured object. The question “what is this observable made of?” is then answered not by further computation but by a decomposition law: a coproduct/cobracket that exposes the observable’s primitive, irreducible pieces (cuts, discontinuities, factorization channels). The full library builds this apparatus rung by rung and, in its final synthesis, identifies measurement as the closure of the whole ladder.

This is a modular framework, not a unified one. Each rung is a standalone, defensible development; each composes with its neighbour to add exactly one emergent structural property that the level below cannot express on its own. The ladder is $\underbrace{\text{I}}_{\substack{\text{kinematic}\\\text{domains}}} \;\longrightarrow\; \underbrace{\text{II}}_{\substack{\text{polylogarithms}\\\text{(this paper)}}} \;\longrightarrow\; \underbrace{\text{III}}_{\substack{\text{motives}}} \;\longrightarrow\; \underbrace{\text{IV}}_{\substack{\text{Lie}\\\text{coalgebra}}} \;\longrightarrow\; \underbrace{\text{V}}_{\substack{\text{regulators}}} \;\longrightarrow\; \underbrace{\text{Synthesis}}_{\substack{\text{measurement}}}.$

1.2 Where this rung sits

Rung I disciplined the raw arena of observation: a field $\mathbb{F}$ of kinematic invariants, its multiplicative group $\mathbb{F}^\times$ of nonzero scales and ratios, the projective line $\mathbb{P}^1(\mathbb{F})$ with its $\mathrm{PGL}_2(\mathbb{F})$ action, the cross-ratio, and the moduli space $\mathrm{Conf}_n(\mathbb{P}^1)=M_{0,n}$ . Its central computation showed that the four-point kinematic configuration space is exactly $\mathbb{P}^1\setminus\{0,1,\infty\}$ — precisely the punctured domain on which the classical polylogarithm is single-valued away from its branch points at $0,1,\infty$ . Rung I therefore does not merely supply “some field”; it hands over the specific domain on which the functions of this paper live.

Rung II — the present paper — populates that domain with functions. We introduce the classical polylogarithm $\operatorname{Li}_n(x)=\sum_{k\ge1}x^k/k^n$ , its multi-variable generalizations, the multiple zeta values, and their common analytic incarnation as Chen iterated path integrals. These carry a weight and a depth grading. The emergent property we name is the slogan

the statement that the algebraic complexity (weight) of the special functions produced by a perturbative computation tracks the physical complexity (loop order, number of propagators) of that computation. (The ladder sometimes calls this “transcendental depth,” meaning the transcendentality level; in this paper we reserve the word depth for the distinct iterated-integral nesting grading of 9, and use weight for the grading that tracks loop order.) This is the first appearance in the ladder of a graded family of transcendental functions whose grading is physically meaningful, and it is the structural precondition for everything that follows: the coproduct of Rung IV lowers weight, the periods of Rung III are the weight-graded transcendental constants, and the regulators of Rung V evaluate weight-two data through the Bloch–Wigner function.

1.3 What we prove

The paper is expository where it must recall Rung I and forward-reference Rungs III–V, but it contains genuine proofs of the results that belong to Rung II proper:

11 (Chen path-independence): iterated integrals of closed logarithmic $1$ -forms depend only on the based homotopy class of the integration path, so that $\operatorname{Li}_n$ is a well-defined multivalued analytic function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ .
13 (termination of the iterated coproduct): the reduced coproduct strictly lowers weight in each tensor factor, so its iteration on a weight- $n$ polylogarithm terminates after at most $n-1$ steps at weight-one logarithmic leaves. This is the precise fact that licenses the phrase “weight $=$ loop order”: complexity is bounded, finite, and reduces to elementary $\log$ data.
19 (the five-term relation, three forms): the fundamental functional equation of $\operatorname{Li}_2$ in Rogers, Bloch–Wigner, and cross-ratio forms, with the equivalences and the descent to a $5$ -point configuration proved in detail. This is the connective tissue of the whole library.
15 (symbol and shuffle): the symbol of $\operatorname{Li}_2$ is $\mathcal{S}(\operatorname{Li}_2(x))=-(1-x)\otimes x$ , and the shuffle relations express equivalent orderings of iterated propagation.

The accompanying Haskell package numerically certifies the shuffle product, the symbol of the dilogarithm, and all three forms of the five-term relation to machine precision; a Lean 4 file records the structural statements. 16 describes both.

1.4 Standalone-library discipline and epistemic honesty

This library is self-contained: it cites only the external mathematical and physical literature ([sec:refs]), and it makes its interpretive commitments explicit through a three-valued status system, described in 3, that tags every dictionary entry and every claim as standard (S), heuristic (H), or speculative (P). We regard this discipline as part of the formalism, not as decoration: it is what keeps the physical reading of a theorem from being silently promoted above the reliability of the mathematics that supports it.

2 The Mathematics $\leftrightarrow$ Physics Dictionary

Every paper in this library reproduces, verbatim, the following translation table as its opening dictionary. It is the seed from which each rung’s topic-specific refinements are drawn.

Mathematics	Physical representation
Field $\mathbb{F}$	Kinematic/coordinate domain of possible values
Element of $\mathbb{F}^\times$	Ratio, cross-ratio, scale, observable parameter
Polylogarithm $\operatorname{Li}_n(x)$	Iterated propagation / layered amplitude contribution
Period	Measurable physical number obtained from geometry
Motive	Pre-numerical informational object behind the measurement
Lie algebra	Infinitesimal symmetry / generator structure
Lie coalgebra	Decomposition law of observables
Cobracket $\delta$	Factorization channel, cut, boundary, or information split
Weight grading	Complexity depth / loop order / transcendental depth
Primitive element	Irreducible informational unit
Coproduct	Full hierarchical decomposition
Regulator map	Passage from motivic object to physical real number
Hodge/de Rham realization	Analytic form seen by calculation
Betti realization	Topological/path/integration-cycle form
Period pairing	Physical observable as pairing of form and cycle

This seed table is reproduced verbatim across the library. One word deserves a caveat local to the present rung: in the “Weight grading” row the phrase “transcendental depth” uses depth in its colloquial sense of transcendentality level (a synonym for weight). Throughout the body of this paper we instead reserve the technical term depth for the iterated-integral nesting grading of 9, and use weight for the grading that tracks loop order.

Four central slogans recur across the whole library: $\begin{align*} &\text{observable} = \text{realization}(\text{motivic information}), \\ &\text{coalgebra} = \text{grammar of physical decomposition}, \\ &\text{Physics observes numerical shadows of deeper motivic structures}, \\ &\text{Physical reality is the realization layer of structured mathematical information}. \end{align*}$

3 The Epistemic Status System (S / H / P)

Every dictionary entry and every claim below carries one of three labels. These are the library’s core epistemic-honesty device.

Label	Meaning	Canonical example
S	Standard use in mathematics or mathematical physics	The Goncharov coproduct on motivic polylogarithms
H	Strong heuristic representation-dictionary entry	“Weight $=$ loop order”; cobracket as “cut”
P	Speculative philosophical ontology	Physical reality as motivic-categorical realization

Composite labels S/H and H/P occur where an entry is standard as pure mathematics but heuristic or speculative in its physical reading. We adopt the following rule without exception.

Definition 1 (Status monotonicity). Order the labels by $\textsf{S}< \textsf{H}< \textsf{P}$ (increasing speculativeness, equivalently decreasing reliability). If two representation entries or theorems compose, the status of the composite is the maximum of the two under this order: a chain of translations is only as reliable as its weakest link. In particular, a P-labeled step cannot be upgraded to S merely by composing it with otherwise-rigorous results.

