// ============================================================================= // Conformal Reconstruction from Projection Kernels: // Minimal Microlocal Signatures for Emergent Lorentzian Geometry // ============================================================================= // // Paper 2 in the Projected Ontology series. // // Central result: a projection kernel on a smooth manifold determines // a Lorentzian conformal class [g]_K from the unique projective quadric // algebraically determined by its singular directions, without assuming // any background metric, connection, or kinetic operator. The conditions // {algebraic saturation, Witt index 1, smooth variation} are shown to // be irredundant: each is independently necessary. // // The conformal class in turn fixes the unoriented characteristic // foliation. Directed propagation — including the retarded/advanced // distinction — requires additional orientation or propagator data. // // The projective cone field provides a concrete microlocal // realization of the abstract characteristic signature introduced // in Paper 1, via two independent routes: operator-relative (D-H // theorem) and intrinsic (algebraic saturation). No bridge axiom // is required. // // Formal companion: theories/pot_causal_reconstruction.kleis (20 Z3 results) // // Pipeline: // kleis test --raw-output --example compile \ // docs/papers/pot_conformal_reconstruction_paper.kleis \ // > pot_conformal_reconstruction_paper.typ // typst compile pot_conformal_reconstruction_paper.typ // // ============================================================================= import "stdlib/prelude.kleis" import "stdlib/templates/arxiv_paper.kleis" // ============================================================================= // Paper Metadata // ============================================================================= define paper_title = "Conformal Reconstruction from Projection Kernels" define paper_authors = [ Author("Engin Atik", "1") ] define paper_affiliations = [ Affiliation(1, "Kleis Research", "https://kleis.io") ] define paper_abstract = "A projection kernel $K$ on a smooth manifold $M$ determines a Lorentzian conformal class $[g]_K$ whenever the projective singular directions extracted from its wavefront set algebraically determine, at each fiber, a unique smooth nondegenerate quadratic line of Witt index one. These three conditions --- algebraic saturation, Witt index one, smooth variation --- form an irredundant sufficient signature: removing any one produces an explicit countermodel. The conformal class fixes the unoriented characteristic foliation; directed propagation requires additional data. The projective cone field realizes the abstract characteristic signature of the preceding paper via two independent routes (operator-relative and intrinsic), eliminating all bridge axioms. Distinguished parametrices of real-principal-type operators provide an important sufficient class. The remaining ambiguity is metric calibration: selecting $g$ from $[g]_K$ requires collective scale information from a coherent kernel family. All results are formalized with 20 Z3-verified examples; dependency graph, countermodels, and epistemic boundaries are explicitly recorded." define paper_keywords = "projected ontology, conformal reconstruction, wavefront set, projective quadric, Witt index, microlocal analysis, cone field, characteristic signature, Lorentzian geometry, formal verification, Z3" // ============================================================================= // Section 1: Introduction — The Problem // ============================================================================= define sec_intro = ArxivSection("Introduction", "=== From common causal structure to conformal geometry The preceding paper in this series [Atik2026a] established a common characteristic signature across physically distinct production kernels: $ C_(\"QED\") = C_(\"GR\") = C_(\"CG\"). $ QED, Einstein gravity, and Mannheim's conformal gravity share the same causal skeleton despite differing in spin, differential order, and field content. The multiplicity invariance theorem ($Z(p) = Z(p^m)$) ensures that a fourth-order operator does not introduce new null directions beyond those of the corresponding second-order theory. That paper also demonstrated the boundary of what characteristic and topological signatures alone can determine: a conformal countermodel shows that $T + C$ are insufficient to select a full metric representative. The residual ambiguity is precisely conformal: $g'_(mu nu) = Omega^2(x) g_(mu nu)$. The present paper addresses the question that Paper 1 left open: #align(center)[_Under what conditions does a projection kernel determine a Lorentzian conformal class?