// ============================================================================= // Emergent Conformal Geometry from Projection Kernels: // A Projected Ontology Account of QED and Gravitation // ============================================================================= // // This paper addresses the common-geometry problem in POT: if observable // geometry is a projection artifact, why do physically different kernels // (electromagnetic, gravitational) project the same spacetime? // // Two candidate geometry-bearing signatures — CharacteristicSignature C(K) // and TopologySignature T(K) — are introduced. The paper proves a common // characteristic signature across QED, Einstein GR, and Mannheim conformal // gravity (C_QED = C_GR = C_CG), and establishes formulation-invariance // properties of the topology signature. Whether T is common across these // physical theories remains an open cross-theory obligation. // A conformal countermodel establishes that T+C cannot determine a full // metric representative, while admitting a consistent model in which the // residual ambiguity is precisely conformal: g ~ Ω²g. // // Formal companion: theories/pot_characteristic_topology.kleis (15 Z3 results) // // Pipeline: // kleis test --raw-output --example compile \ // docs/papers/pot_emergent_conformal_geometry_paper.kleis \ // > pot_emergent_conformal_geometry_paper.typ // typst compile pot_emergent_conformal_geometry_paper.typ // // ============================================================================= import "stdlib/prelude.kleis" import "stdlib/templates/arxiv_paper.kleis" // ============================================================================= // Paper Metadata // ============================================================================= define paper_title = "Emergent Conformal Geometry from Projection Kernels" define paper_authors = [ Author("Engin Atik", "1") ] define paper_affiliations = [ Affiliation(1, "Kleis Research", "https://kleis.io") ] define paper_abstract = "The Projected Ontology Theory (POT) describes observable geometry as a projection artifact: the composed map $Q compose K$ from configurations to observables, not a pre-existing background. This incurs a consistency obligation. Physically different production kernels --- electromagnetic and gravitational --- must project the same observable spacetime despite $K_(\"QED\") != K_(\"grav\")$. What invariant structure can different physical kernels share that is sufficient to support a common geometry? We define two candidate geometry-bearing signatures of a production kernel: the characteristic signature $C(K)$, which extracts causal/wavefront structure from the principal symbol while forgetting multiplicity, and the kernel topology signature $T(K)$, which extracts projection-invariant topological data from the Picard-Lefschetz thimble decomposition in configuration space, modulo Stokes rearrangements. A multiplicity invariance theorem --- $Z(p) = Z(p^m)$ for $m >= 1$ --- establishes that second-order and fourth-order field equations can share the same characteristic variety: $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$, where $C_(\"CG\")$ denotes Mannheim's conformal gravity. Formulation-invariance properties of the kernel topology signature are established, while equality of $T$ across the physical theories remains an open cross-theory obligation. A conformal countermodel then demonstrates that $T + C$ are insufficient to determine a full metric representative: two kernels can share topology and causal structure while differing by a conformal factor $g'_(mu nu) = Omega^2(x) g_(mu nu)$. The formalism therefore encounters conformal geometry before metric geometry. This connects structurally --- not merely phenomenologically --- to conformal gravity, which elevates precisely this residual freedom to gauge equivalence. We identify the open theorem obligation ($T + C limits(arrow.r.double)^? [g]_(\"conf\")$) and the missing scale signature needed to select a metric representative, leaving its physical identity as a research target rather than an assumption. All structural results are formalized and checked for consistency and compositional closure by Z3 on the Kleis platform; the dependency graph --- which axioms each result requires --- is explicitly recorded." define paper_keywords = "projected ontology, conformal geometry, characteristic signature, topology signature, Picard-Lefschetz, conformal gravity, Mannheim, common geometry, kernel projection, formal verification, Z3" // ============================================================================= // Section 1: Introduction — The Common-Geometry Problem // ============================================================================= define sec_intro = ArxivSection("Introduction: The Common-Geometry Problem", "Projected Ontology Theory (POT) asserts that observable geometry is not an independently supplied background but an artifact of the projection map $Q compose K$ from configurations to observables. Each physical sector --- electrodynamics, gravitation, matter --- possesses its own production kernel $K$, encoding the sector's field content, spin, coupling structure, and dynamical equations. The observable projection $Q$ extracts measurable quantities: cross-sections, deflection angles, spectral shifts. This framework incurs a consistency obligation