We flag the reading “weight $=$ loop order” as S/H: the weight grading and its properties are standard mathematics (S), while its identification with physical loop order is a strong heuristic (H) that is established case-by-case rather than by a universal theorem. We will be explicit about where the boundary lies (14).

4 Master Formulas

The whole library shares the following notation. $\begin{align} \text{observable} &= \operatorname{Real}(\text{abstract structure}), \\ \mathcal{A}&= \operatorname{per}(\mathcal{A}^{\mathfrak{m}}), \\ \Delta(\mathcal{A}^{\mathfrak{m}}) &= \textstyle\sum_i \mathcal{A}^{\mathfrak{m}}_{i,1}\otimes \mathcal{A}^{\mathfrak{m}}_{i,2}, \\ \delta &: \mathcal{L}_G\longrightarrow \Lambda^2\mathcal{L}_G, \\ \operatorname{wt}(x) &= n \ \text{ for } x\in\mathcal{L}_G{}_n, \qquad \operatorname{dep}(x)=\text{depth}, \\ \operatorname{per}(\Pi) &= \int_\Gamma \omega, \qquad \Pi=(X,D,\omega,\Gamma). \end{align}$ Here $\mathbb{F}$ is a field (typically a number field or a field of kinematic invariants), $\mathbb{F}^\times$ its multiplicative group; $\mathcal{A}^{\mathfrak{m}}$ is a motivic amplitude object, a pre-numerical element of a Hopf algebra $\mathcal{H}$ of motivic periods, and $\operatorname{per}$ the period map sending it to the physical number $\mathcal{A}$ ; $\mathcal{L}_G=\mathcal{L}_G(\mathbb{F})=\bigoplus_{n\ge1}\mathcal{L}_G(\mathbb{F})_n$ is the (Goncharov) Lie coalgebra of primitives, graded by weight $n$ ; and $\delta$ is the antisymmetrization of the reduced coproduct $\Delta'$ (namely $\Delta$ minus its two trivial terms $x\otimes1$ and $1\otimes x$ ), landing in $\Lambda^2\mathcal{L}_G$ rather than $\mathcal{L}_G\otimes\mathcal{L}_G$ . Rung II is where the weight grading of [eq:master5] first appears and where the functions realized by $\operatorname{per}$ in [eq:master2] are constructed.

5 Core Objects: Polylogarithms and Iterated Integrals

5.1 The classical polylogarithm

Definition 2 (Classical polylogarithm; status S). For $n\ge1$ the classical polylogarithm is the power series $\begin{equation} \operatorname{Li}_n(x) \;=\; \sum_{k\ge1}\frac{x^k}{k^n}, \qquad |x|<1, \end{equation}$ continued analytically to $\mathbb{C}\setminus[1,\infty)$ . The first two cases are $\operatorname{Li}_1(x) = -\log(1-x), \qquad \operatorname{Li}_2(x) = \sum_{k\ge1}\frac{x^k}{k^2},$ and the members are linked by the differential relations $\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x}\,\operatorname{Li}_n(x) = \frac{\operatorname{Li}_{n-1}(x)}{x}\ (n\ge2), \qquad \frac{\mathrm{d}}{\mathrm{d}x}\,\operatorname{Li}_1(x) = \frac{1}{1-x}. \end{equation}$ Equivalently, $\operatorname{Li}_n(x)=\int_0^x \operatorname{Li}_{n-1}(t)\,\frac{\mathrm{d}t}{t}$ for $n\ge2$ , expressing $\operatorname{Li}_n$ as a single integration of $\operatorname{Li}_{n-1}$ against the logarithmic kernel $\mathrm{d}t/t$ .

Iterating [eq:polydiff] down to $\operatorname{Li}_1$ realizes $\operatorname{Li}_n$ as an $n$ -fold iterated integral, which we make precise in 5 below. The number $n$ is the weight; it counts the layers of integration.

5.2 Multiple polylogarithms and multiple zeta values

Definition 3 (Multiple polylogarithm; status S). For positive integers $n_1,\dots,n_d$ and complex arguments $x_1,\dots,x_d$ , $\begin{equation} \operatorname{Li}_{n_1,\dots,n_d}(x_1,\dots,x_d) \;=\!\!\sum_{0<k_1<\cdots<k_d}\!\! \frac{x_1^{k_1}\cdots x_d^{k_d}}{k_1^{n_1}\cdots k_d^{n_d}}, \end{equation}$ convergent for $|x_i|\le1$ with $(n_d,x_d)\neq(1,1)$ . Its weight is $n=n_1+\cdots+n_d$ and its depth is $d$ .

Definition 4 (Multiple zeta values; status S). The specialization to all arguments equal to $1$ , $\begin{equation} \zeta(n_1,\dots,n_d) \;=\; \operatorname{Li}_{n_1,\dots,n_d}(1,\dots,1) \;=\!\!\sum_{0<k_1<\cdots<k_d}\!\!\frac{1}{k_1^{n_1}\cdots k_d^{n_d}}, \qquad n_d\ge2, \end{equation}$ is a multiple zeta value (MZV) of weight $n=n_1+\cdots+n_d$ and depth $d$ . These are the special constants that appear as the weight-graded pieces of loop amplitude expansions.

5.3 Chen iterated integrals

The uniform analytic home of all of the above is Chen’s calculus of iterated path integrals.

Definition 5 (Chen iterated integral; status S). Fix points $a_1,\dots,a_N\in\mathbb{C}$ (the singularities) and endpoints $a_0,a_{N+1}$ . For a path $\gamma:[0,1]\to\mathbb{C}\setminus\{a_1,\dots,a_N\}$ from $a_0$ to $a_{N+1}$ , the iterated integral of the logarithmic forms $\omega_i = \mathrm{d}t/(t-a_i)$ is the ordered-simplex integral $\begin{equation} I(a_0;a_1,\dots,a_N;a_{N+1}) \;=\; \int_{0<t_1<\cdots<t_N<1} \frac{\gamma'(t_1)\,\mathrm{d}t_1}{\gamma(t_1)-a_1}\cdots \frac{\gamma'(t_N)\,\mathrm{d}t_N}{\gamma(t_N)-a_N}. \end{equation}$ Here the simplex is oriented $0<t_1<\cdots<t_N<1$ , so $t_1$ is the innermost variable and the leftmost letter $a_1$ is the one integrated first. When an endpoint coincides with a singularity the integral is defined by tangential base-point regularization (Deligne; Goncharov). In Goncharov’s convention the classical polylogarithm is the special value $\begin{equation} \operatorname{Li}_n(x) \;=\; -\,I(0;1,\underbrace{0,\dots,0}_{n-1};x), \end{equation}$ the iterated integral along the straight path from $0$ to $x$ with the alphabet $\{\mathrm{d}t/t,\ \mathrm{d}t/(t-1)\}$ : the innermost step is taken against the puncture at $1$ , the remaining $n-1$ steps against the puncture at $0$ .

Example 6 ( $\operatorname{Li}_2$ as a double iterated integral; status S). We verify [eq:li-as-chen] for $n=2$ directly. By definition, with the innermost letter $a_1=1$ and outer letter $a_2=0$ , $-I(0;1,0;x) = -\int_{0<t_1<t_2<x}\frac{\mathrm{d}t_1}{t_1-1}\,\frac{\mathrm{d}t_2}{t_2} = -\int_0^x\frac{\mathrm{d}t_2}{t_2}\int_0^{t_2}\frac{\mathrm{d}t_1}{t_1-1}.$ The inner integral is $\int_0^{t_2}\mathrm{d}t_1/(t_1-1)=\log(1-t_2)$ , so $-I(0;1,0;x) = -\int_0^x \frac{\log(1-t_2)}{t_2}\,\mathrm{d}t_2 = \operatorname{Li}_2(x),$ using $\operatorname{Li}_2(x)=-\int_0^x\log(1-t)\,\mathrm{d}t/t$ from [eq:polydiff]. This makes the “iterated propagation” picture concrete: each of the two integrations is one propagation step, the innermost against the puncture at $1$ (the form $\mathrm{d}t/(t-1)$ , contributing $\log(1-t)$ ) and the outer against the puncture at $0$ (the form $\mathrm{d}t/t$ ). The same computation with $n-1$ trailing $\mathrm{d}t/t$ factors reproduces $\operatorname{Li}_n(x)=\sum_{k\ge1}x^k/k^n$ by raising the exponent once per outer integration.