_] === What is not assumed The conventional approach to conformal reconstruction begins with a Lorentzian manifold $(M, g)$, extracts a causal order $prec$, and invokes the Malament--Hawking--King--McCarthy theorem [Malament1977, HKM1976] to recover the conformal class. This is analysis _on_ a given spacetime. Projected Ontology Theory asks a logically prior question. Given a smooth manifold $M$ (no metric, no connection, no volume form) and a distribution $K in cal(D)'(M times M)$, what geometric structure can be _reconstructed_ from $K$? If the answer is a Lorentzian conformal class, then conformal geometry is not assumed but emergent. The only primitive input is $(M, C^infinity)$: a smooth differentiable carrier. Everything else --- null cones, causal structure, conformal metric --- must come from the kernel. === The result We identify the minimal kernel invariant sufficient for Lorentzian conformal reconstruction: the _complete projective characteristic cone field_ $P Sigma_K$. This is a smooth bundle of nonsingular real projective quadrics of Witt index one over $M$, extracted from the wavefront set of $K$. Three conditions are required and shown to be irredundant: - *C1 (Algebraic saturation)*: at each $x in M$, the projective singular directions $D_(K,x) subset P(T^*_x M)$ extracted from $\"WF\"'(K)$ determine a unique nondegenerate quadric: $dim I_2(D_(K,x)) = 1$. - *C2 (Witt index one)*: the nonzero generator $q_x$ of $I_2(D_(K,x))$ has Witt index one. - *C3 (Smooth variation)*: $x arrow.r.bar [q_x]$ varies smoothly over $M$. The hypotheses contain no metric and no differential operator; _Lorentzian_ appears only in the conclusion (Section 3). === Separation of geometry and directed propagation A second result prevents this from being merely another 'null cones determine conformal structure' argument. Unparameterized bicharacteristic curves are conformally invariant (Section 5), so the conformal class determines the full unoriented characteristic foliation: $ Sigma_K arrow.r.double [g]_K arrow.r.double R_K^(\"leaf\") quad \"but\" quad [g]_K arrow.r.double.not R_K^(arrow.r). $ Retarded and advanced Green kernels witness the separation: they share conformal geometry and unoriented leaves but differ in directed propagation. The geometry/dynamics boundary is at the directed/undirected distinction. === Relationship to Paper 1 Paper 1 introduced the characteristic signature $C(K)$ as an abstract invariant: the zero set of the principal symbol up to multiplicity and tensor-projector equivalence. This paper provides a concrete microlocal realization via two independent routes --- operator-relative (a theorem from Duistermaat--Hörmander) and intrinsic (a definition from algebraic saturation) --- eliminating the bridge axiom that connected abstract and concrete levels in earlier versions. The construction is detailed in Section 6. === Structure of the paper Section 2 establishes the mathematical setting: wavefront sets, cone fields, and projective quadrics on a smooth manifold without metric. Section 3 states the conformal reconstruction theorem. Section 4 grounds the algebraic step $P Sigma_K arrow [g]_K$ through the proportionality lemma for quadratic forms (Package A). Section 5 establishes irredundancy of the conditions and the geometry/dynamics separation (Package B). Section 6 constructs the concrete realization of Paper 1's abstract characteristic signature (Package C). Section 7 gives a compact epistemic accounting of all results. Section 8 identifies the remaining geometric arrow --- metric calibration --- and its relationship to the family-coherence formalism developed separately [Atik2026b].") // ============================================================================= // Section 2: Kernel Singularity Structure on a Smooth Manifold // ============================================================================= define sec_singularity = ArxivSection("Kernel Singularity Structure on a Smooth Manifold", "=== The wavefront set Let $M$ be a smooth manifold of dimension $n > 2$. The cotangent bundle $T^* M$ carries a canonical symplectic structure $omega = d theta$, where $theta$ is the Liouville--Poincaré one-form. This structure exists for _any_ smooth manifold; no metric, connection, or volume form enters its definition. For a distribution $K in cal(D)'(M times M)$, the _wavefront set_ $\"WF\"(K) subset T^*(M times M) without 0$ is defined by localization and Fourier analysis in local coordinates [Hoermander1971]. The definition is coordinate-independent: $\"WF\"$ transforms correctly under diffeomorphisms. The _primed wavefront set_ $\"WF\"'(K) subset (T^* M without 0) times (T^* M without 0)$ is the relational form, obtained by flipping the sign of the second covector. === The off-diagonal wavefront relation Decompose the primed wavefront set as $\"WF\"'(K) = Delta^* union C_K$, where $Delta^* = N^* Delta without 0$ is the conormal bundle of the diagonal in $M times M$ (the singularity directions attributable to the distributional identity) and $C_K := \"WF\"'(K) without N^* Delta$ is the _off-diagonal wavefront relation_. For a kernel with a distributional diagonal part $K = delta + K_(\"reg\")$, the conormal subtraction removes the directions arising from $delta$. === Projective singular directions From $C_K$, we extract fiberwise projective singular directions: $ D_(K,x) := P(pi_1(C_K) inter T^*_x M) subset P(T^*_x M), $ where $pi_1$ is the projection to the first factor and $P$ denotes projectivization. This extraction uses only $K$ and the smooth structure of $M$. === Algebraic saturation and the quadric Consider the vector space of quadratic forms vanishing on these directions: $ I_2(D_(K,x)) := { q in S^2(T_x M) : q(xi) = 0 space forall [xi] in D_(K,x) }. $ When $dim I_2(D_(K,x)) = 1$, the extracted directions algebraically determine a unique nondegenerate quadratic cone $Q_(K,x) = Z_P(q_x)$, where $q_x$ is the nonzero generator. This is the _algebraic saturation_ condition (C1). The directions need not densely populate the quadric; they need only be algebraically determining. The cone field $Sigma_K subset T^* M without 0$ and its projectivization $P Sigma_K = Sigma_K slash bb(R)_(> 0) subset P(T^* M)$ then agree with the quadrics $Q_(K,x)$. Projectivization quotients out the scale ambiguity $xi tilde lambda xi$ that is also invisible to conformal rescaling $g tilde Omega^2 g$. === Operator-relative sufficient class For a Green kernel $K$ of a differential operator $L$, the Duistermaat--Hörmander theory [DH1972] relates $\"WF\"(K)$ to the characteristic set $\"Char\"(L) = {(x, xi) in T^* M without 0 : sigma_(\"prin\")(L)(x, xi) = 0}$. For distinguished parametrices, the extracted directions $D_(K,x)$ fill the projective characteristic variety $Z_P(sigma_(\"prin\"))(x)$, so algebraic saturation holds automatically for this class. The argument is given in Section 3. === The bicharacteristic leaf relation On a smooth conic hypersurface $Sigma_K subset T^* M without 0$, the restriction of the canonical symplectic form $omega$ to $T Sigma_K$ has a one-dimensional kernel: $ Xi_K = ker(omega|_(T Sigma_K)). $ The integral curves of $Xi_K$ are the bicharacteristic curves --- defined intrinsically from $Sigma_K$ and the canonical symplectic structure, with no Hamiltonian chosen externally and no metric required. The _leaf relation_ $R_K$ is the equivalence relation on $Sigma_K$ generated by these curves: $(z, z') in R_K$ if and only if $z$ and $z'$ lie on the same characteristic leaf. The off-diagonal wavefront relation $C_K$ of a distinguished parametrix realizes this leaf relation: $C_K = R_K$ restricted to appropriate domains [DH1972].") // ============================================================================= // Section 3: The Conformal Reconstruction Theorem // ============================================================================= define sec_theorem = ArxivSection("The Conformal Reconstruction Theorem", "=== Statement #strong[Theorem (Conformal reconstruction).] _Let $M$ be a smooth manifold with $dim M > 2$ and let $K in cal(D)'(M times M)$. Define_ $ D_(K,x) := [pi_1(C_K) inter T^*_x M] subset P(T^*_x M), $ _the projective singular directions extracted from $\"WF\"'(K)$ at each fiber. Suppose:_ - _*C1 (Algebraic saturation)*: $dim I_2(D_(K,x)) = 1$ at every $x in M$, where $I_2(D_(K,x)) = { q in S^2(T_x M) : q(xi) = 0 space forall [xi] in D_(K,x) }$, and the nonzero generator $q_x$ is nondegenerate._ - _*C2 (Witt index one)*: $\"Witt\"(q_x) = 1$ at every $x in M$._ - _*C3 (Smooth variation)*: $x arrow.r.bar [q_x]$ is a smooth section of the projective quadric bundle._ _Then $K$ determines a unique Lorentzian conformal class $[g]_K$ on $M$._ _Proof._ By C1, the one-dimensional space $I_2(D_(K,x)) = chevron.l q_x chevron.r$ determines, at each $x$, a nondegenerate quadratic form $q_x$ unique up to nonzero scalar. Its projective zero locus $Q_(K,x) = Z_P(q_x)$ is therefore well-defined. By C2, $\"Witt\"(q_x) = 1$, so $q_x$ has Lorentzian signature. By the proportionality lemma (Section 4), the projective quadric determines the quadratic form up to scalar, hence determines a conformal class at each fiber. By C3, the assignment $x arrow.r.bar [q_x]$ is smooth, so the fiberwise conformal classes assemble into a smooth Lorentzian conformal class $[g]_K$ on $M$. $square$ === Vocabulary of the hypotheses The hypotheses contain: smooth manifold; distribution; wavefront set; projective cotangent bundle; symmetric bilinear forms; ideal of vanishing quadratics; nondegeneracy; Witt index one; smoothness. They contain _no metric_ and _no differential operator_. _Lorentzian_ appears only in the conclusion. This is the sense in which the theorem describes emergence: the smooth carrier is primitive input; the conformal Lorentzian structure is output. === The reconstruction chain The proof factors into steps, each with a distinct epistemic character: #figure( table( columns: 3, [*Arrow*], [*Content*], [*Status*], [$K arrow \"WF\"'(K) arrow D_(K,x)$], [Definition (microlocal analysis on smooth $M$)], [Standard], [$D_(K,x) arrow I_2(D_(K,x)) = chevron.l q_x chevron.r$], [C1: algebraic saturation --- directions determine unique quadric], [Obligation on $K$], [$[q_x] arrow Q_(K,x) = Z_P(q_x)$], [Definition (projective zero locus)], [Standard], [$Q_(K,x) arrow [g]_K$], [C2 + C3 + proportionality lemma], [Grounded (Section 4)], ), caption: [The conformal reconstruction chain with per-arrow epistemic status.], ) === C1: where the theorem's content lives The algebraic step $Q_(K,x) arrow [g]_K$ is standard projective geometry (Section 4). The genuinely substantive content is in C1: _for which distributions $K$ do the extracted singular directions algebraically determine a unique nondegenerate quadric?_ This is a nontrivial condition. Too few directions (e.g., a finite set not spanning the null quadric) leave the quadric under-determined: $dim I_2 > 1$. Too many directions (from a degenerate operator) can give $dim I_2 = 0$. The Duistermaat--Hörmander theory [DH1972] provides an important sufficient class. For a distinguished parametrix of a real-principal-type operator with quadratic principal symbol $sigma_(\"prin\")$, the singular directions fill the full projective characteristic variety $Z_P(sigma_(\"prin\"))$. By the proportionality lemma (Section 4), the space of quadratic forms vanishing on a nondegenerate projective quadric is one-dimensional, so $dim I_2 = 1$ automatically. Algebraic saturation is a _theorem_ for this class, not an assumption. Fewster [Fewster2026] extends the underlying microlocal machinery to $V^plus.minus$-decomposable Green-hyperbolic operators whose bicharacteristic curves need not follow spacetime null geodesics. The full intrinsic characterization of the class of distributions satisfying C1 remains open. === Time orientation The conformal class $[g]_K$ is independent of time orientation: if $K_1$ and $K_2$ have the same characteristic cone but opposite temporal orientations, they determine the same conformal class (REC-3 in the companion theory file). The time-orientation ambiguity is an irreducible $bb(Z)_2$ degree of freedom that does not affect conformal reconstruction. === Relationship to classical results The Malament--Hawking--King--McCarthy theorem [Malament1977, HKM1976] establishes that the causal structure of a past-and-future-distinguishing spacetime determines