// ============================================================================= // Projective Metric Reconstruction from Coherent Kernel Families // ============================================================================= // // Paper 3 in the Projected Ontology series. // // Central result (PROJ-1A): on a compact Riemannian 4-manifold in the // Paneitz constant-kernel class, a classically Weyl-invariant coherent // kernel family at one loop has ker H_F = ℝ·1. The family determines // the metric up to a global positive rescaling ℝ₊ (dimensional // calibration). // // The paper contains two theorems: // Theorem 1 (scale selection): conformal scale geometry (mathematics) // Theorem 2 (PROJ-1A): physical realization via BW + Paneitz (physics) // // A semidefinite kernel lemma for future Approach B is in Appendix A. // // The Discussion interprets the result in the Lorentzian realization // reconstructed by Paper 2: nonconstant conformal-mode solutions are // spacetime dynamical excitations, not additional reconstruction // ambiguity. Lorentzian signature is an output of the reconstruction, // not a property of the ontological substrate. // // Formal companion: theories/pot_metric_coherence.kleis // // Pipeline: // kleis test --raw-output --example compile \ // docs/papers/pot_projective_metric_paper.kleis \ // > pot_projective_metric_paper.typ // typst compile pot_projective_metric_paper.typ // // ============================================================================= import "stdlib/prelude.kleis" import "stdlib/templates/arxiv_paper.kleis" // ============================================================================= // Paper Metadata // ============================================================================= define paper_title = "Projective Metric Reconstruction from Coherent Kernel Families" define paper_authors = [ Author("Engin Atik", "1") ] define paper_affiliations = [ Affiliation(1, "Kleis Research", "https://kleis.io") ] define paper_abstract = "The preceding papers in this series established that a projection kernel on a smooth manifold determines a Lorentzian conformal class $[g]_K$ from its projective singular directions. This paper addresses the residual calibration problem: selecting a metric representative from the conformal class. We define a coherent kernel family intrinsically --- as a collection of distributional two-point kernels sharing a common conformal class --- and introduce quantum realizability as a separate derived property enabling effective-action analysis. On a compact Riemannian four-manifold whose Paneitz operator has trivial kernel, the Barvinsky--Wachowski decomposition of the one-loop effective action gives $H_F = H_A prop -a dot Delta_4$, whence $ker H_F = bb(R) dot 1$: only the global Weyl orbit survives (Theorem PROJ-1A). The compact Riemannian realization is an analytic tool for establishing the scale quotient; Lorentzian signature is an output of the reconstruction (Paper 2), not a property of the ontological substrate. In the independently reconstructed Lorentzian realization, nonconstant conformal-mode solutions are spacetime dynamical excitations determined by initial data, not additional reconstruction ambiguity; this interpretation does not modify the quotient theorem. The trilogy therefore terminates at a precise conceptual boundary: individual kernels determine conformal shape; coherent families determine relative scale; one global $bb(R)_+$ calibration remains. All formal implications and consistency examples are checked in a companion Z3-verified theory file." define paper_keywords = "projected ontology, metric reconstruction, conformal scale, Paneitz operator, Weyl anomaly, kernel family, conformal mode, effective action, formal verification, Z3" // ============================================================================= // Section 1: Introduction — The Calibration Problem // ============================================================================= define sec_intro = ArxivSection("Introduction: The Calibration Problem", "=== From conformal class to metric The preceding papers in this series established a reconstruction chain from projection kernels to conformal geometry. Paper 1 [Atik2026a] identified a common characteristic signature across physically distinct production kernels: $C_(\"QED\") = C_(\"GR\") = C_(\"CG\")$. Paper 2 [Atik2026b] proved that a projection kernel on a smooth manifold determines a Lorentzian conformal class $[g]_K$ whenever its projective singular directions satisfy algebraic saturation, Witt index one, and smooth variation. Those three conditions were shown to be irredundant. Paper 2 also identified the precise boundary of individual-kernel geometry. Two distinct enrichments branch from the conformal class: directed propagation (requiring orientation data $O(K)$) and metric calibration (requiring family-level information $S(F)$). The present paper resolves the second branch. The question is: #align(center)[_What collective structure of a coherent physical kernel family selects a metric representative $g in [g]$?