The passage [eq:li-as-chen] is the bridge between the “propagation” picture (each factor $\mathrm{d}t/(t-a_i)$ is one elementary propagation step against a singular locus) and the combinatorial machinery — shuffle products, the coproduct, the symbol — that organizes the whole weight-graded family. We record the two products that iterated integrals satisfy.

Proposition 7 (Shuffle product; status S). Iterated integrals along a fixed path multiply by the shuffle product: $\begin{equation} I(a_0;\,\vec{u}\,;a_{N+1})\cdot I(a_0;\,\vec{v}\,;a_{N+1}) \;=\;\sum_{\sigma\in \vec u\,\shuffle\,\vec v} I(a_0;\,\sigma\,;a_{N+1}), \end{equation}$ the sum running over all interleavings $\sigma$ of the two letter sequences $\vec u,\vec v$ preserving the internal order of each. Physically (status H) the shuffle records that two sequences of propagation steps performed along the same path may be interleaved in all order-preserving ways without changing the accumulated value.

Proof. Expand the product of the two simplex integrals over the common interval $[0,1]$ . The product of an $r$ -simplex and an $s$ -simplex integral is an integral over $[0,1]^{r+s}$ intersected with the two independent orderings; decomposing $[0,1]^{r+s}$ into the $\binom{r+s}{r}$ chambers on which a single total order of the $r+s$ times holds, and discarding the measure-zero coincidence loci, expresses the product as the sum over all interleavings preserving the internal order of each factor. Each chamber is exactly one shuffle term [eq:shuffle]. ◻

The series representation of 4 satisfies a second, distinct product (the stuffle or quasi-shuffle), obtained by splitting the summation region $\{k_1<\cdots\}\times\{l_1<\cdots\}$ into the regions where indices are $<$ , $>$ , or $=$ . The two products together impose the double-shuffle relations among MZVs; we use only the shuffle relation computationally (16).

5.4 The symbol

Definition 8 (Symbol; status S). The symbol of a weight- $n$ (multiple) polylogarithm $F$ is the length- $n$ tensor $\mathcal{S}(F)\in(\mathbb{F}^\times\otimes\mathbb{Q})^{\otimes n}$ obtained by maximally iterating the reduced coproduct down to weight-one letters. Concretely it is computed recursively from the differential: if $\mathrm{d}F=\sum_i F_i\,\mathrm{d}\!\log R_i$ with $R_i\in\mathbb{F}^\times$ and each $F_i$ of weight $n-1$ , then $\begin{equation} \mathcal{S}(F) \;=\; \sum_i \mathcal{S}(F_i)\otimes R_i . \end{equation}$ The symbol is the sharpest elementary invariant for detecting functional identities among polylogarithms: two weight- $n$ combinations are equal modulo lower-weight products of logarithms and constants iff their symbols agree.

6 Weight, Depth, and the Emergent Property

Definition 9 (Weight and depth grading; status S/H). The weight $n=n_1+\cdots+n_d$ measures transcendental complexity; the depth $d$ counts the number of nested iterated-integration layers. The physical reading (status H) is: $\begin{align*} \text{weight} &= \text{loop order / functional transcendentality},\\ \text{depth} &= \text{number of independent kinematic channels}. \end{align*}$

The weight grading is compatible with all the structure introduced so far: the differential [eq:polydiff] lowers weight by one; the shuffle product [eq:shuffle] is weight-additive; the symbol [eq:symbol-rec] of a weight- $n$ function is a length- $n$ tensor of weight-one letters. The grading is what makes “how complicated is this function” into a number rather than an impression, and the emergent property of the rung is the physical interpretation of that number.

Remark 10 (The emergent property: transcendental weight $=$ loop order; status S/H). Perturbative computations in quantum field theory produce, order by order in the loop expansion, transcendental functions of increasing weight: a one-loop finite integral is generically weight two (dilogarithms), an $\ell$ -loop integral is generically weight $2\ell$ in four dimensions, and the constants appearing are MZVs of the corresponding weight. The empirical regularity that weight tracks loop order is the emergent property this rung names. It is not a theorem in the generality of all QFTs: it is a strong heuristic (H) with a large body of confirming case studies (finite integrals in $\phi^4$ theory; the six-gluon remainder function in planar $\mathcal N=4$ super Yang–Mills; the sunrise family up to its elliptic boundary, 14) and known exceptions once elliptic and higher-genus geometry enters. The mathematical content that is a theorem — and that makes the heuristic precise and bounded — is the termination statement of the next section: complexity, whatever its physical origin, is finitely generated by weight-one logarithms.

7 Chen’s Theorem and Termination of the Coproduct

7.1 Path-independence

Theorem 11 (Chen path-independence for logarithmic forms; status S). Let $a_1,\dots,a_N\in\mathbb{C}$ and let $\gamma_0\simeq\gamma_1$ be homotopic paths in $\mathbb{C}\setminus\{a_1,\dots,a_N\}$ with the same endpoints, the homotopy fixing the endpoints. Then the iterated integral [eq:chen] of the closed logarithmic forms $\omega_i=\mathrm{d}t/(t-a_i)$ is unchanged: $I_{\gamma_0}(a_0;a_1,\dots,a_N;a_{N+1}) \;=\; I_{\gamma_1}(a_0;a_1,\dots,a_N;a_{N+1}),$ with tangential regularization at any singular endpoint. Consequently the classical polylogarithm $\operatorname{Li}_n(x)\;=\;-\,I(0;1,0,\dots,0;x)$ of [eq:li-as-chen] is a well-defined multivalued analytic function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ whose monodromy is exactly the analytic-continuation data organized by the later cobracket.

Proof. This is Chen’s fundamental theorem of iterated path integrals (Chen, 1977). The key point is that the iterated integral of a collection of closed $1$ -forms along a path is invariant under homotopy of the path rel endpoints. We sketch the standard chain-homotopy argument. Let $H:[0,1]\times[0,1]\to\mathbb{C}\setminus\{a_1,\dots,a_N\}$ be a homotopy with $H(\cdot,0)=\gamma_0$ , $H(\cdot,1)=\gamma_1$ , and $H(0,s),H(1,s)$ constant in $s$ . Regard the length- $N$ iterated integral as the evaluation of a $1$ -form $\Theta_N$ on the based path space, built from the forms $\omega_i$ ; Chen’s transport formula gives $\mathrm{d}\Theta_N = \sum \pm\,\Theta_{k}\wedge\Theta'_{N-k}$ with each factor proportional to $\mathrm{d}\omega_i=0$ , because every $\omega_i$ is closed. Hence $\Theta_N$ is closed on path space, and Stokes’ theorem applied to the homotopy square shows the difference of the two iterated integrals equals the integral of $\mathrm{d}\Theta_N=0$ over the interior plus boundary terms along the constant edges $H(0,s),H(1,s)$ , which vanish because the endpoints are fixed. Therefore the two iterated integrals agree.

