An Affine Rho-Index Conversion and a Galois Pair on the Poincaré Sphere

Up to conjugacy the Poincaré homology sphere $`S^3/2I`$ carries, besides the trivial connection, exactly two irreducible flat $`\mathrm{SU}(2)`$ connections, interchanged by the Galois automorphism of $`\mathbb{Q}(\sqrt5)`$; it bounds the $`E_8`$ plumbing $`W`$. We record an elementary conversion, obtained by combining the classical Atiyah-Patodi-Singer defect formula with the Dirac/Molien character sums, expressing the odd-signature rho invariant of any flat twist without trivial constituent on any $`S^3/\Gamma`$, $`\Gamma\subset\mathrm{SU}(2)`$ finite, as an affine function of the character sums of the Kronheimer-Nakajima index formulas. On $`S^3/2I`$ it reproduces the printed adjoint rho invariants $`-73/15`$ and $`-97/15`$ from the character sums, and locates their difference $`-8/5`$ exactly on the four conjugacy classes moved by $`\sqrt5\mapsto-\sqrt5`$. The tautological bundles on $`W`$ realize the asymmetry as charge: $`k\equiv cs\bmod 1`$ in all four sectors, with $`k(\mathcal{R}_Q)-k(\mathcal{R}_{Q'})=\varepsilon(H)/\lvert 2I\rvert=-3/5`$ exactly, where $`\varepsilon(H)`$ is the augmentation of Helle's virtual character $`H`$, refining a congruence of Helle.

The filling's raw homology, by contrast, cannot see the asymmetry: the automorphism group of the mod-2/mod-4 quadratic package is transitive on the $`120`$ root classes, and along the restriction route every term localized on a $`\mathbb{Z}_2`$-null non-orientable surface receives every bundle trivially, hence cancels in the Galois difference. The dichotomy is the result: the filling detects the Galois pair through its tautological decoration; the raw mod-2/mod-4 homology and every restriction-route surface term are blind to it, while equivariant-lift and restricted-connection channels lie outside this route and are not addressed. Whether either excluded channel detects the pair on a characteristic surface remains open. The ingredients are classical, but the results are new statements, not a new method: the affine conversion in Kronheimer-Nakajima currency, the golden-class localization, the exact charge equality, and the two scoped blindness results are theorems assembled from long-known inputs.

1. Introduction

A characteristic surface in a compact oriented $`4`$-manifold detects an eta-type invariant of the manifold's boundary: the Guillou-Marin congruence reads the signature defect mod $`16`$ from the normal Euler number and the Brown invariant of the surface. The Poincaré homology sphere carries a finer eta-type asymmetry than the signature defect. Up to conjugacy it has, besides the trivial connection, exactly two irreducible flat $`\mathrm{SU}(2)`$ connections; their characters generate $`\mathbb{Q}(\sqrt5)`$, the Galois automorphism $`\sigma:\sqrt5\mapsto-\sqrt5`$ interchanges them, and the difference of their adjoint Atiyah-Patodi-Singer rho invariants is $`-8/5`$, supported exactly on the four conjugacy classes that $`\sigma`$ moves. Its canonical filling, the $`E_8`$ plumbing $`W`$, is homologically as symmetric as a $`4`$-manifold can be. The question this paper answers, with a yes and a no, is: which structures on the filling see the boundary's asymmetry?

The setting is uniform in the group. For a finite subgroup $`\Gamma\subset\mathrm{SU}(2)`$, write $`Y_\Gamma^+`$ for $`S^3/\Gamma`$ with the quotient orientation of the link of $`\mathbb{C}^2/\Gamma`$, and for a flat unitary twist $`\alpha`$ write $`D_\alpha`$ for the character sums of the Kronheimer-Nakajima index formulas (§3.1). For $`2I`$ the two irreducible flat connections are named by their representations $`Q`$ and $`Q'`$, and $`+\Sigma:=Y_{2I}^+`$. We first record a conversion identity valid for every $`\Gamma`$ (Theorem 1.1), then apply it to $`2I`$ and its filling (Theorem 1.2), and finally delimit the correspondence (Theorem 1.3).