_] === What the answer turns out to be The answer is not the one originally anticipated. We initially sought a family-generated mechanism that selects a unique metric $g_F$ from the conformal class --- that is, a construction $F arrow.r g_F$ with no residual ambiguity. The investigation instead reveals that the intrinsic output of a coherent kernel family is a _projective metric_: $ F arrow.r [sigma_F]_(bb(R)_+) arrow.r [g_F]_(bb(R)_+), $ where $[sigma_F]_(bb(R)_+) = {c sigma_F : c > 0}$ and $[g_F]_(bb(R)_+) = {c^2 g_F : c > 0}$. The family determines all local conformal variation --- the enormous $C^infinity_+(M)$ freedom of local Weyl rescaling is reduced to a single global $bb(R)_+$ orbit. What survives is dimensional calibration: the choice of a unit of length. One experimental measurement then converts the entire relational solution into conventional units. === Two projectivizations This result extends a structural motif already present in Paper 2. Paper 2 projectivized the covector: the characteristic singular direction $xi_x in T^*_x M without 0$ was replaced by its projective equivalence class $[xi_x] in P(T^*_x M)$, removing local covector magnitude. Paper 3 projectivizes the scale: the conformal scale $sigma_F in Gamma(E^+[1])$ is replaced by its $bb(R)_+$-orbit $[sigma_F]$, removing global dimensional normalization. At both levels, what is quotiented out is unphysical magnitude; what survives is relational structure. $ \"Paper 2\":& quad q_x &arrow.r [q_x] quad &\"(projectivize local covector magnitude)\" \ \"Paper 3\":& quad sigma_F &arrow.r [sigma_F]_(bb(R)_+) quad &\"(projectivize global dimensional scale)\" $ === Ontological boundary POT takes the projection kernels considered here to arise from an underlying ontological Hilbert space $cal(H)_(\"ont\")$ equipped with modal flow, as introduced in the preceding POT framework [Atik2026a]. Neither $cal(H)_(\"ont\")$ nor its modal evolution is reconstructed in this paper. In particular, the terms _Riemannian_ and _Lorentzian_ below refer exclusively to geometric realizations on the reconstructed manifold $M$; the modal-flow parameter is not identified with spacetime time. Paper 2 already established that Lorentzian signature emerges from the Witt-index-one condition on the projective characteristic quadric --- it is an output of the reconstruction, not an assumption about the substrate. === Intrinsic families and quantum realizability A kernel family is defined _intrinsically_ as a collection of distributional two-point kernels sharing a common conformal class: $ F = {K_i}_(i in I), quad K_i in cal(D)'(M times M), quad [g]_(K_i) = [g]_(K_j) quad forall i, j. $ This is _conformal coherence_. The family is not defined by reference to a Lagrangian or a quantum field theory. The trilogy puts kernels before the Lagrangian: $cal(L) arrow.r K$ is the conventional direction; POT reverses it. To access effective-action methods, Paper 3 additionally uses a derived property: _quantum realizability_. A family $F$ is quantum-realizable if it admits a common quantum realization from which a collective effective action $Gamma_F$ can be constructed. This is a _realization property_, not part of the definition of the family. The logical architecture is: $ F arrow.r cases([g]_F quad &\"(from the kernels themselves\" \, \"via Paper 2)\", Gamma_F arrow.r H_F quad &\"(through a quantum realization)\") $ The two branches meet: $([g]_F, H_F) arrow.r [g_F]_(bb(R)_+)$. The effective action is a derived object computed from the realization, not the ontological starting point. === Compact Riemannian realization and Lorentzian interpretation The scale-quotient theorem (PROJ-1A) is established on a compact Riemannian realization, where elliptic spectral theory provides the Paneitz constant-kernel result needed to show $ker H_F = bb(R) dot 1$. The compact Riemannian setting is an analytic tool for the reconstruction proof; it is not a claim that the physical universe is Riemannian. Once the quotient $F arrow.r [g_F]_(bb(R)_+)$ is established, the independently reconstructed Lorentzian realization (Paper 2) provides a causal-dynamical interpretation: nonconstant conformal-mode solutions are spacetime excitations, not additional reconstruction ambiguity. This interpretation is discussed in Section 6 but does not modify the quotient theorem. === Structure of the paper Section 2 establishes the mathematical setting: conformal scales as positive sections of a density bundle, and the map from scale to metric representative. Section 3 explains the Barvinsky--Wachowski decomposition of the one-loop effective action and its consequence that $H_B = 0$ for classically Weyl-invariant families; it also discusses the $gamma = 0$ scheme choice. Section 4 states the projective metric reconstruction theorem (PROJ-1A). Section 5 gives epistemic accounting. Section 6 discusses the conceptual content: the two quotients, the Lorentzian interpretation, and geometry as reconstructed structure. A semidefinite kernel lemma for future Approach B extensions is proved in Appendix A.") // ============================================================================= // Section 2: Conformal Scale Geometry // ============================================================================= define sec_scale = ArxivSection("Conformal Scale Geometry", "This section establishes the mathematical infrastructure for metric calibration. All results here are standard conformal geometry, independent of quantum field theory. === Conformal scales Let $(M, [tilde(g)])$ be a smooth manifold equipped with a conformal class of Riemannian (or Lorentzian) metrics. A _conformal scale_ is a positive section $ sigma in Gamma(E^+[1]), $ where $E^+[1]$ denotes the bundle of positive conformal densities of weight $+1$. Under a Weyl rescaling $tilde(g) arrow.r Omega^2 tilde(g)$, a scale of weight $+1$ transforms as $sigma arrow.r Omega sigma$. === Scale-metric reconstruction A conformal scale $sigma$ selects a metric representative from the conformal class via $ g_sigma := sigma^(-2) tilde(g). $ This map is the _scale-metric reconstruction_. It is well-defined: $g_sigma$ is independent of the choice of representative $tilde(g)$ within $[tilde(g)]$, because replacing $tilde(g)$ by $Omega^2 tilde(g)$ and simultaneously replacing $sigma$ by $Omega sigma$ gives $(Omega sigma)^(-2)(Omega^2 tilde(g)) = sigma^(-2) tilde(g) = g_sigma$. This invariance is formalized as result SCL-1 in the companion theory file. === Injectivity If $sigma_1$ and $sigma_2$ are positive conformal scales with $g_(sigma_1) = g_(sigma_2)$, then $sigma_1^(-2) tilde(g) = sigma_2^(-2) tilde(g)$, hence $sigma_1 = sigma_2$ pointwise. Conformal scale selection is therefore _injective_ on positive scales: equal metrics imply equal scales (SCL-4). In the reverse direction, distinct positive scales give distinct metric representatives (SCL-3). === The family-level problem Paper 2 established that an individual kernel $K$ determines a conformal class $[g]_K$ but not a specific metric representative. Selecting $g$ requires a scale $sigma$ not available from any single kernel. A _coherent kernel family_ is defined intrinsically (Section 1): $F = {K_i}_(i in I)$ with $[g]_(K_i) = [g]_(K_j)$ for all $i, j$. The family is not defined by reference to a generating Lagrangian. To access effective-action methods, we additionally require _quantum realizability_: $F$ admits a common realization from which a collective effective action $Gamma_F$ can be constructed. This is a property of the family, not part of its definition. The Lagrangian, if one exists, is a representation used to compute $Gamma_F$; it is not ontologically prior to the kernels. The results of Sections 3 and 4 will show that the family determines a scale _projectively_: not a specific $sigma_F$, but an equivalence class $[sigma_F]_(bb(R)_+) = {c sigma_F : c > 0}$. The $bb(R)_+$-orbit removes only a single global positive constant --- the dimensional normalization. All local variation of $sigma$ is fixed by the family's quantum dynamics.") // ============================================================================= // Section 3: Weyl Anomaly and the Conformal-Mode Hessian // ============================================================================= define sec_anomaly = ArxivSection("Weyl Anomaly and the Conformal-Mode Hessian", "=== The conformal anomaly In a classically Weyl-invariant quantum field theory (conformal scalars, massless fermions, gauge vectors), the trace of the renormalized stress-energy tensor acquires a nonzero expectation value in curved spacetime: $ chevron.l T^mu_mu chevron.r = a E_4 - c W^2 + gamma' square R, $ where $E_4$ is the Euler density, $W^2 = C_(mu nu alpha beta) C^(mu nu alpha beta)$ is the Weyl tensor squared, and $square R$ is a total-derivative term. The coefficients $a$ and $c$ are unambiguous: $a$ (the type-A coefficient) is positive for all unitary matter content. For free fields: #align(center)[#table( columns: 3, [*Field*], [$a$], [$c$], [Real conformal scalar], [$1/(90(8 pi)^2)$], [$3/(90(8 pi)^2)$], [Weyl fermion], [$(11 slash 2)/(90(8 pi)^2)$], [$9/(90(8 pi)^2)$], [Gauge vector], [$62/(90(8 pi)^2)$], [$36/(90(8 pi)^2)$], )] The coefficient $gamma'$ of $square R$ is scheme-dependent: it can be shifted by adding a finite local $R^2$ counterterm to the effective action. This scheme dependence is standard [DeserDuffIsham1976] and reflects the fact that $square R$ is a trivial cocycle in the Weyl cohomology. === Effective action decomposition (Barvinsky--Wachowski) Barvinsky and Wachowski [BW2023] prove, for the one-loop effective action of a classically Weyl-invariant theory in curved spacetime, the decomposition $ Gamma_(\"ren\")[g] = Gamma_A [g] + Gamma^(\"conf\")[g], $ where $Gamma^(\"conf\")$ is _exactly Weyl-invariant_: $g_(mu nu) (delta Gamma^(\"conf\"))/(delta g_(mu nu)) = 0$. The anomalous part $Gamma_A$ generates the trace anomaly; the Weyl-invariant part $Gamma^(\"conf\")$ is obtained by 'conformization' --- restricting the full effective action to the conformally invariant representative of the Weyl orbit. The decomposition is exact at one loop. It is not an approximation, truncation, or perturbative expansion. === Why $H_B = 0$ The conformal-mode Hessian is the second functional derivative of $Gamma_(\"ren\")$ with respect to the conformal factor $phi$, where $g_(mu nu) = e^(2 phi) hat(g)_(mu nu)$ for a fixed background $hat(g)$: $ H_F := (delta^2 Gamma_(\"ren\")))/(delta phi^2). $ Since $Gamma^(\"conf\")$ is Weyl-invariant, its conformal-mode variation vanishes identically: $ (delta)/(delta phi) Gamma^(\"conf\")[e^(2 phi) hat(g)] = 0 quad \"for all\" phi(x). $ Therefore: $ H_F = (delta^2 Gamma_A)/(delta phi^2) = H_A. $ The 'correction' Hessian is $H_B := H_F - H_A = 0$, identically, for any classically Weyl-invariant family in the one-loop domain. This is the critical step: not that $H_B$ is small, or negligible, or bounded --- it vanishes exactly. The entire conformal-mode dynamics of the family is captured by the anomaly. === The anomaly Hessian and the Paneitz operator In the Wess--Zumino form of the anomaly action [Riegert1984], the conformal factor $sigma$ enters through $ Gamma_A supset (2 beta)/(16 pi^2) integral sqrt(macron(g)) space sigma macron(Delta)_4 sigma, $ where $Delta_4 = square^2 + 2 R^(mu nu) nabla_mu nabla_nu - 2/3 R square + 1/3 (nabla^mu R) nabla_mu$ is the Paneitz operator [Paneitz1983] and $beta = -16 pi^2 a$ in the Barvinsky convention. The conformal variation of the anomaly gives, using $delta_omega[sqrt(g) C^2] = 0$ (conformal invariance in four dimensions) and $delta_omega[sqrt(g) cal(E)_4] = 4 sqrt(g) Delta_4 omega$ [BW2023, Eq. 2.18]: $ H_A = (4 beta)/(16 pi^2) Delta_4 + \"(\" gamma\"-dependent terms)\". $ === The $gamma = 0$ scheme choice The $square R$ coefficient $gamma'$ is scheme-dependent and can be set to zero by an appropriate choice of finite $R^2$ counterterm. We adopt this _trivial anomaly scheme_ ($gamma = 0$) throughout, in which: $ H_A = (4 beta)/(16 pi^2) Delta_4 = -(a)/(pi^2) Delta_4. $ Since $a > 0$ for any non-trivial classically Weyl-invariant matter content, the anomaly Hessian is _negative-semidefinite_: $ H_A = -(a)/(pi^2) Delta_4 <= 0. $ This is the well-known conformal factor problem: the conformal mode has a wrong-sign kinetic term. As we will see in Section 4, this does not affect the kernel. We emphasize that $gamma = 0$ is stated as a domain hypothesis of the theorem, not as a claim about physics. Whether the kernel $ker H_F$ remains $bb(R) dot 1$ for all admissible finite $R^2$ scheme changes is left as an open proposition (Section 6).") // ============================================================================= // Section 4: Projective Metric Reconstruction Theorem // ============================================================================= define sec_theorem = ArxivSection("Projective Metric Reconstruction Theorem", "=== The Paneitz constant-kernel class #emph[Definition.] We say a compact four-dimensional Riemannian manifold $(M, hat(g))$ belongs to the _Paneitz constant-kernel class_ if the Paneitz operator $Delta_4$ on $(M, hat(g))$ satisfies $ Delta_4 >= 0, quad ker Delta_4 = bb(R) dot 1 quad (\"constants only\"). $ This is a spectral property of the elliptic Paneitz operator on a compact Riemannian manifold, where $Delta_4$ is self-adjoint on $L^2(M)$ with discrete spectrum. Known sufficient conditions include: - Gursky [Gursky1999]: $R_(hat(g)) >= 0$ and $integral Q_(hat(g)) \"dvol\" >= 0$. - Gursky--Viaclovsky [GV2003]: $R_(hat(g)) > 0$ and $k_(hat(g)) + 1/6 Y[hat(g)]^2 > 0$. - Lai [Lai2014]: three-positive Ricci curvature. Examples: $S^4$, $bb(C) P^2$, $S^2 times S^2$, and many connected sums. === Statement #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt, width: 90%)[ *Theorem (PROJ-1A: Projective Metric Reconstruction).* Let $F$ be a conformally coherent kernel family admitting a quantum realization on a compact Riemannian four-manifold $(M, hat(g))$. Suppose: + $F$ is _classically Weyl-invariant_: its quantum realization is by fields with classically Weyl-invariant action (conformal scalars, massless fermions, gauge vectors). + $F$ is in the _one-loop domain_: the effective action $Gamma_(\"ren\")$ is computed at one loop (quantum matter on a classical gravitational background). + The Barvinsky--Wachowski decomposition holds: $Gamma_(\"ren\") = Gamma_A + Gamma^(\"conf\")$ with $delta_sigma Gamma^(\"conf\") = 0$. + $(M, hat(g))$ belongs to the Paneitz constant-kernel class: $ker Delta_4 = bb(R) dot 1$. + The trivial anomaly scheme is adopted: $gamma = 0$. Then $H_F = H_A prop -a dot Delta_4$ with $a > 0$, and $ ker H_F = bb(R) dot 1. $ ]] === Proof By hypothesis (3), the conformal-mode Hessian decomposes as $H_F = H_A + H_B$ with $H_B = 0$ (Section 3). By hypothesis (5), $H_A = -(a slash pi^2) Delta_4$. By hypothesis (4), $Delta_4 >= 0$ with $ker Delta_4 = bb(R) dot 1$. Therefore $H_F = -(a slash pi^2) Delta_4$ is negative-semidefinite with $ker H_F = ker Delta_4 = bb(R) dot 1$. $square$ === Geometric conclusion The one-dimensional kernel $ker H_F = bb(R) dot 1$ means the only conformal transformations preserving the effective action's stationarity conditions are _global constant rescalings_ $g_(mu nu) arrow.r c^2 g_(mu nu)$, $c in bb(R)_+$. All nonconstant local Weyl transformations are broken by the anomaly. Therefore the family $F$ determines a _projective conformal scale_: an equivalence class $[sigma_F]_(bb(R)_+)$ rather than a specific $sigma_F$. The reconstruction is: $ F arrow.r [sigma_F]_(bb(R)_+) arrow.r [g_F]_(bb(R)_+). $ The residual $bb(R)_+$ is dimensional calibration. After choosing one physical unit of length $ell_0$: $ ([g_F]_(bb(R)_+), ell_0) arrow.r g_F. $ This final step is metrology, not physics. === Physical domain The theorem applies to any conformally coherent family whose quantum realization is by free, massless, classically Weyl-invariant fields on a compact Riemannian four-manifold in the Paneitz constant-kernel class. This includes: - Pure Yang--Mills theory (gauge vectors) with massless fermions. - The Standard Model in its conformal (ultraviolet) limit, where all explicit mass parameters are taken to zero. - Large-$N$ conformal field theories. on backgrounds such as $S^4$, $bb(C) P^2$, $S^2 times S^2$, and their connected sums. === Analytic realization and the Lorentzian bridge The compact Riemannian conformal manifold $(M_E, [g_E])$ used in PROJ-1A is not a Riemannian representative of the Lorentzian conformal class $[g]_(\"Lor\")$ reconstructed in Paper 2; positive conformal rescaling cannot change signature. It is an