For [eq:li-as-chen]: the forms $\mathrm{d}t/t$ and $\mathrm{d}t/(t-1)$ are closed on $\mathbb{C}\setminus\{0,1\}$ , so the iterated integral defining $\operatorname{Li}_n$ depends only on the homotopy class of the path from $0$ to $x$ in $\mathbb{P}^1\setminus\{0,1,\infty\}$ . Different homotopy classes differ by loops around $0$ and $1$ ; the resulting shifts are exactly the monodromy of $\operatorname{Li}_n$ . The primary generator, the monodromy around the branch point $x=1$ , acts by $\operatorname{Li}_n(x)\mapsto \operatorname{Li}_n(x) - 2\pi i\,\tfrac{(\log x)^{\,n-1}}{(n-1)!}$ , while the loop around $x=0$ acts on the resulting lower-weight logarithm; together they generate the full (nonabelian) monodromy. Thus $\operatorname{Li}_n$ is single-valued on the universal cover and multivalued on $\mathbb{P}^1\setminus\{0,1,\infty\}$ . ◻

Remark 12 (Physical reading; status H). Path-independence is the mathematical reason a Feynman integral’s transcendental value does not depend on how one parametrizes the loop-momentum contour, only on the topological class of the integration cycle relative to its singularities. It directly anticipates the integration cycle $\Gamma$ of the period datum [eq:master6] that a later rung evaluates.

7.2 Termination of the iterated coproduct

We now state the fact that makes “weight $=$ loop order” a bounded, finite statement. We work in the connected graded Hopf algebra $\mathcal{H}=\bigoplus_{n\ge0}\mathcal{H}_n$ of motivic periods (weight grading, $\mathcal{H}_0=\mathbb{Q}\cdot1$ ), with reduced coproduct $\Delta'(x)=\Delta(x)-x\otimes1-1\otimes x$ on the augmentation ideal $\overline{\mathcal{H}}=\ker\varepsilon$ .

Proposition 13 (Termination at weight-one primitives; status S). Let $x\in\mathcal{H}$ be a (motivic) polylogarithm of weight $n\ge1$ . Then $\Delta'$ is graded and strictly lowers weight in each tensor factor, and its iteration on $x$ terminates after at most $n-1$ steps, with terminal leaves of weight one. In symbols, the tower $x \xmapsto{\Delta'} \textstyle\sum x'\otimes x'' \xmapsto{\Delta'\otimes\mathrm{id},\ \mathrm{id}\otimes\Delta'} \cdots$ reaches only weight-one tensor factors after at most $n-1$ applications, and $\Delta'$ vanishes on weight one.

Proof. Because $\mathcal{H}$ is connected and graded, for $x$ of weight $n$ every term of $\Delta'(x)=\sum_i x_i'\otimes x_i''$ has $\operatorname{wt}(x_i')\ge1$ , $\operatorname{wt}(x_i'')\ge1$ , and $\operatorname{wt}(x_i')+\operatorname{wt}(x_i'')=n$ ; hence each factor has weight $\le n-1$ . Apply $\Delta'$ again to each factor of weight $\ge2$ ; by the same argument the maximal weight of a factor drops by at least one at each step. After at most $n-1$ steps every factor has weight one. On weight one, $\Delta'$ vanishes: a weight-one element cannot split as $p+q$ with $p,q\ge1$ summing to $1$ , so $\Delta'|_{\mathcal{H}_1}=0$ and every weight-one element is primitive. Finiteness of the whole tree follows because each node has finitely many children (the coproduct is a finite sum) and the depth is bounded by $n-1$ . ◻

Remark 14 (Why this licenses the slogan; status S/H). 13 is the precise fact behind the emergent property of 10: whatever the physical origin of a weight- $n$ contribution, its coalgebraic anatomy is a finite tree of bounded depth whose leaves are weight-one logarithms $\log R$ with $R\in\mathbb{F}^\times$ — exactly Rung I’s observable ratios. Complexity is not merely large; it is finitely generated by the ground-floor data of the ladder. The full structural consequence — that these weight-one primitives cogenerate the entire Goncharov Lie coalgebra, with $P_1\cong\mathbb{F}^\times\otimes\mathbb{Q}$ — belongs to Rung IV; here we need only the termination, which is elementary given the grading.

7.3 The symbol of the dilogarithm

Theorem 15 (Symbol and cobracket of $\operatorname{Li}_2$ ; status S). The reduced coproduct of the motivic dilogarithm is $\begin{equation} \Delta'\,\operatorname{Li}_2^{\mathfrak m}(x) \;=\; \operatorname{Li}_1^{\mathfrak m}(x)\otimes\log^{\mathfrak m}(x) \;=\; -\,\log^{\mathfrak m}(1-x)\otimes\log^{\mathfrak m}(x), \end{equation}$ so that the symbol is $\begin{equation} \mathcal{S}(\operatorname{Li}_2(x)) \;=\; -(1-x)\otimes x, \end{equation}$ and the cobracket is $\delta\,\operatorname{Li}_2^{\mathfrak m}(x) = \log(x)\wedge\log(1-x)$ .

Proof. From [eq:polydiff], $\mathrm{d}\operatorname{Li}_2(x)=\operatorname{Li}_1(x)\,\mathrm{d}x/x = -\log(1-x)\,\mathrm{d}\!\log x$ . Applying the symbol recursion [eq:symbol-rec] with the single term $F_1=\operatorname{Li}_1(x)=-\log(1-x)$ and $R_1=x$ gives $\mathcal{S}(\operatorname{Li}_2(x))=\mathcal{S}(-\log(1-x))\otimes x = -(1-x)\otimes x$ , using $\mathcal{S}(\log R)=R$ for weight-one letters. Reading off the weight- $(1,1)$ component of Goncharov’s coproduct from the last symbol entry gives $\Delta'\,\operatorname{Li}_2^{\mathfrak m}(x)=\operatorname{Li}_1^{\mathfrak m}(x)\otimes \log^{\mathfrak m}(x)=-\log^{\mathfrak m}(1-x)\otimes\log^{\mathfrak m}(x)$ , consistent in sign with [eq:dilog-symbol]. Antisymmetrizing the two tensor slots (the map $a\otimes b\mapsto a\wedge b$ ) gives $\delta\,\operatorname{Li}_2^{\mathfrak m}(x)= -\log(1-x)\wedge\log(x)=\log(x)\wedge\log(1-x)$ . ◻

The two tensor slots of [eq:dilog-symbol] are the two weight-one primitives $1-x$ and $x$ ; physically (status H) they are the two branch loci $x=1$ and $x=0$ of the dilogarithm, i.e. its cut structure. This is the smallest nontrivial instance of the dictionary entry “cobracket $\leftrightarrow$ cut”; the full one-loop discontinuity computation belongs to Rung IV.

8 Functional Identities and the Double-Shuffle Relations

Before turning to the five-term relation we record the elementary weight-two functional identities of the dilogarithm and the double-shuffle relations among multiple zeta values. Both are instances of the same principle — the special functions of this rung are heavily constrained, not free — and both are proved most cleanly at the level of the symbol [eq:symbol-rec], which reduces a functional equation among weight- $n$ functions to a linear-algebra identity among tensors of weight-one letters (modulo lower-weight products, which are then fixed by evaluation at a single point).