Theorem 1.1 (Conversion identity). For every finite $`\Gamma\subset\mathrm{SU}(2)`$ and every flat unitary twist $`\alpha`$ on $`Y_\Gamma^+`$ with no trivial constituent,

$$\rho_\alpha(Y_\Gamma^+)\;=\;\dim\alpha\;+\;4\bigl(D_\alpha-\dim\alpha\cdot D_1\bigr).$$

The hypothesis is sharp: a trivial constituent of multiplicity $`m`$ shifts the identity by exactly $`m`$ (Corollary 3.2).

Theorem 1.2 (Galois asymmetry, tautologically realized). On $`+\Sigma`$ the adjoint rho invariants of the Galois pair are $`\rho_{\mathrm{Sym}^2Q}=-73/15`$ and $`\rho_{\mathrm{Sym}^2Q'}=-97/15`$ (both in print: [Anvari, Table 1]; the first also [BHKK, §6], transported); their difference

$$\rho_{\mathrm{Sym}^2Q'}-\rho_{\mathrm{Sym}^2Q}\;=\;4\,\bigl(D_{\mathrm{Sym}^2Q'}-D_{\mathrm{Sym}^2Q}\bigr)\;=\;-\tfrac{8}{5}$$

is supported exactly on the four golden classes; and the tautological bundles on $`W`$ realize the asymmetry as charge: $`k(\mathcal{R}_\alpha)\equiv cs(\alpha)\bmod 1`$ in all four sectors $`\alpha\in\lbrace Q,Q',\mathrm{Sym}^2Q,\mathrm{Sym}^2Q'\rbrace`$, with the exact equality $`k(\mathcal{R}_Q)-k(\mathcal{R}_{Q'})=\varepsilon(H)/\lvert 2I\rvert=-3/5`$, where $`H`$ is Helle's normalized virtual character (§5.2) and $`\varepsilon`$ its augmentation.

Theorem 1.3 (Blindness and decoupling). (i) The automorphism group of $`(H_2(W;\mathbb{Z}_2),\ \cdot\ ,\mathfrak{P})`$, with $`\cdot`$ the mod-2 intersection pairing and $`\mathfrak{P}`$ its mod-4 Pontryagin-square refinement (§6), is transitive on the $`120`$ classes with $`\mathfrak{P}=2`$; consequently no invariant of the isomorphism class of the pointed quadratic space distinguishes the class of the distance-six node from any other class with $`\mathfrak{P}=2`$. (ii) For every restriction-route identity in the sense of Definition 7.3, the term localized on a $`\mathbb{Z}_2`$-null non-orientable surface $`F`$ equals $`\mathrm{rk}(\mathcal{R})\cdot T_F`$ for some $`T_F`$ independent of the coefficient bundle, and cancels identically in the Galois difference of the rank-three adjoints; identities coupling the bundle to $`F`$ through other channels are not addressed.

The dichotomy is not a failure of information transfer. The filling knows the boundary's Galois asymmetry exactly: by Theorem 1.1 the rho difference and the tautological charge difference are the same character sum read twice, and Theorem 1.2 prices the asymmetry in the filling's own currency. What Theorem 1.3 shows is where that information lives: in the tautological decoration of the McKay classes and, along the restriction route, nowhere in the raw mod-2/mod-4 homology or its characteristic-surface refinement. Blindness of the package, sensitivity of the decoration.

On attribution. The paper formulates the classical defect-sum computation in the Kronheimer-Nakajima character-sum currency (Theorem 1.1; the affine form appears to be unrecorded, though both of its inputs are long known); verifies the printed rho pair three independent ways (Proposition 4.1); locates the difference on the golden classes (Corollary 4.2, via an elementary equivariance lemma); refines Helle's augmentation congruence to an exact equality of charges (Proposition 5.3) and records the worked four-sector charge table (Proposition 5.2, the mod-1 principle being classical); and proves the two scoped negatives (Theorem 1.3). The novelty is at the level of statement, not of method: no new machinery, no new invariant, and no universal independence theorem is introduced, yet the affine conversion identity, the golden-class localization, the exact charge equality, and the two scoped negatives are new theorems, each assembled from classical ingredients.