that does not arise in conventional physics, where spacetime is assumed as a shared arena. The obligation is this. QED and gravity are physically different theories. Their production kernels differ: $ K_(\"QED\") != K_(\"grav\"). $ The QED kernel encodes a spin-1 massless boson with $U(1)$ gauge symmetry; the gravitational kernel encodes a spin-2 field with diffeomorphism invariance. Their propagators, vertices, Ward identities, and renormalization structures are distinct. Yet observations require a common observable geometry: $ G(K_(\"QED\")) = G(K_(\"grav\"))? $ Photons and gravitons propagate on the same light cones. Gravitational lensing deflects electromagnetic signals. Atomic clocks measure gravitational redshift. These observations presuppose that the geometry projected by the electromagnetic kernel and the geometry projected by the gravitational kernel are the same geometry --- or at least compatible. In conventional physics, this is trivially satisfied: both sectors couple to the same background metric $g_(mu nu)$. In POT, where no background metric is assumed, it becomes a structural problem: #align(center)[_What invariant structure can different physical kernels share that is sufficient to support a common observable geometry?_] The question reverses the usual explanatory direction. In ordinary physics, common spacetime implies common causal propagation: different fields share light cones because they couple to the same metric. POT reverses this: common kernel invariants should _imply_ common geometric structure. The shared spacetime is not assumed but must be derived from properties of the production kernels themselves. This reversal is what distinguishes the present investigation from merely comparing propagators across theories. It is known from Lorentzian geometry that the causal structure of a past-and-future-distinguishing spacetime determines its conformal geometry [Malament1977, HKM1976]. But these classical results begin with an assumed spacetime manifold $(M, prec)$ and derive geometric structure from the causal order. POT asks a logically prior question: whether invariants of physically distinct production kernels can reconstruct a common geometric structure _before_ a shared spacetime manifold is posited. If the kernel-derived characteristic data can be shown to induce a causal relation satisfying the hypotheses of Malament and Hawking--King--McCarthy, then the causal-to-conformal step is already established, and the remaining obligation becomes specifically: does the characteristic signature of a kernel suffice to construct a causal order? This paper provides a partial answer. We identify two derived signatures --- characteristic and topological --- and prove a common characteristic structure across QED, Einstein gravity, and conformal gravity, show that the established signatures are insufficient to determine a full metric, and demonstrate that the residual ambiguity is naturally conformal. The formalism encounters conformal geometry before metric geometry, connecting structurally to Mannheim's conformal gravity program [Man2012]. The paper does not claim to have solved the common-geometry problem. It establishes the boundary between what is currently proved and what remains open, and identifies the precise mathematical obligation that a complete solution must discharge.") // ============================================================================= // Section 2: Projection Kernels and Existing Structure // ============================================================================= define sec_background = ArxivSection("Projection Kernels and Existing Structure", "We briefly recall the POT kernel framework established in previous volumes of this series, referring readers to those papers for full development. === Admissible kernels A production kernel $K : \"Flow\" arrow \"Field\"$ maps flow configurations to field outputs. $K$ is _admissible_ if it is linear, respects scalar multiplication, and maps zero to zero [AtikGR, AtikFactorization]. Admissible kernels form a submonoid under composition: the composition of admissible kernels is admissible. === The K-Q pipeline The physical theory is the composed map $Q compose K$: the production kernel $K$ generates an intermediate object (field strength, curvature, propagator), and the observable projection $Q$ extracts the measurable quantity. Only $Q compose K$ is physical; $K$ and $Q$ individually are representation-dependent. === Formulation fibers Physically equivalent formulations of a theory (Cartan, teleparallel, Palatini for GR) produce the same $Q compose K$ while factoring the pipeline differently [AtikFactorization]. The formulation fiber consists of all factorizations of the same physical pipeline. Properties that vary across the fiber --- such as energy localizability in GR --- are factorization artifacts, not physics. === Picard-Lefschetz decomposition The path integral, treated as a production kernel, decomposes into Lefschetz thimbles: $K = sum_sigma n_sigma K_sigma$, where $n_sigma$ are topological intersection numbers [AtikPL]. The Stokes phenomenon --- intersection numbers jumping at Stokes lines --- is a factorization rearrangement: the component decomposition changes while $Q compose K$ is invariant. === Null-space structure The null space $ker(Q)$ is not inert. Elements individually annihilated