associated analytic realization of the quantum family --- denoted $F_E$ --- on which the one-loop effective action, the Barvinsky--Wachowski decomposition, and the Paneitz spectral problem are defined. PROJ-1A therefore establishes the projective scale quotient within this Riemannian realization: $ F_E arrow.r [g_(F,E)]_(bb(R)_+). $ Identifying the precise continuation or correspondence by which this quotient applies to the physical Lorentzian realization is a separate _realization bridge_ question and is not proved here. The paper establishes the scale-quotient theorem; the realization bridge between Riemannian and Lorentzian settings is an open obligation, consistent with the trilogy's policy of recording epistemic boundaries explicitly. This is analogous to standard practice in quantum field theory, where one-loop determinants and spectral results are computed in a Riemannian (Euclidean) setting and applied to physics via analytic continuation. The difference is that POT makes the logical gap explicit rather than treating it as routine.") // ============================================================================= // Section 5: Epistemic Accounting // ============================================================================= define sec_epistemic = ArxivSection("Epistemic Accounting", "=== Formally checked results The formal companion file #emph[theories/pot_metric_coherence.kleis] contains formally checked implications and consistency examples, verified by the Z3 SMT solver. Z3 confirms that the formal implications are correctly stated within the axiom system; it does not verify the underlying physics (Barvinsky--Wachowski, anomaly coefficients) or analysis (Paneitz spectral theorem), which are externally grounded by the cited literature. #align(center)[#table( columns: 4, [*Label*], [*Result*], [*Type*], [*Status*], [QR-1], [Quantum realizability $arrow.r.double$ effective action], [Formal implication], [Checked], [SCL-1], [Scale invariance: $(Omega sigma)^(-2)(Omega^2 tilde(g)) = sigma^(-2) tilde(g)$], [Algebraic identity], [Checked], [SCL-2], [Scale-metric reconstruction: $([g], sigma) arrow.r g_sigma$], [Definition], [Checked], [SCL-3], [Equal scales give equal representatives], [Algebraic identity], [Checked], [SCL-4], [Positive injectivity: $g_(sigma_1) = g_(sigma_2) arrow.r.double sigma_1 = sigma_2$], [Formal theorem], [Checked], [COH-1], [Metric coherence $arrow.r.double$ conformal coherence], [Formal implication], [Checked], [COH-2], [Conformal coherence $arrow.r.double.not$ metric coherence], [Countermodel], [Checked], [COH-3], [Field diversity compatible with coherence], [Consistency example], [Checked], [COH-4], [Family scale + conformal class $arrow.r$ metric], [Axiomatized], [Checked], [COH-5], [Admissibility $arrow.r.double.not$ coherence], [Countermodel], [Checked], [CAND-1], [Individual mass fails family generation], [Countermodel], [Checked], [CAND-2], [Arbitrary RG scale fails uniqueness], [Countermodel], [Checked], [REL-CM-1], [$|S_F| = 0$: incompatible family], [Countermodel], [Checked], [REL-CM-2], [$|S_F| > 1$: under-determined (residual Weyl)], [Countermodel], [Checked], [REL-CM-3], [$|S_F| = 1$: metric-coherent (unique scale)], [Countermodel], [Checked], [ZERO-BL], [$ker H_A = \"span\"{1}$ (compact Riemannian Paneitz class)], [Externally grounded], [Checked], [ZERO-1], [$H_(\"break\") = 0 arrow.r$ residual $bb(R)_+$], [Countermodel], [Checked], [ZERO-2], [$H_(\"break\") != 0$, external $arrow.r$ imported scale], [Countermodel], [Checked], [ZERO-3], [$H_(\"break\") != 0$, family-generated $arrow.r$ POT closure], [Open obligation], [Checked], [PROJ-1], [Semidefinite kernel lemma (sign-independent)], [Formal theorem], [Checked], [PROJ-1A], [One-loop compact Riemannian realization], [Formal theorem], [Checked], )] === Dependency structure PROJ-1A depends on: - _Externally grounded physics_ (not Z3-verified): the Barvinsky--Wachowski one-loop decomposition [BW2023]; standard anomaly coefficients for free fields. - _Externally grounded mathematics_ (not Z3-verified): the Paneitz constant-kernel spectral theorem on compact Riemannian 4-manifolds [Gursky1999, GV2003, Lai2014]. - _Formally checked mathematics_ (Z3-verified): the scale-metric reconstruction (SCL-1--4); family definition and quantum realizability (QR-1). The Z3 verification confirms that the formal implications are correctly composed: if the five hypotheses hold, then $ker H_F = bb(R) dot 1$ and the projective reconstruction follows. === The zero-mode trichotomy The theory file records three exhaustive cases for the source of scale: - *ZERO-1*: No Weyl breaking ($H_(\"break\") = 0$). The metric is determined only up to global rescaling. _This is the case realized by PROJ-1A._ - *ZERO-2*: External Weyl breaking (masses, external VEVs). The metric is fully determined, but the absolute scale is imported from outside the family, not generated by it. - *ZERO-3*: Family-generated Weyl breaking. The absolute scale emerges from the family's own dynamics. _This remains an open obligation._ PROJ-1A realizes ZERO-1 on its stated domain. The residual $bb(R)_+$ is the unavoidable consequence of having no external dimensionful scale in a classically Weyl-invariant family. This is not an incompleteness of the reconstruction; it is the correct physical answer for this class of families.") // ============================================================================= // Section 