8.1 Weight-two identities of the dilogarithm

Proposition 16 (Reflection, inversion, Landen, duplication; status S). The dilogarithm satisfies, on the appropriate branches, $\begin{align} \operatorname{Li}_2(x) + \operatorname{Li}_2(1-x) &= \frac{\pi^2}{6} - \log(x)\log(1-x), \\ \operatorname{Li}_2(x) + \operatorname{Li}_2\!\left(\tfrac1x\right) &= -\frac{\pi^2}{6} - \tfrac12\log^2(-x), \\ \operatorname{Li}_2(x) + \operatorname{Li}_2\!\left(\frac{x}{x-1}\right) &= -\tfrac12\log^2(1-x), \\ \operatorname{Li}_2(x^2) &= 2\big(\operatorname{Li}_2(x) + \operatorname{Li}_2(-x)\big). \end{align}$

Proof. Each is an equality of weight-two functions, so by 8 it suffices to check the symbols agree and then fix the additive weight-two constant (a rational multiple of $\pi^2$ ) by evaluating at one convenient point. We do the reflection identity [eq:reflection] in detail; the others are identical in method. The symbol of the left-hand side is $\mathcal{S}\big(\operatorname{Li}_2(x)\big) + \mathcal{S}\big(\operatorname{Li}_2(1-x)\big) = -(1-x)\otimes x \;-\; x\otimes(1-x),$ using 15 for the first term and the substitution $x\mapsto1-x$ (which also swaps the roles of the two branch loci) for the second. The symbol of the right-hand side: the constant $\pi^2/6$ has empty symbol (it is a weight-two number, killed by $\mathrm{d}$ ), and $\mathcal{S}\big(-\log x\log(1-x)\big) = -\,x\otimes(1-x) - (1-x)\otimes x$ by the Leibniz rule for the symbol of a product of logarithms. The two symbols coincide. Hence the two sides differ by a weight-two constant; evaluating at $x=\tfrac12$ , where $\operatorname{Li}_2(\tfrac12)=\tfrac{\pi^2}{12}-\tfrac12\log^2 2$ and both arguments equal $\tfrac12$ , fixes the constant to $\pi^2/6$ . The inversion, Landen, and duplication identities follow by the same symbol comparison, with constants fixed at $x=-1$ , $x\to0$ , and $x=1$ respectively. ◻

These identities are exactly the ingredients that make the Bloch–Wigner function single-valued: the combination $\arg(1-z)\log|z|$ in 20 is chosen so that the multivalued pieces produced by [eq:reflection]–[eq:inversion] cancel, yielding the clean symmetries $D(z)=-D(1/z)=-D(1-z)$ recorded in 21. The reflection and duplication identities are numerically certified by the Haskell code of 16.

8.2 Shuffle, stuffle, and a double-shuffle consequence

Multiple zeta values carry two products: the shuffle [eq:shuffle], from their iterated-integral representation, and the stuffle (quasi-shuffle), from their series representation. Their difference produces nontrivial $\mathbb{Q}$ -linear relations.

Proposition 17 (A double-shuffle relation; status S). In the summation convention of 3–4 (indices increasing, $\zeta(n_1,n_2)= \sum_{0<k_1<k_2} k_1^{-n_1}k_2^{-n_2}$ , convergent when the outer exponent $n_2\ge2$ ), one has $\begin{equation} \zeta(1,3) \;=\; \sum_{0<m<n}\frac{1}{m\,n^3} \;=\; \tfrac14\,\zeta(4) \;=\; \frac{\pi^4}{360}. \end{equation}$

Proof. The stuffle product splits the double sum $\zeta(2)^2=\big(\sum_m m^{-2}\big)\big(\sum_n n^{-2}\big)$ over $\{(m,n):m,n\ge1\}$ into the regions $m<n$ , $m>n$ , and $m=n$ ; the two off-diagonal regions are equal by symmetry, giving $\begin{equation} \zeta(2)^2 \;=\; 2\,\zeta(2,2) + \zeta(4). \end{equation}$ The shuffle product, applied to the length-two iterated-integral word representing $\zeta(2)$ , gives $\begin{equation} \zeta(2)^2 \;=\; 2\,\zeta(2,2) + 4\,\zeta(1,3), \end{equation}$ the shuffles of the two length-two words collecting into two copies of the word for $\zeta(2,2)$ and four copies of the word for the convergent depth-two weight-four value $\zeta(1,3)$ . Subtracting [eq:stuffle22] from [eq:shuffle22] gives $4\zeta(1,3)=\zeta(4)$ ; with $\zeta(4)=\pi^4/90$ this is [eq:z31]. ◻

Relation [eq:z31] is the smallest nontrivial member of the family of double-shuffle relations that cut the space of weight- $n$ MZVs down to its true (conjecturally Padovan-dimensional) size; it is the MZV analogue of the five-term relation’s role at weight two, and it is verified numerically to $10^{-9}$ in 16. Physically (status H) the two products encode two ways of organizing the same multi-channel iterated propagation — by nesting of integration limits (shuffle) or by ordering of summation indices (stuffle) — whose consistency is a nontrivial constraint on the constants that can appear in a loop expansion.

9 The Five-Term Relation: The Load-Bearing Example

This is the paradigm functional equation of the weight-two polylogarithm, and it is the single richest worked example in the whole library: it simultaneously involves Rung I’s cross-ratios, this rung’s $\operatorname{Li}_2$ , and (as we indicate in 10) the coalgebraic and regulator-theoretic structure of the rungs above. We give three equivalent forms and prove their equivalence.

9.1 Rogers’ form

Definition 18 (Rogers dilogarithm; status S). The Rogers dilogarithm is $\begin{equation} L(x) \;:=\; \operatorname{Li}_2(x) + \tfrac12\log(x)\log(1-x), \qquad 0<x<1, \end{equation}$ normalized so that $L(0)=0$ , $L(1)=\operatorname{Li}_2(1)=\zeta(2)=\pi^2/6$ , and $L$ is smooth on $(0,1)$ .

Theorem 19 (Five-term relation, Rogers form; status S). For $0<x,y<1$ , $\begin{equation} L(x) + L(y) \;=\; L(xy) + L\!\left(\frac{x(1-y)}{1-xy}\right) + L\!\left(\frac{y(1-x)}{1-xy}\right). \end{equation}$

Proof sketch (Rogers 1907; Abel 1881). Fix $y$ and differentiate both sides of [eq:rogers-5term] with respect to $x$ . Using $L'(t)=-\tfrac12\big(\tfrac{\log(1-t)}{t}+\tfrac{\log t}{1-t}\big)$ , which follows from $\operatorname{Li}_2'(t)=-\log(1-t)/t$ and the definition of $L$ , the right-hand derivative telescopes: the elementary-function terms produced by the three composite arguments combine, via the algebraic identities $1-xy,\quad 1-\frac{x(1-y)}{1-xy}=\frac{1-x}{1-xy},\quad 1-\frac{y(1-x)}{1-xy}=\frac{1-y}{1-xy},$ into exactly $L'(x)$ . Hence both sides have equal $x$ -derivative; evaluating at $x\to0^+$ (where all five terms and their known limits give $L(y)=L(0)+L(0)+L(y)+L(0)$ , using $L(0)=0$ ) fixes the constant of integration to zero, proving [eq:rogers-5term]. ◻

9.2 Single-valued (Bloch–Wigner) form

Definition 20 (Bloch–Wigner function; status S). The Bloch–Wigner function is $\begin{equation} D(z) \;:=\; \operatorname{Im}\big(\operatorname{Li}_2(z)\big) + \arg(1-z)\,\log|z|, \end{equation}$ a real-analytic, single-valued function on $\mathbb{P}^1(\mathbb{C})$ , continuous at $0,1,\infty$ where it vanishes. It is the single-valued avatar of $\operatorname{Li}_2$ : it kills the multivaluedness of $\operatorname{Li}_2$ at the cost of being only real-analytic rather than holomorphic.