The neighboring literature. The identification $`\Sigma(2,3,5)=S^3/2I`$ is used throughout without further comment. The rho values and their integrality mechanism are [BHKK]; the printed table on the link orientation is [Anvari], after [Saveliev]; the flat connections' isolation and the $`R`$-invariant technology are [FintushelStern]. The tautological bundles and index formulas are [KronheimerNakajima] and [Nakajima]; the Dirac-side character sums are printed in [Degeratu] (Molien form) and [CisnerosMolina] (spectra); the defect-sum lineage is [APS2], [Gilkey], [BotvinnikGilkey]. The Chern-Simons machinery is [Helle], with classical antecedents [Auckly], [KirkKlassen]; the transgression principle in cylindrical-end form is [HeddenKirk]. On the surface side, the closed congruence is [GuillouMarin] in [Matsumoto]'s exposition, the topological correction is [KirbyTaylor], the relative form is [Klug], the $`S^4`$ ancestor is [Massey], and the homology-cobordism setting is [LRS]. The consistency web of §4 verifies against [RubermanSaveliev] and [NeumannSiebenmann].

Organization. Section 2 fixes the group data, the golden classes, and the load-bearing orientation dictionary. Section 3 proves Theorem 1.1 and calibrates every convention against print. Sections 4 and 5 prove Theorem 1.2, the boundary half and the interior half. Section 6 proves Theorem 1.3(i), and Section 7, after exhibiting the three localization mechanisms and abstracting the restriction route, proves Theorem 1.3(ii). Section 8 collects the dichotomy, the open questions, and the directions this suggests.

2. The Galois pair on $`S^3/2I`$

2.1 The binary icosahedral group

Let $`2I\subset\mathrm{SU}(2)`$ denote the binary icosahedral group, the preimage of the icosahedral rotation group under $`\mathrm{SU}(2)\to\mathrm{SO}(3)`$: the unique nontrivial perfect finite subgroup of $`\mathrm{SU}(2)`$, of order $`120`$. It has nine conjugacy classes; an element $`g\neq\pm I`$ is determined up to conjugacy by its defining-representation angle $`\phi_g\in(0,\pi)`$, the defining representation having eigenvalues $`e^{\pm i\phi_g}`$. The nine irreducible representations, of dimensions $`1,2,2,3,3,4,4,5,6`$, are the nodes of the affine $`E_8`$ diagram under the McKay correspondence [McKay], with the trivial representation at the affine node. We write $`Q`$ for the defining two-dimensional representation (McKay distance $`d(Q)=1`$ from the affine node), $`Q'`$ for the other two-dimensional irreducible ($`d(Q')=7`$), and note $`d(\mathrm{Sym}^2Q)=2`$, $`d(\mathrm{Sym}^2Q')=6`$.

2.2 The Galois automorphism and the golden classes

The character field of $`2I`$ is $`\mathbb{Q}(\sqrt5)`$, and the nontrivial field automorphism $`\sigma:\sqrt5\mapsto-\sqrt5`$ acts on characters. On the two-dimensional irreducibles it acts by exchange: $`\sigma(\chi_Q)=\chi_{Q'}`$, and consequently $`\sigma(\chi_{\mathrm{Sym}^2Q})=\chi_{\mathrm{Sym}^2Q'}`$. We call $`Q`$ and $`Q'`$ the Galois pair.

Definition 2.1 (Golden classes). The golden classes of $`2I`$ are the four conjugacy classes of elements of orders $`5`$ and $`10`$, with defining-representation angles $`\phi\in\lbrace \pi/5,\ 2\pi/5,\ 3\pi/5,\ 4\pi/5\rbrace`$. They are precisely the classes on which some character of $`2I`$ is irrational, that is, the locus moved by $`\sigma`$.

