TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Equivariant elliptic-complex index character and fixed-component formula

Statement

Assume full AC. Let $T = (R / Z)^{r}$ , $r \geq 0$ , act smoothly by isometries on a compact smooth manifold $M$ without boundary. Let $E^{0}, \dots, E^{N}$ be smooth finite-rank Hermitian complex bundles and $d_{j} : C^{\infty} (M, E^{j}) \to C^{\infty} (M, E^{j + 1})$ first-order differential maps with $d_{j + 1} d_{j} = 0$ , setting $d_{- 1} = d_{N} = 0$ . Their positive-frequency symbols use $\partial_{x}^{α} \mapsto (i ξ)^{α}$ . Assume the symbol sequence is exact for every nonzero real cotangent vector. A jointly smooth unitary bundle action has maps $ρ_{t}^{j} (x) : E_{x}^{j} \to E_{t \cdot x}^{j}$ , with $ρ_{e}^{j} (x) = I$ and $ρ_{s t}^{j} (x) = ρ_{s}^{j} (t \cdot x) ρ_{t}^{j} (x)$ ; set $(U_{t}^{j} s) (x) = ρ_{t}^{j} (t^{- 1} \cdot x) s (t^{- 1} \cdot x)$ and require $U_{t}^{j + 1} d_{j} = d_{j} U_{t}^{j}$ .

Write $d = ⨁_{j} d_{j}$ , let $d^{*}$ be its formal adjoint for the invariant metrics and Riemannian volume, and put $D = d + d^{*}$ . Its even-to-odd symbol defines the compact-support class $σ \in K_{T, c}^{0} (T^{*} M)$ . The cohomology groups $H^{j} = \ker d_{j} / im d_{j - 1}$ are finite dimensional. Define ${Ind}_{T} (E^{∙}) = \sum_{j} (- 1)^{j} [H^{j}] \in R (T)$ , where $R (T)$ is the Grothendieck ring of finite-dimensional continuous complex representations.

For any $g \in T$ , write $M^{g} = ∐_{a} F_{a}$ for its finitely many connected fixed components, $ν_{a} = (T F_{a})^{⊥} \subset T M ∣_{F_{a}}$ , $π_{a} : T^{*} F_{a} \to F_{a}$ , and $i_{a} : T^{*} F_{a} ↪ T^{*} M$ for extension of covectors by zero on $ν_{a}$ . Put $ν_{a, C} = ν_{a} \otimes_{R} C$ and $e_{a} = λ_{- 1} (ν_{a, C}^{*}) = \sum_{k} (- 1)^{k} [Λ^{k} ν_{a, C}^{*}]$ . Then

$χ_{{Ind}_{T} (E^{∙})} (g) = \sum_{a} \int_{T^{*} F_{a}} \frac{{ch}_{g, c} (i_{a}^{*} σ)}{π_{a}^{*} {ch}_{g} (e_{a})} π_{a}^{*} Td (T F_{a} \otimes_{R} C) .$

For a symbol on the $g$ -fixed base, ${ch}_{g, c}$ means the compact-support relative Chern character weighted by the eigenvalue of $g$ on each eigenbundle; for an absolute virtual bundle $V$ , ${ch}_{g} (V) = \sum_{λ} λ ch (V_{λ})$ . The Todd class is $\prod_{j} x_{j} / (1 - e^{- x_{j}})$ in formal Chern roots, truncated in each degree. The integral is the oriented compact-support cohomological pairing on the cotangent bundle, oriented by $d θ = \sum_{j} d ξ_{j} \land d x_{j}$ . The denominator has nonzero degree-zero term $\det_{R} (I - d g ∣_{ν_{a}})$ , so its positive-degree part is inverted by a finite nilpotent series. Its full characteristic class is retained for a positive-dimensional component. The degree-zero value of ${ch}_{g}$ is the trace at $g$ ; it is distinct from the degree-zero rank of a Borel equivariant Chern character.