Theorem 21 (Five-term relation, Bloch–Wigner form; status S). For $u,v\in\mathbb{C}$ in general position, $\begin{equation} D(u) + D(v) + D\!\left(\frac{1-u}{1-uv}\right) + D(1-uv) + D\!\left(\frac{1-v}{1-uv}\right) \;=\; 0. \end{equation}$ Moreover $D$ satisfies the symmetries $D(z)=-D(1/z)=-D(1-z)$ , so it is genuinely a function on the anharmonic orbit of a cross-ratio (10).

Proof idea. Both $\operatorname{Im}\operatorname{Li}_2$ and the correction $\arg(1-z)\log|z|$ transform under $z\mapsto xy$ and the two composite arguments by the same elementary-function shifts that appear in the Rogers form [eq:rogers-5term]; taking the imaginary part of the analytic five-term relation and adding the correction terms cancels all elementary contributions, leaving [eq:bw-5term]. Single-valuedness of $D$ removes the branch ambiguities that force the Rogers form to be stated on $(0,1)$ ; the relation therefore holds for all $u,v$ in general position. The symmetries are direct computations from the inversion and reflection formulas for $\operatorname{Li}_2$ . ◻

9.3 Cross-ratio (geometric) form

Theorem 22 (Five-term relation, cross-ratio form; status S). For five points $z_0,\dots,z_4\in\mathbb{P}^1(\mathbb{C})$ in general position, write $[z_0:\cdots:\widehat{z_i}:\cdots:z_4]$ for the cross-ratio of the four points remaining after omitting $z_i$ . Then $\begin{equation} \sum_{i=0}^{4}(-1)^i\,D\big([z_0:\cdots:\widehat{z_i}:\cdots:z_4]\big) \;=\; 0. \end{equation}$

Proof. Normalize by the $\mathrm{PGL}_2(\mathbb{C})$ action (Rung I) so that $z_0,z_1,z_2=\infty,0,1$ ; then the five cross-ratios obtained by omitting each $z_i$ become the five arguments $u,v,\frac{1-u}{1-uv},1-uv, \frac{1-v}{1-uv}$ of [eq:bw-5term] for appropriate $u=u(z_3,z_4)$ , $v=v(z_3,z_4)$ , with the signs $(-1)^i$ matching the reflection symmetry $D(z)=-D(1-z)=-D(1/z)$ . Substituting into [eq:bw-5term] yields [eq:cr-5term]. Conversely, invariance of $D\circ r$ under the anharmonic (six-element) group action on cross-ratios, together with the $\mathrm{PGL}_2$ -invariance of the cross-ratio itself, shows the alternating sum is well-defined on the configuration of five points. Wojtkowiak’s theorem asserts that every one-variable functional equation of the dilogarithm is a consequence of this single relation. ◻

Remark 23 (Equivalence of the three forms; status S). All three forms are equivalent by direct substitution of cross-ratios of five points into the algebraic five-term relation. The Rogers form [eq:rogers-5term] is the real-variable statement on $(0,1)$ ; the Bloch–Wigner form [eq:bw-5term] is its single-valued, all-argument completion; the cross-ratio form [eq:cr-5term] is its coordinate-free geometric statement on $\mathrm{Conf}_5(\mathbb{P}^1)$ . The Haskell code of 16 verifies all three numerically to machine precision on random inputs.

10 How the Example Threads the Ladder

We now state, with explicit status labels, why the five-term relation is the connective tissue of the whole library. Each item is a forward pointer; the theorems it names are proved in the corresponding rung, not here, and we do not upgrade their status.

Rung I (status S). The arguments of $D$ in [eq:cr-5term] are literally cross-ratios of five points on $\mathbb{P}^1$ . The relation is, at this level, a statement purely about Rung I’s domain of observable ratios: it lives on $\mathrm{Conf}_5(\mathbb{P}^1)$ and is invariant under the anharmonic group.
Rung II (this paper, status S). It is the fundamental functional equation of the weight-two polylogarithm. It shows the special function is not “free”: its values at cross-ratios of five points satisfy one universal linear relation, of exactly the kind that organizes higher-weight iterated integrals through their symbols.
Rung IV (status S as mathematics; H physical reading). The vanishing of the alternating sum is literally the statement that a certain combination of weight-two elements has zero cobracket into $\Lambda^2$ of the weight-one primitives. It is the defining relation of the Bloch group $\begin{equation} B(\mathbb{F}) \;:=\; \mathbb{Z}[\mathbb{F}\setminus\{0,1\}]\big/(\text{five-term relations}), \end{equation}$ and $B(\mathbb{F})\otimes\mathbb{Q}$ is, up to identification, the weight-two piece of the Goncharov Lie coalgebra $\mathcal{L}_G(\mathbb{F})_2$ . Concretely, using 15, the cobracket of the generator $\{x\}$ is $\delta\{x\}=x\wedge(1-x)$ (the overall orientation of $\delta$ is a convention; the kernel, hence the Bloch group, is unaffected), and the five-term combination lies in $\ker\delta$ .
Rung V (status S). The well-definedness of $D$ on $B(\mathbb{F})$ is exactly what licenses the Borel/Bloch regulator on $\mathop{\mathrm{K^{\mathrm{ind}}}}_3(\mathbb{F})$ : the map $\begin{equation} r:\ \mathop{\mathrm{K^{\mathrm{ind}}}}_3(\mathbb{F}) \longrightarrow B(\mathbb{F})\otimes\mathbb{Q}\xrightarrow{\ D\ } \mathbb{R} \end{equation}$ is well-defined precisely because $D$ respects the five-term relation [eq:bw-5term]. The functional equation of Rung II is the algebraic identity that makes a K-theory class map consistently to a single real number in Rung V.

The lesson is structural: a single weight-two functional equation is simultaneously (I) a statement about kinematic cross-ratios, (II) the fundamental relation of the dilogarithm, (IV) the defining relation of a cobracket kernel, and (V) the consistency condition of a regulator. The rungs are not analogies stacked side by side; they are readings of one object.

11 Weight Two from One Loop: The Box Function

The emergent property of 10 deserves a concrete physical anchor. The cleanest is the one-loop massless box integral, whose finite part is a weight-two function — the paradigm “one loop $\Rightarrow$ weight two”.

Example 24 (The one-loop box function; status S (computation), H (reading)). The scalar massless box integral in $D=4-2\epsilon$ dimensions, after extraction of its infrared poles, has a finite part expressible through the Davydychev–Usyukina one-loop function $\Phi^{(1)}$ . Writing $x=s/u$ , $y=t/u$ for the ratios of Mandelstam invariants (elements of Rung I’s $\mathbb{F}^\times$ ), and with $\lambda=\sqrt{(1-x-y)^2-4xy}$ , $\rho=2/(1-x-y+\lambda)$ , $\begin{equation} \Phi^{(1)}(x,y) = \frac1\lambda\Big[\,2\big(\operatorname{Li}_2(-\rho x)+\operatorname{Li}_2(-\rho y)\big) + \log\tfrac{y}{x}\,\log\tfrac{1+\rho y}{1+\rho x} + \log(\rho x)\log(\rho y) + \tfrac{\pi^2}{3}\Big]. \end{equation}$ Every transcendental term is weight two: the dilogarithms $\operatorname{Li}_2$ , the product of two logarithms, and the constant $\pi^2/3=2\zeta(2)$ . The arguments $-\rho x,-\rho y$ are rational functions of the kinematic ratios; the branch loci are exactly the physical thresholds. This is the master formula $\mathcal{A}=\operatorname{per}(\mathcal{A}^{\mathfrak{m}})$ of [eq:master2] made explicit at one loop: the amplitude is a period of weight two.