7: Discussion — Two Quotients and the Trilogy Result // ============================================================================= define sec_discussion = ArxivSection("Discussion", "=== The reconstruction chain The three papers of this series compose into a single reconstruction chain: #align(center)[#table( columns: 4, [*Paper*], [*Input*], [*Output*], [*What is determined*], [1], [Kernel $K$], [$C(K)$], [Characteristic signature], [2], [Signature $C(K)$], [$[g]_K$], [Conformal class], [3], [$([g]_K, F)$], [$[g_F]_(bb(R)_+)$], [Projective metric], )] Each stage extracts geometric structure from the kernel data by quotienting out unphysical degrees of freedom. The surviving structure at each level is relational, not absolute. === Two quotients, two levels The trilogy involves two distinct quotient operations, operating at different geometric levels. *Paper 2: local covector magnitude.* The characteristic singular direction at $x in M$ is a covector $xi_x in T^*_x M without 0$. Projectivization removes the local positive function $lambda(x)$ relating $xi_x$ and $lambda(x) xi_x$: $ xi_x tilde lambda(x) xi_x, quad lambda(x) > 0, $ producing the projective singular direction $[xi_x] in P(T^*_x M)$. This quotient is _local_: a separate positive factor is removed at each point. The surviving data is the projective cone field $P Sigma_K$, which determines the conformal class $[g]_K$. *Paper 3: global dimensional scale.* After the anomaly has eliminated all nonconstant Weyl modes as reconstruction ambiguity, the conformal scale $sigma_F$ is determined up to a single global constant $c in bb(R)_+$: $ sigma_F tilde c sigma_F, quad c > 0. $ This quotient is _global_: a single positive number, the same at all points, is the residual freedom. It is dimensional calibration. The progression from Paper 2 to Paper 3 is therefore a reduction of ambiguity: #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt, width: 80%)[ $ underbrace([g]_K, \"local\" C^infinity_+(M) \"freedom\") arrow.r^(\"anomaly\") underbrace([g_F]_(bb(R)_+), \"global\" bb(R)_+ \"freedom\") $ ]] The anomaly converts the infinite-dimensional group $C^infinity_+(M)$ of local Weyl rescalings to the one-dimensional group $bb(R)_+$ of global rescalings. === What the surviving $bb(R)_+$ means physically The residual global rescaling $g_(mu nu) arrow.r c^2 g_(mu nu)$ changes the dimensional normalization of the metric --- the numerical value of lengths, areas, and curvature scalars in a given unit system --- without changing any dimensionless ratio. It is the transformation that relates 'measuring in meters' to 'measuring in feet.' For a classically Weyl-invariant family with no external dimensionful parameters, no intrinsic mechanism can distinguish $c = 1$ from $c = 2$. Asking the theory to determine $c$ would be analogous to asking a theory of angles to determine whether a triangle has side length one meter. The request is not unanswered; it is meaningless within the theory's ontology. One external measurement --- any single dimensionful observable --- calibrates the entire structure: $ ([g_F]_(bb(R)_+), ell_0) arrow.r g_F, $ where $ell_0$ is one measured length. This is metrology, not physics. === Lorentzian realization: dynamics versus reconstruction PROJ-1A establishes the scale quotient $F arrow.r [g_F]_(bb(R)_+)$ on a compact Riemannian realization. Independently, Paper 2's characteristic-quadric reconstruction produces a _Lorentzian_ conformal class $[g]_K$ whenever the Witt index is one. This raises the question: what do nonconstant conformal-mode solutions mean in the Lorentzian realization? On a globally hyperbolic Lorentzian spacetime, the Paneitz operator $Delta_4$ is a fourth-order operator whose Cauchy problem is well-posed (with four pieces of initial data on a spacelike surface) when $Delta_4$ is Green-hyperbolic [Bar2015]. The homogeneous conformal-mode equation $Delta_4 delta phi = 0$ then admits infinitely many nonconstant solutions, parametrized by Cauchy data. This is radically different from the compact Riemannian setting, where the $L^2$ kernel contains only constants. The key observation is that these nonconstant Lorentzian solutions are _spacetime dynamical excitations_ --- conformal-mode waves determined by initial conditions --- not additional reconstruction ambiguity. The Lorentzian interpretation therefore does not establish, strengthen, or modify the quotient theorem. It explains what nonconstant conformal solutions mean once the reconstructed geometry is given: - The scale quotient $[g_F]_(bb(R)_+)$ is established by PROJ-1A. - Nonconstant conformal modes in the Lorentzian realization change the _physical state_, not the _reconstructed geometry_. - Only the global calibration $delta phi = c$ everywhere represents a different choice of ruler. Paper 2 already separated geometry from directed propagation: $[g]_K$ determines conformal shape but not temporal orientation. Paper 3 now separates the scale quotient from conformal-mode dynamics: $[g_F]_(bb(R)_+)$ determines the projective metric but not which conformal excitations are present. The trilogy's final architecture is richer than 'kernels reconstruct the metric': #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt, width: 80%)[ Kernel relations $arrow.r$ geometry + propagation data + state data + metrological calibration. ]] Only the first of these is tensorial geometry. === Signature as output, not substrate