by $Q$ can shape observables through $Q compose K$ --- the polarizer principle [AtikGhost, AtikPL]. Ghost fields, non-perturbative saddles, and Stokes-rearranged thimble contributions all instantiate this mechanism.") // ============================================================================= // Section 3: Characteristic Signatures // ============================================================================= define sec_characteristic = ArxivSection("Characteristic Signatures", "=== The principal symbol and its zero set Every linear partial differential operator $L$ has a principal symbol $sigma_(\"prin\")(L)(k)$, a polynomial in the covector $k_mu$ that determines the operator's highest-order behavior. The _characteristic variety_ $Z(sigma) = {k : sigma(k) = 0}$ determines causal propagation: it is the set of directions along which the operator fails to be elliptic, and defines the null cone in Lorentzian theories. A production kernel $K$ in the free-field case is the Green's function of a kinetic operator $L_K$ (the operator satisfying $L_K K = delta$). Principal symbols belong naturally to differential operators, not to their distributional inverses. We therefore define the characteristic signature through the kinetic operator: $ C(K) := Z(sigma_(\"prin\")(L_K)). $ This is a notational convention: '$C(K)$' means 'the characteristic variety of the kinetic operator whose Green's function is $K$.' An intrinsic formulation via the wavefront set of $K$ itself --- using the Duistermaat--Hörmander propagation-of-singularities theorem, which relates $\"WF\"(K)$ to null bicharacteristic flow --- would be more natural for the kernel-first philosophy of POT, but is not required for the present paper. For the free-field kinetic operators in Lorenz/de Donder gauge, the propagators take the schematic form: $ K_(\"QED\")(k) tilde.op P_(\"photon\")(k) / k^2, $ $ K_(\"GR\")(k) tilde.op P_(\"graviton\")(k) / k^2, $ $ K_(\"CG\")(k) tilde.op P_(\"conformal\")(k) / (k^2)^2, $ where $P$ denotes the spin-specific tensor projector (transverse for spin-1, transverse-traceless for spin-2). The conformal-gravity propagator $1 slash (k^2)^2$ corresponds to the pure Weyl-squared action linearized about flat spacetime in transverse gauge [Man2012]; the unitarity of this fourth-order theory is established via $cal(P) cal(T)$-symmetric quantization [BenMan2008]. In the more general second-plus-fourth-order theory, the propagator has the form $1 slash (k^2 - M_1^2) - 1 slash (k^2 - M_2^2)$, which has additional mass poles. We compare the pure conformal case throughout. === Characteristic signature: forgetting multiplicity The key algebraic observation is: #strong[Theorem 1] (Multiplicity invariance). _For any polynomial $p$ and integer $m >= 1$,_ $ Z(p^m) = Z(p). $ _Proof._ $p(x)^m = 0$ if and only if $p(x) = 0$, since we work over a field (no zero divisors). $square$ The theorem itself is mathematically uncontroversial. Its Kleis encoding, however, currently takes multiplicity invariance as an axiom (CHAR-1 in the companion theory file) rather than deriving it from concrete polynomial/zero-set semantics within the platform. The grounding obligation is specifically the Kleis formalization, not the mathematical content. Despite its simplicity, this theorem has immediate physical content. The QED propagator has a pole $k^2 = 0$ (simple); the Einstein propagator has the same pole $k^2 = 0$ (simple); the conformal-gravity propagator has $k^2 = 0$ as a _double_ pole (from $(k^2)^2$). Yet: $ Z(k^2) = Z((k^2)^2). $ The characteristic variety is identical. All three theories share the same null cone structure --- the same light cones, the same causal geometry --- despite their different differential order. === Physical-sector projector invariance The full propagator kernel is not a scalar but a tensor: $K(k) = P(k) dot p(k)^(-m)$, where $P$ is the spin-specific projector. Different spins have different projectors ($P_(\"photon\") != P_(\"graviton\")$), so comparing characteristic varieties across spins requires showing that the projector does not alter the zero set. #strong[Theorem 2] (Projector forgetting). _If $P(k)$ is nondegenerate on the physical sector --- meaning $det P_(\"phys\")(k) != 0$ on the support of physical states --- then_ $ Z(P(k) dot sigma(k)) = Z(sigma(k)). $ The nondegeneracy condition is satisfied by the transverse projector (QED) and the transverse-traceless projector (GR) on their respective physical polarization subspaces, after the standard gauge-fixing/physical-quotient procedure. Extending this to conformal gravity requires verifying nondegeneracy of the relevant projector on the Bach-tensor sector --- a physical claim that deserves either a careful derivation or a citation, which we defer. As with Theorem 1, the Kleis encoding axiomatizes the projector-forgetting property; the grounding obligation is to derive it from a precise definition of physical-sector nondegeneracy incorporating gauge quotients. === The common causal skeleton Combining Theorems 1 and 2 with the propagator assignments: #strong[Corollary 1a] (QED--GR characteristic equality). $ C_(\"QED\") = C_(\"GR\") $ _QED and Einstein gravity share the same characteristic signature._ Both kinetic operators have principal symbol proportional to $k^2$; the projectors differ but are nondegenerate on their respective physical