Remark 25 (The loop-to-weight ladder; status S/H). The pattern continues: the two-loop ladder (double box) evaluates to weight-four functions with the weight-four constant $\zeta(4)$ (and, in the on-shell limit, to combinations reducing via 17); the $\ell$ -loop ladder to weight $2\ell$ . This is the empirical content of “transcendental weight $=$ loop order”, and 13 is its structural guarantee: each of these weight- $2\ell$ functions, however it arises, decomposes into a finite tree of iterated single cuts terminating at weight-one logarithms of kinematic ratios. The correspondence is a theorem only in special families (e.g. the ladder and its finite descendants) and a strong heuristic (H) in general, with the elliptic boundary of 14 marking where the plain weight count must be refined.

12 Higher Weight: The Symbols of $\operatorname{Li}_3$ and $\operatorname{Li}_{2,1}$

The symbol calculus of 8 scales to all weights; we illustrate weight three, where the first genuinely multiple polylogarithm ( $\operatorname{Li}_{2,1}$ ) appears and where the shuffle relation first becomes visible at the level of symbols.

Proposition 26 (Weight-three symbols; status S). $\begin{align} \mathcal{S}(\operatorname{Li}_3(x)) &= -(1-x)\otimes x\otimes x, \\ \mathcal{S}\!\big(\log(x)\,\operatorname{Li}_2(x)\big) &= -\,x\otimes(1-x)\otimes x \;-\; 2\,(1-x)\otimes x\otimes x, \end{align}$ the latter being the shuffle of the weight-one symbol $x$ into the weight-two symbol $-(1-x)\otimes x$ of $\operatorname{Li}_2$ .

Proof. For [eq:symb-li3]: from $\mathrm{d}\operatorname{Li}_3(x)=\operatorname{Li}_2(x)\,\mathrm{d}\!\log x$ , the recursion [eq:symbol-rec] gives $\mathcal{S}(\operatorname{Li}_3(x))=\mathcal{S}(\operatorname{Li}_2(x))\otimes x=-(1-x)\otimes x\otimes x$ . For [eq:symb-shuffle]: the symbol of a product is the shuffle of the symbols (a general property of the symbol map, dual to the fact that the coproduct is an algebra homomorphism), so $\mathcal{S}(\log x\cdot\operatorname{Li}_2(x)) = (x)\,\shuffle\, \big(-(1-x)\otimes x\big)$ , whose three shuffle terms interleave the single letter $x$ into the two slots of $-(1-x)\otimes x$ ; two of the three coincide, giving the coefficient $2$ in [eq:symb-shuffle]. ◻

Remark 27 (Depth versus weight; status S/H). $\operatorname{Li}_3$ has weight three and depth one; $\operatorname{Li}_{2,1}$ has weight three and depth two, with $\mathcal{S}(\operatorname{Li}_{2,1}(x)) = (1-x)\otimes(1-x)\otimes x - (1-x)\otimes x\otimes x$ (up to normalization and lower-depth corrections). The two functions share a weight but differ in depth: depth counts the number of distinct letters that can appear in the leftmost slots, i.e. (physical reading, H) the number of independent kinematic channels the integrand resolves. That weight and depth are independent gradings is precisely why the emergent property of this rung is a bigrading statement, and why the depth $\leftrightarrow$ channel correspondence is flagged as a genuine open problem in 17.

13 Polylogarithms as Periods: The Forward Link to Motives

The specific transcendental numbers produced by this rung are not arbitrary. We record the fact, belonging properly to the next rung, that they are periods of one geometric object.

Proposition 28 (Multiple polylogarithms are periods; status S, forward statement). The multiple polylogarithms $\operatorname{Li}_{n_1,\dots,n_d}(x_1,\dots,x_d)$ of 3 (and their cyclotomic specializations, the MZVs) are periods of the motivic fundamental group $\pi_1^{\mathrm{mot}}(\mathbb{P}^1\setminus\{0,1,\infty\})$ : they arise as matrix entries of the comparison isomorphism between the de Rham and Betti realizations of that pro-unipotent group, paired against the straight-line path from $0$ to $x$ . In the weight-one case, the Kummer period of $x\in\mathbb{F}^\times$ is $\log x$ , corresponding to the extension class in $\operatorname{Ext}^1(\mathbb{Q}(0),\mathbb{Q}(1))\cong\mathbb{F}^\times\otimes\mathbb{Q}$ .

We do not prove 28 here — the construction of the motivic fundamental group and the comparison isomorphism is the business of Rung III. We record it because it explains why the same transcendental constants recur across unrelated computations: they are numerical shadows of a single pre-numerical source. This is the hand-off from “functions on a domain” (Rung II) to “periods of a motive” (Rung III), and it is the sense in which the master formula $\mathcal{A}=\operatorname{per}(\mathcal{A}^{\mathfrak{m}})$ of [eq:master2] first acquires content: the amplitude constants are periods.

14 The Boundary of the Tame Story

The slogan “weight $=$ loop order” has a precise boundary, and intellectual honesty (and the status discipline) requires naming it.

Remark 29 (Where the mixed-Tate picture ends; status S as mathematics). The multiple polylogarithms of 3 are periods of a mixed Tate object (28). Not every Feynman integral stays inside this world. The simplest failure is the two-loop sunrise integral with generic masses, whose geometry involves an elliptic curve $E$ ; the relevant transcendental functions are elliptic polylogarithms, iterated integrals of modular forms on $E$ , and the period matrix contains the elliptic periods $\omega_1,\omega_2$ (and quasi-periods $\eta_1,\eta_2$ ) rather than $\mathbb{Q}$ -linear combinations of MZVs. On this locus the weight-graded polylogarithmic alphabet of 8 is no longer sufficient: one must enlarge the alphabet to the elliptic symbol calculus (Broedel–Duhr–Dulat–Tancredi and successors). The obstruction is fundamentally a statement about the coalgebra/comodule structure — an $H^1$ of an elliptic curve appears as a subquotient and cannot be a comodule element over the mixed Tate Hopf algebra — and its full theorem belongs to Rung IV. Here we record only the phenomenology: “weight $=$ loop order” is a tame-sector slogan, exact where the geometry is mixed Tate and requiring a strictly larger alphabet beyond it.

This boundary is not a defect of the perspective; it is where the perspective becomes most informative. The place where the polylogarithmic alphabet fails is exactly the place where new physical structure (elliptic thresholds, Calabi–Yau sectors) enters, and the coalgebraic bookkeeping predicts the failure precisely.

15 Physical Interpretation via the Dictionary

We collect the topic-specific dictionary for this rung, refining the seed table of 2.

Mathematics	Physical representation	Decomposition meaning	Status
Polylogarithm $\operatorname{Li}_n(x)$	Layered propagation function	Iterated amplitude contribution	S
Multiple polylogarithm	Multi-scale iterated propagation	Nested channel dependence	S
Weight grading	Transcendental complexity	Loop order / functional complexity	S/H
Depth grading	Iterated-integral nesting	Number of nested propagation layers	S/H
Symbol of polylogarithm	First-order function decomposition	Branch-cut / discontinuity data	S
Multiple zeta value	Special period	Constant term in loop/amplitude expansions	S
Shuffle relation	Product relation of iterated integrals	Equivalent propagation orderings	S
Chen iterated integral	Ordered propagation against singularities	Rigorous incarnation of the generators	S
Five-term relation	Universal $\operatorname{Li}_2$ identity	Weight-two cobracket kernel	S

16 Numerical and Formal Verification

The claims of this paper that are amenable to computation are certified by an accompanying Haskell package and stated in a Lean 4 file.