The Riemannian/Lorentzian distinction in this paper concerns the signature of the _reconstructed geometric realization_, not the evolution parameter of ontological Hilbert space $cal(H)_(\"ont\")$. POT's logical ordering is: $ cal(H)_(\"ont\") \"modal flow\" arrow.r {K, F, \"projection relations\"} arrow.r^(\"reconstruction\") (M, [g]). $ Lorentzian causal evolution --- null cones, Cauchy surfaces, retarded/advanced propagation --- becomes meaningful only _after_ the last arrow. The modal-flow parameter that orders states in $cal(H)_(\"ont\")$ cannot initially mean coordinate time, proper time, or a causal parameter, because none of those structures exists yet. Three notions should be distinguished: #align(center)[#table( columns: 2, [*Concept*], [*Character*], [Modal parameter], [Orders/evolves states in $cal(H)_(\"ont\")$; no spacetime causal structure], [Compact Riemannian realization], [Analytic tool for the scale-quotient proof (PROJ-1A)], [Lorentzian causal evolution], [Emergent spacetime dynamics on the reconstructed $(M, [g_F])$], )] There is no reason for the first to be either of the latter two. $cal(H)_(\"ont\")$ does not need to be Lorentzian; saying so would be close to a category mistake unless some independent structure of $cal(H)_(\"ont\")$ warranted it. Lorentzian signature belongs to the reconstructed manifold. It is therefore not merely $g_(mu nu)$ that emerges from projection-kernel structure. The causal-temporal interpretation that $g_(mu nu)$ makes possible emerges with it. The full reconstruction chain is: $ (cal(H)_(\"ont\"), \"modal flow\") arrow.r K arrow.r C(K) arrow.r P Sigma_K arrow.r [g]_K arrow.r^F [g_F]_(bb(R)_+). $ From the reconstructed Lorentzian geometry one then obtains: $ [g_F]_(bb(R)_+) arrow.r {\"null cones, causal ordering, Cauchy surfaces, spacetime propagation\"}. $ Within POT, modal evolution is posited at the ontological level, whereas spacetime causal evolution belongs to the reconstructed Lorentzian realization. The trilogy has established the second half of this structural ordering; it has not derived modal flow itself. === Open extensions Three directions extend PROJ-1A without weakening it: *Scheme independence (SCH-?).* The theorem is stated in the $gamma = 0$ anomaly scheme. A finite $R^2$ counterterm shifts $gamma$ and adds lower-order terms to $H_A$ beyond the Paneitz operator. Whether $ker H_F$ remains $bb(R) dot 1$ across all admissible scheme changes is an open proposition. If established, the scheme restriction in hypothesis (5) can be removed. *Higher loops.* The Barvinsky--Wachowski decomposition is proved at one loop. Whether $Gamma^(\"conf\")$ acquires conformal-mode dependence at higher loops is a separate question. Wess--Zumino consistency constrains the form of anomalies at all orders but does not by itself imply that higher-loop contributions to the Weyl-invariant part are absent. *Approach B (beyond Weyl invariance).* For families with explicit mass parameters or spontaneous symmetry breaking, $H_B != 0$. The semidefinite kernel lemma (Appendix A) then becomes the operative tool: if $H_B$ has the same sign as $H_A$ and $1 in ker H_B$, the projective reconstruction survives. Establishing this for specific symmetry-breaking models (Ghilencea--Stueckelberg, Coleman--Weinberg, the Standard Model Higgs sector) is a model-by-model question, deferred to future work. === The trilogy result The three-paper program terminates at a precise conceptual boundary: #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt, width: 80%)[ Individual kernels determine conformal shape. Coherent families determine relative scale. One global $bb(R)_+$ calibration remains. ]] Each paper achieves a successive demotion of a conventional primitive: #align(center)[#table( columns: 3, [*Paper*], [*Demotion*], [*Reconstruction*], [1], [Lagrangian/operator formulation is not fundamental], [$K arrow.r C(K)$], [2], [The metric tensor is not fundamental], [$C(K) arrow.r [g]_K$], [3], [Absolute metric scale is not fundamental], [$([g]_K, F) arrow.r [g_F]_(bb(R)_+)$], )] What remains primitive throughout is _relations encoded by projection kernels_: not coordinates, not a metric, not a tensor field, not an absolute scale. The tensor language appears because a sufficiently rich collection of projective singular directions happens to admit a unique quadratic representative up to scale. In that sense, $g_(mu nu)$ becomes a compressed representation of relational kernel structure. === Geometry as reconstructed structure The three results of this series reverse the usual explanatory order. In conventional field theory, geometric structure is supplied first: $ (M, g) arrow.r cal(L)[g, Phi] arrow.r L arrow.r K = L^(-1). $ POT begins from the opposite end: $ K arrow.r \"WF\"'(K) arrow.r P Sigma_K arrow.r [g]_K arrow.r^F [g_F]_(bb(R)_+). $ Paper 1 showed that physically relevant structure can survive changes of formulation at the kernel level. Paper 2 identified the projective characteristic cone field $P Sigma_K$ as sufficient to reconstruct a Lorentzian conformal class. The present paper adds the family level: coherent kernel-family dynamics removes the local conformal freedom, leaving only a global positive rescaling. The tensor $g_(mu nu)$ is not part of the primitive data. Tensorial geometry appears as a representation of relational structure already encoded in the kernel family. The two projectivizations make this explicit. At