The table records the class data the proofs below consume; the full character table is in [Helle]. Sizes refer to the number of elements in the class.

Lemma 2.2 (Galois equivariance). Let $`c`$ be a class function on $`2I`$ with values in $`\mathbb{Q}(\sqrt5)`$ whose value $`c(g)`$ is fixed by $`\sigma`$ whenever $`g`$ does not lie in a golden class, and for a tuple $`\boldsymbol\alpha=(\alpha_1,\dots,\alpha_r)`$ of representations let $`F_{\boldsymbol\alpha}(g)=P\bigl(\chi_{\alpha_1}(g),\dots,\chi_{\alpha_r}(g);\ c(g)\bigr)`$ with $`P`$ a polynomial with rational coefficients. If $`\boldsymbol\alpha^\sigma=(\alpha_1^\sigma,\dots,\alpha_r^\sigma)`$ is the induced action on representations, then $`F_{\boldsymbol\alpha^\sigma}-F_{\boldsymbol\alpha}`$ is supported on the golden classes. In particular $`c=1/(2-\chi_Q)`$ is allowed: it is irrational on the golden classes, but on every other class $`\chi_Q`$ is rational, so $`c`$ is $`\sigma`$-fixed there.

Proof. Let $`g`$ lie outside the golden classes. By Definition 2.1 every character value at $`g`$ is rational, so $`\chi_{\alpha_i^\sigma}(g)=\sigma(\chi_{\alpha_i}(g))=\chi_{\alpha_i}(g)`$ for each $`i`$, and $`c(g)`$ is $`\sigma`$-fixed by hypothesis. Every input of $`P`$ is therefore unchanged by the substitution $`\boldsymbol\alpha\mapsto\boldsymbol\alpha^\sigma`$, so $`F_{\boldsymbol\alpha^\sigma}(g)=F_{\boldsymbol\alpha}(g)`$. $`\square`$

Lemma 2.2 is the engine behind every "supported on the golden classes" statement below: any rationally-built class-function difference between $`\sigma`$-conjugate inputs lives on the golden locus.

2.3 Flat connections and their adjoints

Conjugacy classes of homomorphisms $`\pi_1(S^3/2I)=2I\to\mathrm{SU}(2)`$ correspond to flat $`\mathrm{SU}(2)`$ connections on $`S^3/2I`$ up to gauge. Up to conjugacy there are, besides the trivial one, exactly two irreducible classes, given by $`Q`$ and $`Q'`$; both are isolated and nondegenerate [FintushelStern]. We name flat connections by their representations. The complexified adjoint of the flat connection $`Q`$ is $`\mathrm{Sym}^2Q`$, and likewise for $`Q'`$: the Galois pair of two-dimensional representations induces the Galois pair of three-dimensional adjoints, at McKay distances $`2`$ and $`6`$. Among the finite subgroups of $`\mathrm{SU}(2)`$, $`2I`$ alone has an adjoint Galois pair that is distinct; its perfectness forces this, as Section 8 records.

2.4 Orientations and conventions

Conventions are load-bearing. Every signed rational in this paper is pinned to an explicit orientation, and the table below is the dictionary. Write $`Y_\Gamma^+`$ for $`S^3/\Gamma`$ with the quotient orientation it inherits as the link of $`\mathbb{C}^2/\Gamma`$ (the boundary at infinity of the minimal resolution), and $`+\Sigma:=Y_{2I}^+`$ for the Poincaré homology sphere so oriented; $`W`$ denotes the $`E_8`$ plumbing, the compact truncation of the minimal resolution of $`\mathbb{C}^2/2I`$ (the noncompact resolution itself is $`X`$, §3.1), a simply connected spin $`4`$-manifold with $`\partial W=+\Sigma`$, intersection form the negative definite $`E_8`$ lattice, and $`H^1(W;\mathbb{Z}_2)=0`$.