At an element where the fixed locus changes, the formula uses that element's actual fixed components and normal bundles. It applies also at $g = e$ , when $F = M$ and the normal denominator is $1$ ; it asserts no transversally elliptic distribution formula. If the fixed components are isolated points, it specializes to

$χ_{{Ind}_{T} (E^{∙})} (g) = \sum_{x \in M^{g}} \frac{\sum_{j} (- 1)^{j} tr (ρ_{g}^{j} (x) ∣_{E_{x}^{j}})}{∣ \det_{R} (I - d g_{x}) ∣} .$

This isolated-point assertion also holds assuming only countable choice, provided every fixed point is simple and the nonnegative selfadjoint Hodge realization of $Δ = D^{2}$ has discrete spectrum, smooth eigensections, trace-class heat operators with smooth kernels for positive time, finite-dimensional cohomology, and zero eigenspace canonically isomorphic to that cohomology. Thus the earlier spectral isolated-point case retains its weaker choice assumption.

Statement references

Equivariant elliptic-complex index and character

Equivariant compact-support symbol classes and the normal Koszul class

Analytic symbol index, normal embedding compatibility and ordinary pushforward

Fixed-inclusion localization for equivariant compact symbol classes

Trace-weighted compact symbol character and the cotangent pushforward

Atiyah–Singer I, §§4–9: analytic symbol index and the embedding method

Segal, §§2–4: fixed-inclusion localization in equivariant K-theory

Yi, Theorem 1.1 and §§2–5: the separate simple isolated-point heat-kernel formula

Facts & Assumptions

Given: The equivariant first-order elliptic complex and invariant metrics in the Statement; full AC for the general fixed-component assertion, or the separately stated countable-choice Hodge heat hypotheses for its isolated-point branch.

[F1]

The index and its character are the alternating cohomology representation and trace. Equivariant elliptic-complex index and character

[F2]

The positive-frequency even-to-odd symbol gives a compact-support equivariant triple, and the doubled real normal uses the exterior-multiplication Koszul convention. Equivariant compact-support symbol classes and the normal Koszul class

[L1]

Compact-symbol analytic realization identifies the complex's index with $a_{T, M} (σ)$ . Analytic Gysin compatibility holds for a real equivariant embedding. On a base fixed by $g$ , evaluation is the eigenvalue-weighted ordinary topological pushforward. The full analytic calculus, controlled excision, positive-order associated product and orthogonal Bott normalization are proved in its prerequisite chain. Analytic symbol index, normal embedding compatibility and ordinary pushforward

[L2]

After localization at $S_{g} = {s \in R (T) : s (g) \neq 0}$ , the fixed inclusion reconstructs every class by $σ = \sum_{a} i_{a!} ((π_{a}^{*} e_{a})^{- 1} i_{a}^{*} σ)$ . It proves smoothness and finiteness of the actual fixed components, the normal Koszul self-intersection, and invertibility of the normal class. Fixed-inclusion localization for equivariant compact symbol classes

[L3]

The trace-weighted ordinary compact-symbol pushforward is the cotangent integral of its relative Chern character times $Td (T F_{C})$ , with precisely the stated orientation. It respects localized quotients and retains the full normal characteristic denominator. Trace-weighted compact symbol character and the cotangent pushforward

[L4]

With countable choice and the stated Hodge heat data, simple fixed points have heat-supertrace limit $\sum_{x} Str (ρ_{g} (x)) / ∣ \det_{R} (I - d g_{x}) ∣$ . Relative tangent-groupoid heat-trace limit at simple fixed points

[A1]

Full AC is inherited from the new analytic and topological prerequisites for the general formula. The Axiom of Choice

[A2]

Only countable choice is used through the original local heat supplier in the isolated-point spectral branch. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

technique · Construct the general analytic-symbol reduction and fixed-inclusion reconstruction before evaluating the relative characteristic pushforward; retain the independent isolated heat argument

Given: The hypotheses and exact conventions in the Statement.

1.1givenF1F2L1A1algebra

Put $b = σ_{1} (d)$ at a nonzero real covector. The complex identity gives $b^{2} = 0$ . The positive-frequency symbol of the formal adjoint is $b^{*}$ by integration by parts, so the symbol of $D$ is $b + b^{*}$ . Its square is $b b^{*} + b^{*} b$ ; a vector in its kernel is killed by both $b$ and $b^{*}$ , hence belongs to $\ker b = im b$ and is orthogonal to that image, so is zero. Thus its even-to-odd block is invertible off the zero section and represents $σ$ in [F2]. By [L1] the cohomology is finite dimensional and its alternating representation is exactly $a_{T, M} (σ)$ . This uses the general matrix-symbol analytic chain and does not assume a Clifford-module reduction.