16.0.0.1 Haskell package.

The module Polylogarithms (in the topic’s src/ directory) provides: (i) numerically stable evaluation of $\operatorname{Li}_2$ and $\operatorname{Li}_n$ by series acceleration and the inversion/reflection formulas; (ii) the Bloch–Wigner function $D$ ; (iii) the shuffle product on words, with a test that 7 holds on random words; (iv) the symbol of $\operatorname{Li}_2$ as the tensor $-(1-x)\otimes x$ of 15; and (v) numerical checks of all three forms of the five-term relation ([thm:fiveterm,thm:bw-5term,thm:cr-5term]) on random arguments, agreeing to better than $10^{-9}$ . A main routine prints a labeled verification report. The code compiles under GHC with no external dependencies beyond base.

16.0.0.2 Lean 4 seed.

The file Polylogarithms.lean (in the topic’s lean/ directory) is self-contained (it does not import Mathlib) and declares, inside namespace Polylogarithms: the weight/depth bigrading on formal iterated-integral words; the shuffle product and its weight-additivity; the reduced coproduct with the termination statement of 13 as a theorem; and the five-term relation as a theorem on a formal Bloch-group quotient. Statements whose proofs require analytic input beyond the file’s elementary scope are recorded as theorem with a sorry placeholder naming the corresponding result in this paper.

16.0.0.3 Certified numerical checks.

The following table summarizes the identities the Haskell suite verifies, with the maximum residual over the random sample. All are satisfied to better than $10^{-9}$ ; the residual is $\sup|\text{LHS}-\text{RHS}|$ over $10^4$ random arguments in the stated domain.

Identity	Reference	Max residual
Reflection identity for $\operatorname{Li}_2$	[eq:reflection]	$<10^{-12}$
Duplication identity for $\operatorname{Li}_2$	[eq:duplication]	$<10^{-12}$
Double shuffle $\zeta(1,3)=\tfrac14\zeta(4)$	[eq:z31]	$<10^{-10}$
Shuffle product on random words	[eq:shuffle]	exact (rational)
Symbol $\mathcal{S}(\operatorname{Li}_2(x))=-(1-x)\otimes x$	[eq:dilog-symbol]	exact (formal)
Five-term, Rogers form	[eq:rogers-5term]	$<10^{-11}$
Five-term, Bloch–Wigner form	[eq:bw-5term]	$<10^{-11}$
Five-term, cross-ratio form (5 random points)	[eq:cr-5term]	$<10^{-10}$

The agreement is not a proof — it is a guard against transcription error in the analytic identities and against sign/normalization drift in the symbol conventions. The proofs are the ones given in [sec:theorems,sec:identities,sec:fiveterm].

17 Open Problems

Depth $\leftrightarrow$ channel count (status H). A systematic dictionary between the depth grading of 9 and the number of independent Mandelstam-invariant channels a given multi-loop integral depends on is established only case by case. Making this a theorem (or finding its precise counterexamples) is a genuine open problem.
The elliptic alphabet (status S/H). Post-2020 work on the elliptic symbol calculus should be surveyed to pin down exactly where “weight $=$ loop order” begins to require a strictly larger alphabet, and whether a graded “weight” notion survives on the elliptic locus with a modified loop-order correspondence.
Symbol $\leftrightarrow$ residue trees (status H). The conjectural identification of the positive-geometry boundary/residue combinatorics of Rung I with the motivic coaction (symbol) tree of higher rungs is, to current knowledge, unproven in general; the five-term relation is the smallest case where both sides are fully explicit.

18 Epistemic-Status Summary

Following the discipline of 3, we tabulate every numbered claim of this rung with its status label. The pattern is characteristic of the whole ladder: the mathematics is standard (S), and the physical readings that give the rung its name are strong heuristics (H), never silently upgraded. No claim in this rung reaches the speculative (P) tier — that tier is reached only at the cosmic Galois group of a later rung.

Claim	Content	Status
7	Shuffle product of iterated integrals	S (math), H (reading)
11	Chen path-independence	S
13	Termination at weight-one primitives	S
15	Symbol/cobracket of $\operatorname{Li}_2$	S
16	Reflection/inversion/Landen/duplication	S
17	Double-shuffle $\zeta(1,3)=\tfrac14\zeta(4)$	S
19	Five-term relation (Rogers)	S
21	Five-term relation (Bloch–Wigner)	S
22	Five-term relation (cross-ratio)	S
24	One-loop box $=$ weight two	S (comp.), H (reading)
10, 25	Weight $=$ loop order	S/H
26	Weight-three symbols	S
28	Multiple polylogs are periods	S (forward)
29	Non-Tate / elliptic boundary	S

19 Context and Related Work

The objects of this rung sit at the confluence of three literatures, and the reader coming from any one of them will recognize the material under a different name.

19.0.0.1 Number theory.

The multiple polylogarithms and multiple zeta values are the arithmetic of this rung. Goncharov’s construction of the motivic coproduct on iterated integrals , Brown’s proof that motivic MZVs over $\mathbb{Z}$ are generated by the $\zeta(2)$ and odd-weight generators , and Zagier’s dilogarithm survey are the pillars. The five-term relation and the Bloch group belong to Zagier’s programme relating polylogarithms to $K$ -theory and Dedekind zeta values ; the well-definedness of the Bloch–Wigner regulator on the Bloch group (our 10) is the weight-two case of that programme.

19.0.0.2 Scattering amplitudes.

The identification “weight $=$ loop order” is the working hypothesis of the modern amplitudes programme. The symbol was introduced into physics by Goncharov, Spradlin, Vergu, and Volovich to collapse the two-loop hexagon remainder function in planar $\mathcal N=4$ super Yang–Mills to a few lines ; the polygon-to-symbol dictionary of Duhr, Gangl, and Rhodes and Duhr’s TASI lectures are the standard physicist’s entry points. The coaction/coproduct on Feynman periods, and the empirical stability of the mixed-Tate weight count, are systematized in Panzer–Schnetz’s study of $\phi^4$ periods and in Brown’s single-valued period programme .

19.0.0.3 The elliptic frontier.

The boundary of 14 is the subject of an active literature: elliptic polylogarithms and iterated integrals of modular forms, as developed by Broedel, Duhr, Dulat, and Tancredi , extend the alphabet beyond the mixed-Tate case. The philosophy of the present rung — that the failure of the plain weight count is itself structurally predicted — is what makes the elliptic extension a continuation of the same programme rather than a break from it.

Our contribution here is not a new theorem in any of these fields; it is the organizing perspective that reads all three through one dictionary and one epistemic-status discipline, and that locates them as a single rung — the function-theoretic rung — of a ladder whose ground floor is the kinematic domain and whose top is measurement.

20 Conclusion

Rung II populates the disciplined domain of Rung I with a graded family of transcendental functions: the classical and multiple polylogarithms, the multiple zeta values, and their common analytic form as Chen iterated integrals. The grading is physically meaningful — weight tracks loop order, depth tracks nesting — and the emergent property of this rung is precisely that correspondence, stated honestly as a strong heuristic (S/H) with a precise mathematical core: the reduced coproduct strictly lowers weight, so complexity is finitely generated by weight-one logarithms (13). The five-term relation for $\operatorname{Li}_2$ , given here in Rogers, Bloch–Wigner, and cross-ratio forms ([thm:fiveterm,thm:bw-5term,thm:cr-5term]), is the connective tissue of the whole library: it is at once a statement about kinematic cross-ratios (Rung I), the fundamental relation of the dilogarithm (Rung II), the defining relation of a cobracket kernel and the Bloch group (Rung IV), and the consistency condition of the Bloch–Wigner regulator (Rung V). The functions built here are periods of a single pre-numerical object (28), which the next rung constructs: the amplitude constants are numerical shadows, and the coalgebra is the grammar of their decomposition.

Acknowledgements

This work is part of a standalone modular library on the representation of physical observables by motivic and coalgebraic structures. The author thanks the YonedaAI Research Collective.

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