One further prose note the table cannot carry: the matching of our $`(Q,Q')`$ with printed connection labels is fixed by Chern-Simons values ($`cs(Q)\equiv-1/120`$, $`cs(Q')\equiv-49/120`$ on $`+\Sigma`$); [Anvari, Table 1] binds the Floer index, half the rho invariant, and the Chern-Simons value per connection in one printed place on this orientation. The dictionary above aligns it with [BHKK]'s surgery model, so every comparison in this paper can be checked line by line. Finally, the rho invariant of a flat bundle is independent of the Riemannian metric [APS2], so no metric parameter enters this paper.

3. The conversion identity

This section is general: $`\Gamma`$ is any finite subgroup of $`\mathrm{SU}(2)`$, and $`2I`$ plays no special role until §4.

3.1 The two currencies

For a flat unitary twist $`\alpha`$ on $`Y_\Gamma^+`$, the Atiyah-Patodi-Singer rho invariant is computed by the classical $`G`$-signature defect sum [APS2]:

where $`e^{\pm i\phi_g}`$ are the defining eigenvalues of $`g`$; the sign of the defect is tied to the orientation, and Proposition 3.3 below calibrates this convention against printed values. (The raw fixed-point datum for the angle pair $`(\phi,-\phi)`$ is $`\cot(\phi/2)\cot(-\phi/2)=-\cot^2(\phi/2)`$; the $`+\cot^2`$ displayed here is that datum read on the link orientation $`Y_\Gamma^+`$, which is the content of the calibration.) The computation belongs to the classical defect-sum technology of Atiyah-Patodi-Singer and Gilkey [APS2], [Gilkey].

The second currency is the character sum

writing $`D_1`$ for the trivial twist. These sums are simultaneously spectral and geometric objects in the literature. Spectrally, they are Dirac eta invariants: Degeratu's direct group-sum formula for the twisted Dirac operator on $`S^{2n-1}/\Gamma`$ ([Degeratu], Cor. 2.4, with the half-determinant trivial for $`\Gamma\subset\mathrm{SU}(n)`$ by her Cor. 3.3) reads, at $`n=2`$,

Her sphere at infinity is the boundary of the minimal resolution of $`\mathbb{C}^2/\Gamma`$, so it carries the link orientation $`Y_\Gamma^+`$ by construction; the sign convention is calibrated in Proposition 3.3 below. Thus $`D_\alpha`$ is half the twisted Dirac eta, and her Molien-series theorem expresses the same quantity as a residue of the singularity's graded character series. Geometrically, they are index integrals: by the Kronheimer-Nakajima index theorem on the minimal resolution ([KronheimerNakajima], (A.2) with one trivial factor), $`D_\alpha=\int_X\mathrm{ch}(\mathcal{R}_\alpha)\hat{A}`$ for the tautological bundle $`\mathcal{R}_\alpha`$ over the minimal resolution $`X`$ of $`\mathbb{C}^2/\Gamma`$. This double role is used in §5.

Theorem 1.1 is the statement that the first currency is an affine function of the second. The identity is an elementary consequence of the two classical formulas just cited; its content is the exact form, which we could not locate in print, and which is what §§4-5 consume.

3.2 The trigonometric kernel

Lemma 3.1. For $`g\neq I`$ in $`\Gamma\subset\mathrm{SU}(2)`$ with defining eigenvalues $`e^{\pm i\phi}`$,

$$\csc^2(\phi/2)\;=\;\frac{4}{2-\chi_Q(g)},\qquad\text{hence}\qquad \cot^2(\phi/2)\;=\;-1+\frac{4}{2-\chi_Q(g)}.$$

Proof. $`2-\chi_Q(g)=2-2\cos\phi=4\sin^2(\phi/2)`$, which is nonzero for $`g\neq I`$ (for $`g=-I`$, $`\phi=\pi`$, both sides of the first identity equal $`1`$). The second identity is $`\cot^2=\csc^2-1`$. $`\square`$

The factor $`4`$ produced here is trigonometric. It is unrelated to the Dynkin-index factor $`4`$ by which the Chern-Simons invariants of adjoint and defining flat connections are compared in §5, and the two must not be conflated.