2.1L1L2step 1.1algebra

Apply [L2] to this actual $g$ . Let $b_{a} = (π_{a}^{*} e_{a})^{- 1} i_{a}^{*} σ$ in the localized compact-support group of $T^{*} F_{a}$ . Its reconstruction is $σ = \sum_{a} i_{a!} b_{a}$ . The analytic index in [L1] is $R (T)$ -linear and its Gysin compatibility applies to the real embedding $F_{a} ↪ M$ , with exactly the doubled-real Koszul normal used by [L2]. Applying that homomorphism, and then evaluating the localized representations at $g$ , gives $χ_{{Ind}_{T} (E^{∙})} (g) = \sum_{a} χ_{a_{T, F_{a}} (b_{a})} (g)$ . The sum is finite; all denominators lie in $S_{g}$ , so evaluation is defined.

3.1L1L3step 2.1algebra

The closure of the cyclic group generated by $g$ fixes $F_{a}$ pointwise. The weightwise assertion in [L1] expresses each character on the right of step 2.1 as the weighted ordinary topological pushforward of $b_{a}$ . Apply [L3]. The relative character is multiplicative by an absolute bundle class, so ${ch}_{g, c} (b_{a}) = {ch}_{g, c} (i_{a}^{*} σ) / (π_{a}^{*} {ch}_{g} (e_{a}))$ . Its compact support is inherited from the restricted symbol; the other factors are pulled back from compact $F_{a}$ . The resulting cotangent integral is precisely the displayed term. Sum over $a$ to prove the general formula.

4.1F2L2L3step 3.1algebra

The normal has no $g$ -fixed vector by [L2], so the constant term of ${ch}_{g} (e_{a})$ is the nonzero determinant $\det_{C} (I - g ∣_{ν_{a, C}^{*}}) = \det_{R} (I - d g ∣_{ν_{a}})$ . Orthogonality makes it positive: conjugate nonreal eigenvalues give positive paired factors, and the only possible real eigenvalue different from $1$ is $- 1$ . Positive-degree cohomology of compact $F_{a}$ is nilpotent by its finite dimension, so dividing by that constant and taking a finite geometric series defines the inverse. If $F_{a}$ is a point its Todd factor is $1$ and its symbol triple is the virtual graded fiber, so step 3.1 gives the isolated formula. Empty fixed loci give the empty sum, rank-zero normals give denominator $1$ , and the identity element uses $F = M$ . At an element with larger fixed locus the same proof uses its larger tangent fixed subspace, so no zero normal eigenvalue is retained as a denominator.

5.1givenF1L4A2algebra∎

For the separate countable-choice branch, assume only its stated spectral data. The unitary action commutes with $d$ and hence with $d^{*}$ and $D$ . On each positive $Δ$ -eigenspace with eigenvalue $λ$ , the map $λ^{- 1 / 2} D$ is an equivariant isomorphism between even and odd parity. Their traces therefore cancel. Trace-classness makes the heat trace series absolutely convergent, so these cancellations can be summed. The zero eigenspace is the cohomology by the branch's Hodge hypothesis, giving $Str (U_{g} e^{- t Δ}) = χ_{{Ind}_{T} (E^{∙})} (g)$ for every $t > 0$ . The limit in [L4] proves the isolated-point formula under [A2], independently of [A1] and the general-branch steps. Thus the original weaker-choice spectral assertion is retained as well as the full fixed-component claim.

Proof sources

Atiyah–Singer I, §§4–9, and Segal, §§2–4, organize the normal-index and fixed-inclusion methods. Their required analytic, topological and characteristic calculations are supplied by the declared local proof chain.

Yi, §§2–5: the independent simple-point heat-supertrace argument

Equivariant elliptic-complex index character and fixed-component formula

Statement

Statement references

Facts & Assumptions

Proof

Proof sources

Depends on

Dependency tree · 2 levels

Sources