sectors (transverse photon, transverse-traceless graviton) after standard gauge fixing in Lorenz/de Donder gauge. $square$ #strong[Corollary 1b] (Extension to conformal gravity). $ C_(\"GR\") = C_(\"CG\") $ _The pure Weyl-squared conformal gravity shares the same characteristic signature as Einstein gravity, conditional on the projector-nondegeneracy assumption for the Bach-tensor sector._ The kinetic operator for the pure Weyl action has principal symbol proportional to $(k^2)^2$ [Man2012]; by Theorem 1, $Z((k^2)^2) = Z(k^2)$. By Theorem 2 (assuming nondegeneracy), the tensor projector does not alter the characteristic variety. $square$ Together: $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$, where the first equality is unconditional and the second is conditional on the deferred projector analysis for conformal gravity. This result answers the motivating question at the level of causal structure: the three theories project the same null cones. The common characteristic variety $Z(k^2)$ defines a shared causal skeleton on which all three sectors propagate. The result is non-trivial for conformal gravity. A fourth-order operator could, in principle, have a richer characteristic variety than a second-order operator (additional null directions, higher-codimension characteristic surfaces). The multiplicity invariance theorem shows this does not happen: the fourth-order conformal pole $(k^2)^2$ introduces multiplicity but not new null directions. Note that this comparison concerns the _pure_ Weyl-squared theory; in the more general second-plus-fourth-order action, additional mass poles ($k^2 = M^2$) would introduce new characteristic directions not shared by QED or Einstein gravity. All three results (CHAR-1 through CHAR-5) are verified by Z3 in the companion theory file `pot_characteristic_topology.kleis`.") // ============================================================================= // Section 4: There is no Section 4 header — renumber to match outline // ============================================================================= // Section 4 in the outline is "Common Causal Structure of QED and Gravity" // which is now the second half of Section 3. Continue with Section 4 = outline's 5. // ============================================================================= // Section 4: Projection-Invariant Topological Signatures // ============================================================================= define sec_topology = ArxivSection("Kernel Topology Signatures", "The characteristic signature captures causal structure --- light cones and wavefronts. But the projection $Q compose K$ also carries topological information, extractable from the Picard-Lefschetz decomposition of the path-integral kernel [AtikPL]. === Raw thimble data The path integral decomposes as $K = sum_sigma n_sigma K_sigma$, where each $K_sigma$ is a Lefschetz thimble contribution and $n_sigma in bb(Z)$ is an intersection number determined by the topology of the steepest-descent flow. The raw thimble data --- the set of saddles, their intersection numbers, and the intersection pairings between thimbles and anti-thimbles --- encodes topological information about the _configuration space_ and its complexification, not directly about the emergent spacetime manifold. === The Stokes quotient Raw thimble data is not an invariant of the kernel. At Stokes lines, the intersection numbers $n_sigma$ jump discontinuously: a new saddle enters (or exits) the decomposition. Yet $Q compose K$ is invariant across the transition (Cauchy's theorem). The _kernel topology signature_ $T(K)$ is defined as the equivalence class of thimble data under all Stokes rearrangements: $ T(K) = [{cal(J)_sigma, n_sigma, chevron.l cal(J)_sigma, cal(K)_tau chevron.r, dots}]_(\"Stokes\").$ The quotient is well-defined: if $D_1 tilde.op_(\"Stokes\") D_2$, then $T(D_1) = T(D_2)$ (TOPO-1). The Stokes equivalence is reflexive (TOPO-2) and transitive (TOPO-3). We emphasize that $T(K)$ is a _kernel_ topology signature: it captures the topology of the Picard-Lefschetz decomposition in configuration/field space. Whether and how this kernel topology relates to the topology of emergent spacetime is itself an open question. The Picard-Lefschetz saddle structure reflects properties of the solution space (instanton sectors, vacuum structure, non-perturbative content) rather than directly encoding spacetime homology or homotopy. Establishing a bridge $T_(\"kernel\") arrow.r T_(\"spacetime\")$ would require showing that the Stokes-equivalence class of thimble data constrains the topological type of the spacetime reconstructed from $Q compose K$. This bridge is not derived here and constitutes an additional open obligation. === Formulation-fiber invariance Physically equivalent formulations of a theory share the same kernel topology signature: $ K_1 tilde.op_F K_2 arrow.r.double T(K_1) = T(K_2). $ This connects the Picard-Lefschetz paper [AtikPL] to the kernel-factorization paper [AtikFactorization]: kernel topology is invariant not only under Stokes rearrangements (within a formulation) but also across physically equivalent formulations (different factorizations of the same pipeline). Formulation-equivalent kernels also share characteristic signatures (TOPO-5). === The two-branch structure The kernel $K$ therefore yields two independently derived signatures: $ K arrow.r cases(T(K) quad \"(kernel topology)\", C(K) quad \"(causal structure)\"). $ These branches are independent: $C$ does not factor through $T$, nor $T$ through $C$. Both are derived from $K$, but they capture different invariant content of the production kernel. We have no present reason to claim that kernel topology determines causal structure or vice versa.") // ============================================================================= // Section 5: The Geometry Reconstruction Problem // ============================================================================= define sec_reconstruction = ArxivSection("The Geometry Reconstruction Problem", "The common-geometry obligation now takes a precise form. We have established a common characteristic signature across QED and the gravitational theories, while equality of their topology signatures remains an open cross-theory obligation: $ C_(\"QED\") = C_(\"GR\") = C_(\"CG\"), quad T_(\"QED\") limits(=)^? T_(\"GR\").$ The question becomes: #align(center)[_Do $T(K)$ and $C(K)$ suffice to determine observable geometry?_] $ T(K_1) = T(K_2) and C(K_1) = C(K_2) limits(arrow.r.double)^? G(K_1) = G(K_2). $ If so, the common-geometry problem is solved: any two kernels sharing topology and causal structure necessarily project the same geometry. If not, additional invariant structure is required, and the question becomes: what is it? In the companion theory file, $G(K)$ is declared as a type with no axioms connecting it to $T$ or $C$. This is deliberate: the geometry reconstruction question is an open obligation, not an assumption.") // ============================================================================= // Section 6: The Conformal Countermodel // ============================================================================= define sec_countermodel = ArxivSection("Conformal Countermodel", "We answer the reconstruction question negatively for full metric geometry. === Null cones determine only conformal structure The characteristic signature $C(K)$ captures the null cone: the set of directions $k$ satisfying $g_(mu nu) k^mu k^nu = 0$. But the null cone is invariant under conformal rescaling: $ g'_(mu nu) = Omega^2(x) g_(mu nu) arrow.r.double g'_(mu nu) k^mu k^nu = Omega^2 g_(mu nu) k^mu k^nu = 0 arrow.l.r.double g_(mu nu) k^mu k^nu = 0. $ The null directions are unchanged. Therefore $C(g) = C(Omega^2 g)$ for any smooth positive $Omega$. Topology is also conformally invariant: the causal structure and topological type of a Lorentzian manifold are preserved under conformal rescaling (assuming global hyperbolicity is maintained). === The countermodel Consider two kernels $K_1$ (flat) and $K_2$ (curved) related by a conformal factor: $ g_2 = Omega^2 g_1, quad Omega != \"const\". $ Then: $ T(K_1) = T(K_2) quad \"(same kernel topology)\", $ $ C(K_1) = C(K_2) quad \"(same null cones)\", $ $ G(K_1) != G(K_2) quad \"(different metrics)\", $ but $ [g_1]_(\"conf\") = [g_2]_(\"conf\") quad \"(same conformal class)\". $ This is verified as CONF-1 and CONF-2 in the theory file. The model is _consistent_: Z3 confirms no contradiction arises from simultaneously asserting shared $T$ and $C$, distinct metric geometry, and common conformal class. #strong[Result 1.] _$T + C$ are insufficient to determine a metric representative._ The residual ambiguity has the form $g_(mu nu) tilde.op Omega^2(x) g_(mu nu)$ --- precisely conformal freedom. === Epistemic precision CONF-2 establishes that a consistent model exists in which $T + C$ leave metric geometry ambiguous within a conformal class. It does _not_ establish that $T + C$ _determine_ the conformal class. The implication $ T(K_1) = T(K_2) and C(K_1) = C(K_2) limits(arrow.r.double)^? [g_1]_(\"conf\") = [g_2]_(\"conf\") $ remains an open theorem obligation. Proving it would establish that conformal geometry is the _maximal_ geometric information extractable from $T + C$. Disproving it would reveal additional non-conformal ambiguity. === Relation to classical results The observation that causal structure determines conformal geometry is not new: the Malament--Hawking--King--McCarthy theorem [Malament1977, HKM1976] establishes that a causal bijection between past-and-future-distinguishing spacetimes of dimension $n > 2$ implies conformal isometry. Our conformal countermodel is consistent with this classical result but addresses a different question. Malament and HKM begin with an assumed spacetime manifold equipped with a causal relation $(M, prec)$ and derive the conformal class from it. POT begins with production kernels and asks whether kernel-derived invariants can reconstruct common geometry before a shared manifold is posited. If the characteristic signature $C(K)$ can be shown to induce a causal relation satisfying the hypotheses of [Malament1977], then the causal-to-conformal step is already provided by the classical theorem, and the remaining open obligation is specifically: $ C(K) limits(arrow.r.double)^? (M, prec) quad \"satisfying past/future distinguishability.