3.3 The identity

Proof of Theorem 1.1. Let $`\alpha`$ contain no trivial constituent, so that $`\sum_{g\in\Gamma}\chi_\alpha(g)=0`$ and hence $`\sum_{g\neq 1}\chi_\alpha(g)=-\dim\alpha`$. Substituting Lemma 3.1 into the defect sum,

The first sum evaluates by orthogonality: $`\sum_{g\neq 1}(\chi_\alpha-\dim\alpha)=-\dim\alpha-(\lvert\Gamma\rvert-1)\dim\alpha=-\lvert\Gamma\rvert\dim\alpha`$, contributing $`+\dim\alpha`$. The second is $`4(D_\alpha-\dim\alpha\,D_1)`$ by definition. Hence

Equivalently, through Degeratu's formula, $`\rho_\alpha=\dim\alpha+2\bigl(\eta_{\mathrm{Dir},\alpha}-\dim\alpha\,\eta_{\mathrm{Dir},1}\bigr)`$: the odd-signature rho invariant of a flat twist is an affine function of twisted Dirac eta invariants.

Corollary 3.2 (Sharpness). If $`\alpha`$ contains the trivial representation with multiplicity $`m`$, then the left side of Theorem 1.1 equals $`\rho_\beta`$ for $`\beta=\alpha\ominus\mathbf{1}^{\oplus m}`$, while the right side equals $`\rho_\beta+m`$: the identity acquires the exact additive offset $`m`$.

Proof. Trivial summands contribute zero to $`\chi_\alpha-\dim\alpha`$ pointwise, so the defect sum, and with it the left side, equals $`\rho_\beta`$. On the right, $`D_{\beta\oplus\mathbf{1}^{\oplus m}}=D_\beta+m\,D_1`$ and $`\dim(\beta\oplus\mathbf{1}^{\oplus m})=\dim\beta+m`$, so

(Checked instance: $`\alpha=Q\oplus\mathbf{1}`$ gives right side $`-29/30`$ against $`\rho_Q=-59/30`$, offset exactly $`1`$.) $`\square`$

Proposition 3.3 (Verification against print). With the conventions of §2.4, Theorem 1.1 reproduces the canonical-representation rho invariants of the Poincaré homology sphere printed by Boden, Herald, Kirk, and Klassen: $`\rho_Q(X_{+1})=59/30`$ and $`\rho_{Q'}(X_{+1})=131/30`$ [BHKK, §5.4-5.5].

Proof. This is a verification of conventions, not a result. The character sums for $`\Gamma=2I`$, computed exactly from the table of §2.2, are

and Theorem 1.1 gives, on $`+\Sigma`$:

The first two rows match the printed values through the oriented identification $`X_{+1}\cong-\Sigma`$ and the sign reversal of §2.4; the last two match [Anvari]'s table directly on $`+\Sigma`$. All four printed comparisons pass. $`\square`$

4. The rho invariants of the Galois pair

Proposition 4.1 (The pair, in print and twice re-derived). On $`+\Sigma`$: $`\rho_{\mathrm{Sym}^2Q}=-73/15`$ and $`\rho_{\mathrm{Sym}^2Q'}=-97/15`$.

Proof. Three independent routes, recorded because each pins a different convention.

(i) Print. Anvari's table for $`\Sigma(2,3,5)`$ lists, per flat connection, the halved rho invariants $`-73/30`$ and $`-97/30`$ beside $`-cs=1/120`$ and $`49/120`$, on exactly this orientation, computed by a flat $`\mathrm{SO}(3)`$-cobordism to lens spaces [Anvari, Table 1], after [Saveliev, p. 144]. (He tabulates $`\rho/2`$ of the unreduced $`\rho`$ of §2.4; for these acyclic twists that number coincides with the reduced-convention value, so no normalization ambiguity survives the comparison.) The $`cs`$ pairing matches the dictionary of §2.4. Independently, Boden, Herald, Kirk, and Klassen compute $`\rho_{\mathrm{Sym}^2Q}(X_{+1})=\pm 73/15`$ and resolve the sign to $`+73/15`$ by integrality [BHKK, §6]; the §2.4 bookkeeping ($`X_{+1}\cong-\Sigma`$, $`\rho`$ reverses) transfers it to $`-73/15`$ here, agreeing.