the kernel level, projectivization removes covector magnitude while preserving characteristic directions. At the family level, the anomaly removes local Weyl freedom while leaving a single global $bb(R)_+$ normalization. What remains unfixed is not local spacetime geometry but dimensional calibration. #align(center)[#emph[Projection kernels determine geometry up to the choice of ruler.]] Within its stated domain, the reconstruction reverses the conventional relation: #align(center)[#emph[Geometry is not the stage on which projection kernels live. Within the reconstructed domain of POT, geometry is an invariant organization of the kernels themselves.]] The reconstruction chain: $ K arrow.r C(K) arrow.r [g]_K arrow.r^F [g_F]_(bb(R)_+) arrow.r^(ell_0) g_F. $ The first three arrows are physics. The last is metrology.") // ============================================================================= // References // ============================================================================= define ref_atik_paper1 = ArxivReference("Atik2026a", "Atik, E. Emergent Conformal Geometry from Projection Kernels: A Projected Ontology Account of QED and Gravitation. Kleis Research (2026).") define ref_atik_paper2 = ArxivReference("Atik2026b", "Atik, E. Conformal Reconstruction from Projection Kernels: Minimal Microlocal Signatures for Emergent Lorentzian Geometry. Kleis Research (2026).") define ref_bw = ArxivReference("BW2023", "Barvinsky, A. O. and Wachowski, W. Notes on conformal anomaly, nonlocal effective action, and the metamorphosis of the running scale. Phys. Rev. D 108, 045014 (2023). arXiv:2306.03780.") define ref_gursky = ArxivReference("Gursky1999", "Gursky, M. J. The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE. Commun. Math. Phys. 207, 131--143 (1999).") define ref_gv = ArxivReference("GV2003", "Gursky, M. J. and Viaclovsky, J. A. A fully nonlinear equation on four-manifolds with positive scalar curvature. J. Differential Geom. 63, 131--154 (2003). arXiv:math/0301350.") define ref_lai = ArxivReference("Lai2014", "Lai, Y. A remark on the nonnegativity of the Paneitz operator. Proc. Amer. Math. Soc. 143, 4893--4900 (2015).") define ref_riegert = ArxivReference("Riegert1984", "Riegert, R. J. A non-local action for the trace anomaly. Phys. Lett. B 134, 56--60 (1984).") define ref_ddi = ArxivReference("DeserDuffIsham1976", "Deser, S., Duff, M. J. and Isham, C. J. Non-local conformal anomalies. Nucl. Phys. B 111, 45--55 (1976).") define ref_paneitz = ArxivReference("Paneitz1983", "Paneitz, S. M. A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. SIGMA 4, 036 (2008). Originally circulated as a preprint (1983).") define ref_bar = ArxivReference("Bar2015", "Bär, C. Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333, 1585--1615 (2015). arXiv:1310.0738.") define ref_kleis = ArxivReference("Kleis", "Atik, E. Kleis: A Verification Platform for Mathematical Knowledge Production. https://kleis.io (2025).") // ============================================================================= // Appendix A: Semidefinite Kernel Lemma // ============================================================================= define sec_appendix_lemma = ArxivSection("Appendix A: Semidefinite Kernel Lemma", "This appendix proves a result about self-adjoint operators on a Hilbert space. It is not used by the main theorem (PROJ-1A, where $H_B = 0$), but provides the mathematical infrastructure for future Approach B extensions (Section 6) in which $H_B != 0$. === Statement #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt, width: 85%)[ *Lemma (common-sign kernel intersection).* Let $H_A$, $H_B$ be self-adjoint operators on $L^2(M)$ and let $s in {+1, -1}$. Suppose: (i) $s dot H_A >= 0$ with $ker H_A = bb(R) dot 1$, (ii) $s dot H_B >= 0$ with $1 in ker H_B$. Then $ker(H_A + H_B) = bb(R) dot 1$. ]] === Proof Let $u in ker(H_A + H_B)$. Then $chevron.l u, (H_A + H_B) u chevron.r = 0$. Multiply by $s$: $ s chevron.l u, H_A u chevron.r + s chevron.l u, H_B u chevron.r = 0. $ By hypotheses (i) and (ii), each summand is nonnegative. A sum of nonnegative reals vanishes only if each summand vanishes: $ chevron.l u, H_A u chevron.r = 0 quad \"and\" quad chevron.l u, H_B u chevron.r = 0. $ Therefore $u in ker H_A inter ker H_B$. By (i), $ker H_A = bb(R) dot 1$, so $u in bb(R) dot 1$. Conversely, $1 in ker H_A$ by (i) and $1 in ker H_B$ by (ii), so $1 in ker(H_A + H_B)$. Therefore $ker(H_A + H_B) = bb(R) dot 1$. $square$ === Remarks The case $s = +1$ covers positive-semidefinite operators. The case $s = -1$ covers negative-semidefinite operators, which is the physically relevant case (Section 3). Condition (ii) requires only semidefiniteness of $H_B$ with the same sign as $H_A$, and $1 in ker H_B$. It does _not_ require strict definiteness of $H_B$ on $\"span\"{1}^perp$. The operator $H_A$ already supplies the spectral gap there. In particular, $H_B = 0$ satisfies condition (ii) trivially.") // ============================================================================= // Paper Assembly // ============================================================================= define paper_sections = [ sec_intro, sec_scale, sec_anomaly, sec_theorem, sec_epistemic, sec_discussion, ref_atik_paper1, ref_atik_paper2, ref_bw, ref_gursky, ref_gv, ref_lai, ref_riegert, ref_ddi, ref_paneitz, ref_bar, ref_kleis, sec_appendix_lemma ] 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)) }