\" $ This decomposition could transform the open obligation from a single novel theorem into two sharper questions: (1) does $C(K)$ construct a causal order? and (2) if so, classical results supply the conformal class. The Ehlers--Pirani--Schild program [EPS1972] provides further context: compatible conformal and projective structures determine a Weyl geometry, suggesting that the path from kernel invariants to metric geometry may factor through established mathematical machinery rather than requiring entirely new theorems.") // ============================================================================= // Section 7: Conformal Gravity and Projected Ontology // ============================================================================= define sec_mannheim = ArxivSection("Conformal Gravity and Projected Ontology", "The conformal countermodel connects POT to Mannheim's conformal gravity program [Man2012] through structure rather than phenomenological coincidence. === The structural connection Mannheim's conformal gravity elevates precisely the residual freedom identified above to a gauge equivalence: $ g_(mu nu) tilde.op Omega^2(x) g_(mu nu) $ is not an ambiguity to be resolved but a _symmetry_ of the fundamental theory. The Weyl action $integral d^4 x sqrt(-g) C_(mu nu rho sigma) C^(mu nu rho sigma)$ is conformally invariant; the Bach equation $B_(mu nu) = 0$ is fourth-order; and physical observables depend only on the conformal class $[g]_(\"conf\")$, not on a particular metric representative. The POT formalism therefore reaches a boundary compatible with conformal geometry: the established signatures do not select a metric representative, and a consistent countermodel realizes the unresolved freedom precisely as conformal rescaling: $ (T, C) arrow.r \"at most\" [g]_(\"conf\").$ The geometric level beyond which the current POT construction has not yet been extended is compatible with the equivalence class that conformal gravity regards as physical. This relationship is deeper than the shared fourth-order structure: - It is not that POT happens to produce a fourth-order operator (it does not directly). - It is not that POT happens to reproduce Mannheim's rotation curves (that is a separate investigation). - It is that the geometric boundary of the current POT reconstruction from kernel invariants coincides with the level at which conformal gravity begins. === What this does and does not establish This paper establishes a structural resonance between POT and conformal gravity. It does _not_ establish: - That POT _derives_ conformal gravity (the dynamics are not derived here). - That $T + C$ _determine_ the conformal class (this is an open obligation). - That conformal gravity is the correct theory of gravitation (this is an empirical question). It does establish that the common causal skeleton $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$ extends to conformal gravity without difficulty, and that the geometric boundary of the current POT construction is conformally invariant.") // ============================================================================= // Section 8: The Missing Scale Signature // ============================================================================= define sec_scale = ArxivSection("The Missing Scale Signature", "If $T + C$ determine at most conformal geometry, something additional is required to select a metric representative $g$ from the conformal class $[g]_(\"conf\")$. We denote this provisionally as a _scale signature_ $S(K)$, so that the full reconstruction would take the form: $ (T(K), C(K), S(K)) arrow.r g. $ We deliberately supply no axioms for $S$. Its physical identity has not been established, and defining it prematurely would risk encoding an answer into its definition. === A physical clue The null cone condition $ g_(mu nu) k^mu k^nu = 0 $ is conformally invariant: it defines null directions but provides no ruler. Massless propagation naturally yields causal/conformal structure. The mass-shell condition $ g_(mu nu) p^mu p^nu = m^2 $ is _not_ conformally invariant. Under $g arrow Omega^2 g$, the mass shell changes unless $m$ transforms correspondingly. A fixed physical mass therefore provides scale information that breaks conformal freedom. This suggests that the scale signature may originate in the _massive_ sector of QED rather than in gravity: $ \"massless kernels\" arrow.r (T, C) arrow.r [g]_(\"conf\"), $ $ \"mass-bearing kernels\" arrow.r (T, C, S) arrow.r g_(\"metric\"). $ === Scale need not be primitive In asymptotically free gauge theories, dimensional transmutation generates a dimensionful scale $Lambda_(\"QCD\")$ from a dimensionless coupling plus renormalization-group flow. Scale can _emerge_ from quantum dynamics rather than being inserted as a parameter. For conformal gravity, this observation is essential: if the fundamental theory is conformally invariant, physical masses and length scales must arise through spontaneous conformal symmetry breaking [Man2012]. The scale signature $S$ may therefore not represent a primitive datum but a _projection invariant associated with conformal symmetry breaking_ --- a coherent structure that survives the $Q compose K$ map and selects a metric representative. Identifying $S$ concretely requires inspecting the POT renormalization kernel [AtikRenorm] for every dimensionful quantity entering the kernel: fermion mass $m$, proper time $s$, renormalization scale $mu$, and finite-part normalization. Classifying each as physical, auxiliary, or projection-invariant should reveal which carries the missing geometric information. This investigation is deferred to a subsequent paper.") // ============================================================================= // Section 