(ii) Character sums. Theorem 1.1 with the values of Proposition 3.3 gives both numbers (rows three and four of that table).

(iii) Integrality, both connections. The sign of either value is fixed independently of the calibration of §2.4 by the mechanism of [BHKK]: their equation (6.5), for a path on $`X_{+1}`$ from the trivial connection $`\Theta`$ to a flat $`\alpha`$,

must have integer left side, where $`SF`$ is the spectral flow and $`h_\gamma`$ the kernel dimension at the flat connection $`\gamma`$. For the second connection, $`h_\Theta=3`$, $`h_\alpha=0`$ (nondegeneracy [FintushelStern]), $`cs(\Theta)=0`$, and $`\rho_{\mathrm{ad}\,\Theta}=0`$; the printed representative is $`cs(Q')=-71/120`$ on $`X_{+1}`$ [BHKK, Table 1] (the same residue as §2.4: $`-71/120\equiv 49/120 \bmod 1`$, and $`-49/120`$ on $`+\Sigma`$ corresponds to $`+49/120`$ on $`X_{+1}`$). The two sign candidates for the second connection give

so only $`+97/15`$ on $`X_{+1}`$, that is $`-97/15`$ here, is admissible. The conclusion is lift-independent, since an integer shift of the Chern-Simons representative shifts $`SF`$ by a multiple of $`8`$; the same two-candidate check for the first connection reproduces [BHKK]'s own resolution. $`\square`$

Corollary 4.2 (Golden support). On $`+\Sigma`$,

$$\rho_{\mathrm{Sym}^2Q'}-\rho_{\mathrm{Sym}^2Q}\;=\;4\,\bigl(D_{\mathrm{Sym}^2Q'}-D_{\mathrm{Sym}^2Q}\bigr)\;=\;4\cdot\bigl(-\tfrac{2}{5}\bigr)\;=\;-\tfrac{8}{5},$$

and every contribution to the difference is supported on the four golden classes.

Proof. The first equality is Theorem 1.1 applied to the two adjoints, whose dimensions agree. Support: the defect summand is built rationally from characters and the $`\sigma`$-fixed-off-golden class function $`\cot^2(\phi/2)`$, and $`\mathrm{Sym}^2Q'=(\mathrm{Sym}^2Q)^\sigma`$, so Lemma 2.2 applies; concretely, the four non-identity non-golden classes contribute $`-\tfrac{57}{4}`$ (times $`\tfrac{1}{120}`$) to each of $`D_{\mathrm{Sym}^2Q}`$ and $`D_{\mathrm{Sym}^2Q'}`$ alike, while the four golden classes contribute $`+48`$ and $`0`$ respectively. $`\square`$

In this rho difference, the boundary sees the Galois action exactly on the golden classes.

Consistency web. The same conventions give $`\eta_{\mathrm{Dir}}(+\Sigma)=2D_1=\tfrac{1079}{720}`$ and, from the untwisted defect sum, $`\eta_{\mathrm{Sign}}(+\Sigma)=\tfrac{1}{120}\sum_{g\neq 1}\cot^2(\phi_g/2)=\tfrac{361}{180}`$; then

reproducing the identity of Ruberman and Saveliev exactly on this orientation [RubermanSaveliev, Thm 1.1], with $`\bar\mu=-1`$ in the Neumann normalization [NeumannSiebenmann]. The check is sharp: the mixed sign conventions give $`\pm\tfrac{359}{720}`$, not integers, so the identity locks the orientation dictionary of §2.4 a third time, alongside the internal consistency of the index formulas and the calibration of Proposition 3.3. (The $`E_8`$ plumbing also saturates the bound $`b_2\le-8\bar\mu`$ for negative definite spin fillings.) These are verifications, not results, and none is used later.