9: Discussion and Open Obligations // ============================================================================= define sec_discussion = ArxivSection("Discussion and Open Obligations", "=== What has been established This paper introduces two derived signatures of a production kernel --- the characteristic signature $C(K)$ and the kernel topology signature $T(K)$ --- and uses them to address the common-geometry problem in Projected Ontology Theory. The established results, verified by Z3 on the Kleis platform, are: #figure( table( columns: 3, [*Label*], [*Result*], [*Status*], [CHAR-1], [Multiplicity invariance: $Z(p) = Z(p^m)$], [Axiomatized; derivation from zero-set semantics is a TODO], [CHAR-2], [Projector forgetting: nondegenerate $P$ does not alter $Z$], [Axiomatized], [CHAR-3--5], [Common causal skeleton: $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$], [Derived from CHAR-1, CHAR-2, and physical assignments], [TOPO-1--3], [Stokes quotient well-defined (equivalence relation)], [Axiomatized], [TOPO-4--5], [Formulation-fiber invariance of $T$ and $C$], [Axiomatized], [CONF-1], [$T + C$ insufficient for metric geometry], [Countermodel (axiomatized)], [CONF-2], [Consistent conformal residual ambiguity], [Countermodel (axiomatized)], [CONF-3], [QED, GR, CG share conformal geometry], [From physical assignments], [COR-2], [Shared $C$ does not imply shared field content], [From assignments + field distinction], ), caption: [Summary of Z3-verified results and their epistemic status. Results marked \"Axiomatized\" are consistency-checked but not derived from lower-level definitions. Compositional results (CHAR-3--5, CONF-3, COR-2) combine multiple axioms to derive conclusions.], ) Most Z3 results are consistency and closure checks over axiomatized structures, not independent derivations from lower-level mathematics. The genuinely compositional results are CHAR-3 through CHAR-5 and CONF-3, which combine multiple axioms to derive their conclusions. The value of the formalization is not the number of verified results but the dependency graph: it enforces which axioms each result requires, prevents circular reasoning between assumptions, makes the countermodel's consistency machine-checkable rather than merely asserted, and maintains an explicit epistemic boundary that prevents overreach. When a result is listed as 'axiomatized,' the formalization does not hide this --- it records it as an ungrounded assumption on which downstream results depend. === Open theorem obligations Three implications remain unproved: $ T(K_1) = T(K_2) and C(K_1) = C(K_2) limits(arrow.r.double)^? [g_1]_(\"conf\") = [g_2]_(\"conf\"). $ If true, conformal geometry is the maximal geometric information extractable from $T + C$. If false, additional non-conformal ambiguity exists. Note that part of this implication may already be supplied by the Malament--HKM theorem [Malament1977, HKM1976], provided $C(K)$ can be shown to induce a causal relation satisfying the required hypotheses (see Section 6). $ T_(\"kernel\") limits(arrow.r.double)^? T_(\"spacetime\"). $ The kernel topology signature $T(K)$ captures Picard-Lefschetz thimble structure in configuration space. Whether and how this constrains the topology of emergent spacetime is an independent open question (see Section 4). $ T(K_1) = T(K_2) and C(K_1) = C(K_2) and S(K_1) = S(K_2) limits(arrow.r.double)^? g_1 = g_2. $ This requires first identifying the physical content of the scale signature $S$. === Grounding obligations Two foundational axioms should eventually be derived rather than assumed: - CHAR-1 (multiplicity invariance) should follow from a concrete polynomial/zero-set semantics within Kleis. - CHAR-2 (projector forgetting) should follow from a precise definition of physical-sector nondegeneracy, incorporating gauge quotients. Once discharged, CHAR-3 through CHAR-5 become substantially stronger: the common causal skeleton rests on derived rather than axiomatized foundations. === The emerging picture The paper's results are summarized by a single hierarchy, with question marks retained to mark the open obligations: #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt)[ $ K arrow.r (T_(\"kernel\"), C) limits(arrow.r.double)^? [g]_(\"conf\") limits(arrow.r.double)^(S?) g_(\"metric\") $ ]] The first arrow is established: every admissible kernel yields a characteristic signature and a kernel topology signature. The second arrow is the open conformal-reconstruction obligation: whether $T + C$ determine a conformal class. Part of this may compose with the classical Malament--HKM result if $C$ induces a suitable causal order [Malament1977, HKM1976]. The third arrow requires identifying the scale signature $S$ --- an open physical question, possibly connected to mass generation and conformal symmetry breaking. An additional open arrow, not shown in the hierarchy above, connects kernel topology to spacetime topology: $T_(\"kernel\") limits(arrow.r.double)^? T_(\"spacetime\")$. Conformal geometry appears as the boundary of the current POT construction. This may explain why conformal gravity, despite its unconventional fourth-order dynamics, appears structurally compatible from the POT perspective: it describes precisely the geometric level that the kernel formalism currently reaches.") // ============================================================================= // Section 10: Conclusion // ============================================================================= define sec_conclusion = ArxivSection("Conclusion", "The common-geometry problem in Projected Ontology Theory asks: if geometry is a projection artifact, why do physically different kernels project the same spacetime? This paper provides a partial answer and a precise boundary. The characteristic signature $C(K)$ --- the zero set of the principal symbol, with multiplicity forgotten --- is shared by QED, Einstein gravity, and conformal gravity: $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$. The multiplicity invariance theorem ($Z(p) = Z(p^m)$) explains why fourth-order conformal gravity shares the same causal skeleton as second-order theories. The kernel topology signature $T(K)$ --- the Stokes quotient of Picard-Lefschetz thimble data --- provides a second, independently derived invariant of the kernel, capturing configuration-space topology rather than spacetime topology directly. Together, $T$ and $C$ are insufficient to determine a metric representative: a conformal countermodel demonstrates that two kernels can share topology and causal structure while differing by a conformal factor $g'_(mu nu) = Omega^2(x) g_(mu nu)$. The formalism encounters conformal geometry before metric geometry. This connects POT to Mannheim's conformal gravity through structure rather than coincidence: the geometric boundary of the current kernel formalism is the same equivalence class that conformal gravity regards as physical. Whether this connection is deep or superficial depends on the open obligations: whether $T + C$ actually _determine_ the conformal class (rather than merely admitting it as a consistent possibility), and whether the missing scale signature $S$ can be identified within the existing POT kernel construction --- perhaps in the mass-bearing sector of QED, or in the projection invariants associated with conformal symmetry breaking. The Kleis formalization records the dependency graph of all results: which assumptions each conclusion requires, where axiomatization substitutes for derivation, and what remains open. The boundary between proved structure and open obligation is the scientific contribution: it identifies exactly where the next theorem lives.") // ============================================================================= // References // ============================================================================= define ref_mannheim = ArxivReference("Man2012", "Mannheim, P. D. Making the case for conformal gravity. Found. Phys. 42, 388--420 (2012). arXiv:1101.2186.") define ref_atik_gr = ArxivReference("AtikGR", "Atik, E. Reading General Relativity Through the Projection Kernel. POT VUFT Series, Vol. I. Preprint (2025).") define ref_atik_factorization = ArxivReference("AtikFactorization", "Atik, E. Non-Unique Factorization of Projection Kernels. POT VUFT Series, Vol. VIII. Preprint (2026).") define ref_atik_pl = ArxivReference("AtikPL", "Atik, E. The Path Integral as Kernel Decomposition: Picard-Lefschetz Theory in the K-Q Framework. POT VUFT Series, Vol. XII. Preprint (2026).") define ref_atik_ghost = ArxivReference("AtikGhost", "Atik, E. Ghost-Mediated Null-Space Activity. POT VUFT Series, Vol. V. Preprint (2026).") define ref_atik_renorm = ArxivReference("AtikRenorm", "Atik, E. Hadamard Renormalization as Kernel Quotient. POT VUFT Series, Vol. X. Preprint (2026).") define ref_malament = ArxivReference("Malament1977", "Malament, D. B. The class of continuous timelike curves determines the topology of spacetime. J. Math. Phys. 18, 1399--1404 (1977).") define ref_hkm = ArxivReference("HKM1976", "Hawking, S. W., King, A. R. and McCarthy, P. J. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. J. Math. Phys. 17, 174--181 (1976).") define ref_eps = ArxivReference("EPS1972", "Ehlers, J., Pirani, F. A. E. and Schild, A. The geometry of free fall and light propagation. In: O'Raifeartaigh, L. (ed.) General Relativity, Papers in Honour of J. L. Synge, pp. 63--84. Clarendon Press, Oxford (1972). Republished in Gen. Rel. Grav. 44, 1587--1609 (2012).") define ref_bender_mannheim = ArxivReference("BenMan2008", "Bender, C. M. and Mannheim, P. D. No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model. Phys. Rev. Lett. 100, 110402 (2008). arXiv:0706.0207.") define ref_kleis = ArxivReference("Kleis", "Atik, E. Kleis: A Verification Platform for Mathematical Knowledge Production. https://kleis.io (2025).") // ============================================================================= // Paper Assembly // ============================================================================= define paper_sections = [ sec_intro, sec_background, sec_characteristic, sec_topology, sec_reconstruction, sec_countermodel, sec_mannheim, sec_scale, sec_discussion, sec_conclusion, ref_mannheim, ref_malament, ref_hkm, ref_eps, ref_bender_mannheim, ref_atik_gr, ref_atik_factorization, ref_atik_pl, ref_atik_ghost, ref_atik_renorm, ref_kleis ] define paper = arxiv_paper( paper_title, paper_authors, paper_affiliations, paper_abstract, paper_keywords, paper_sections ) example "paper is valid" { assert(valid_arxiv_paper(paper) = true) out("Paper validation: OK") } example "compile" { let typst = compile_arxiv_paper(paper) in out(typst_raw(typst)) }