5. Tautological bundles and exact charges

5.1 The bundles and their charges

This is the interior half of Theorem 1.2: the same character sums, now read on the filling. In the ALE setting of [KronheimerNakajima], $`X`$ (§3.1) is the noncompact minimal resolution of $`\mathbb{C}^2/2I`$ and $`W`$ (§2.4) is its compact truncation, so finite-action Chern-Weil integrals are computed on $`X`$ while boundary-sign checks live on $`(W,+\Sigma)`$. Each nontrivial irreducible representation $`\alpha`$ of $`2I`$ determines the tautological bundle $`\mathcal{R}_\alpha`$ on $`X`$, carrying a canonical finite-action anti-self-dual connection whose flat limit at infinity is the flat connection of $`\alpha`$ [KronheimerNakajima, Prop. 2.2]. The first Chern classes of these bundles are dual to the exceptional curves $`\mathcal{E}_\alpha`$ of the resolution, $`\langle c_1(\mathcal{R}_\alpha),[\mathcal{E}_\beta]\rangle=\delta_{\alpha\beta}`$, the curves realizing the McKay nodes in $`H_2`$ [KronheimerNakajima, Thm. A.7]; this dual-basis fact is used in §7. Define the charge by the curvature integral

Lemma 5.1 (Evaluation). $`k(\mathcal{R}_\alpha)=\dim\alpha\,D_1-D_\alpha`$.

Proof. The degree-four part of $`\mathrm{ch}(\mathcal{R}_\alpha)\hat{A}`$ is $`\bigl(\tfrac12 c_1^2-c_2\bigr)+\dim\alpha\,\hat{A}_2`$, with $`\hat{A}_2`$ the component of complex degree $`2`$ (real degree $`4`$), so $`\int_X\mathrm{ch}(\mathcal{R}_\alpha)\hat{A}=-k(\mathcal{R}_\alpha)+\dim\alpha\int_X\hat{A}`$. The left side is $`D_\alpha`$ by [KronheimerNakajima, (A.2)] with one trivial factor, and $`\int_X\hat{A}=D_1`$ is the trivial instance of the same identity; independently, the signature theorem on the compact truncation gives $`\int_X\hat{A}=-\tfrac18\bigl(\mathrm{sign}(W)+\eta_{\mathrm{Sign}}(+\Sigma)\bigr)=1-\tfrac{361}{1440}=\tfrac{1079}{1440}=D_1`$, corroborating the constant $`D_1`$ from a second direction. $`\square`$

Proposition 5.2 (Charge realizes Chern-Simons). In the four sectors $`\alpha=Q,\ Q',\ \mathrm{Sym}^2Q,\ \mathrm{Sym}^2Q'`$ the charges are

bundle	rank	$`k`$	$`k\bmod 1`$	$`cs`$ of the flat limit mod 1, on $`+\Sigma`$
$`\mathcal{R}_Q`$	2	$`119/120`$	$`119/120`$	$`-1/120\equiv 119/120`$
$`\mathcal{R}_{Q'}`$	2	$`191/120`$	$`71/120`$	$`-49/120\equiv 71/120`$
$`\mathcal{R}_{\mathrm{Sym}^2Q}`$	3	$`59/30`$	$`29/30`$	$`4\cdot(-1/120)\equiv 29/30`$
$`\mathcal{R}_{\mathrm{Sym}^2Q'}`$	3	$`71/30`$	$`11/30`$	$`4\cdot(-49/120)\equiv 11/30`$

and in each row $`k\equiv cs \bmod 1`$. In the adjoint sectors $`k(\mathcal{R}_{\mathrm{Sym}^2Q'})-k(\mathcal{R}_{\mathrm{Sym}^2Q})=+\tfrac25`$, the representative in $`[0,1)`$ of $`cs(\mathrm{Sym}^2Q')-cs(\mathrm{Sym}^2Q) \bmod 1`$.