its conformal geometry. Our theorem addresses a logically prior question: whether the _singularity structure of a distribution_ can supply the null cone data from which causal and conformal structure follow. The classical result becomes corroborating context rather than the essential conformal-reconstruction step. The essential step is $\"WF\"'(K) arrow P Sigma_K$, which is specific to the kernel-first philosophy of Projected Ontology.") // ============================================================================= // Section 4: Projective-Quadric Grounding (Package A) // ============================================================================= define sec_package_a = ArxivSection("Projective-Quadric Grounding", "The algebraic content of the reconstruction is concentrated in the step $P Sigma_K arrow [g]_K$: a smooth fiberwise nonsingular Witt-index-one projective quadric bundle determines a Lorentzian conformal class. This section makes that step explicit. === Projective quadrics and quadratic forms A _projective quadric_ in $P(V)$ is the projective zero locus $Z_P(q) = {[xi] in P(V) : q(xi) = 0}$ of a nondegenerate quadratic form $q$ on a vector space $V$. The quadric is independent of the particular defining form up to a nonzero scalar: $Z_P(lambda q) = Z_P(q)$ for $lambda != 0$. === The proportionality lemma #strong[Lemma (Proportionality of quadratic forms).] _Let $q_1, q_2$ be nondegenerate quadratic forms on a vector space $V$ with $dim V >= 3$. If $Z_P(q_1) = Z_P(q_2)$, then $q_2 = lambda q_1$ for some $lambda != 0$._ _Proof._ Set $r = q_2 - (q_2(e) slash q_1(e)) q_1$ for any $e in.not Z_P(q_1)$, so $r$ vanishes on all of $Z_P(q_1)$ and also at $e$. The form $r$ is a homogeneous quadratic polynomial on $V$. In the graded coordinate ring of $P(V)$, the ideal of a nondegenerate quadric hypersurface is principal, generated by the defining form $q_1$. (A nondegenerate quadric in $P^(n-1)$ with $n >= 3$ is irreducible and not contained in any hyperplane, so any quadratic form in its vanishing ideal is a scalar multiple of $q_1$.) Therefore $r = mu q_1$ for some $mu$. But $r(e) = 0$ and $q_1(e) != 0$, so $mu = 0$ and $r = 0$. Hence $q_2 = lambda q_1$. $square$ The dimension condition $dim V >= 3$ corresponds to $dim M > 2$ in the reconstruction theorem. === From quadrics to conformal classes A nondegenerate quadratic form on $T^*_x M$ is a (co)metric tensor. Two proportional forms $q_2 = lambda q_1$ with $lambda > 0$ define the same conformal class. The projective quadric $Z_P(q)$ determines $q$ up to scalar (by the proportionality lemma), and hence determines the conformal class. This gives the grounding chain: $ P Sigma_(K,x) = Z_P(q_x) arrow.r.double q_x \"determined up to scalar\" arrow.r.double [q_x]_(\"conf\") \"determined.\" $ Fiberwise, $P Sigma_K$ determines a field of conformal (co)metric tensors. Under C2, this field has Witt index one (Lorentzian signature). Under C3, the field varies smoothly. Together: a smooth Lorentzian conformal class $[g]_K$. === Deriving REC-2 and REC-5 Two consequences follow from the proportionality lemma: - *REC-2*: Kernels with the same characteristic cone have the same conformal class. (Same cone $arrow$ same projective quadric at each fiber $arrow$ proportional forms $arrow$ same conformal class.) - *REC-5*: Characteristic coherence implies conformal coherence. If $C(K_1) = C(K_2)$, then $[g_1]_(\"conf\") = [g_2]_(\"conf\")$. These were axiomatized in the companion theory file; the proportionality lemma provides their algebraic grounding (GRD-1 through GRD-4 in the companion file). === Epistemic status The proportionality lemma is standard projective algebraic geometry ($dim >= 3$). Its Kleis encoding axiomatizes the result; a full derivation from polynomial-algebra primitives within the platform would require additional algebraic infrastructure. The mathematical content is not in doubt; only the internal Kleis grounding is deferred.") // ============================================================================= // Section 5: Minimality and Separation (Package B) // ============================================================================= define sec_package_b = ArxivSection("Minimality and Separation", "This section establishes two results. First, the conditions ${\"C1\"_(\"alg\")\, \"C2\"\, \"C3\"}$ form an irredundant sufficient signature for conformal reconstruction: each is independently necessary within this architecture. Second, the geometry/dynamics boundary falls at the directed/undirected distinction, not at the cone/leaf distinction. === Algebraic saturation suffices (MIN-1) The reconstruction chain (Section 3) arrives at a conformal class via $ D_(K,x) arrow I_2(D_(K,x)) = chevron.l q_x chevron.r arrow Z_P(q_x) arrow [g]_K. $ MIN-1 in the companion file verifies: algebraic saturation (C1) + Witt index one (C2) + smooth variation (C3) gives $[g]_K$. The full bicharacteristic leaf relation is unnecessary for conformal reconstruction. === Irredundancy: three countermodels Each condition is independently necessary within this reconstruction architecture. The formal countermodels below establish _logical irredundancy_ within the Kleis axiomatization; for each, we also describe a concrete mathematical scenario that illustrates the failure mode. #strong[MIN-3 (Algebraic saturation is necessary).] _Z3 model:_ consistent assignment with $dim I_2(D_(K,x)) > 1$ and wrong conformal class. _Mathematical scenario:_ if the singular directions sample only a proper subset of the null cone (e.g., a great circle on a sphere rather than the full $(n-2)$-quadric), then multiple non-proportional quadratic forms vanish on that subset and the conformal class is not determined. #strong[MIN-4 (C2 is necessary).] _Z3 model:_ consistent assignment with algebraic saturation, smooth variation, and non-Lorentzian output. _Mathematical scenario:_ a smooth cone field whose fiberwise quadric has Witt index 2 yields ultrahyperbolic geometry (signature $(2, n-2)$), not Lorentzian. C2 selects Lorentzian conformal geometry from the broader class of conformal structures associated with nondegenerate quadrics. #strong[MIN-5 (C3 is necessary).] _Z3 model:_ consistent assignment with algebraic saturation, Witt index 1, and no global conformal class. _Mathematical scenario:_ on $M = bb(R)^4$, set $g_x = \"diag\"(-1, 1, 1, 1)$ for $x^0 <= 0$ and $g_x = \"diag\"(1, -1, 1, 1)$ for $x^0 > 0$. Each fiber is Lorentzian, but the timelike direction jumps discontinuously at $x^0 = 0$. No smooth global conformal class exists. Together: ${\"C1\"_(\"alg\")\, \"C2\"\, \"C3\"}$ is an irredundant sufficient signature for conformal reconstruction within this architecture. === The geometry/dynamics separation (MIN-2) The conformal class determines more than the cone field alone. On a smooth conic hypersurface $Sigma_K subset T^* M without 0$, the characteristic line field $Xi_K = ker(omega|_(T Sigma_K))$ generates bicharacteristic curves _intrinsically_. Crucially, these curves are conformally invariant: if $p' = f p$ with $f > 0$, then on ${p = 0}$, $ H_(p') = f H_p $ (the extra $p H_f$ term vanishes on the characteristic set). So conformal rescaling reparameterizes bicharacteristic curves without changing their images. The _unoriented_ characteristic foliation $R_K^(\"leaf\")$ is determined by $[g]_K$. However, conformal geometry does not determine _directed propagation_: #strong[MIN-2 (Conformal class does not determine directed propagation).] The retarded and advanced Green kernels of the same operator share: - the same cone field $Sigma_K$, - the same conformal class $[g]_K$, - the same unoriented leaf structure $R_K^(\"leaf\")$, but differ in directed propagation $R_K^(arrow.r)$ (propagation orientation/boundary-condition structure). Therefore: $ Sigma_K arrow.r.double [g]_K arrow.r.double R_K^(\"leaf\") quad \"(all conformal/geometric)\", $ $ [g]_K arrow.r.double.not R_K^(arrow.r) quad \"(directed propagation is extra)\". $ The geometry/dynamics boundary falls at the directed/undirected distinction. Four distinct layers of kernel information emerge: #figure( table( columns: 3, [*Layer*], [*Content*], [*Determines*], [Cone field $Sigma_K$], [Fiberwise characteristic directions], [Conformal shape $[g]_K$], [Unoriented leaves $R_K^(\"leaf\")$], [Unparameterized bicharacteristic curves], [Determined by $[g]_K$], [Directed propagation $R_K^(arrow.r)$], [Propagation orientation/boundary conditions], [Additional directed data], [Family scale $S(F)$], [Collective calibration across kernels], [Metric representative $g$], ), caption: [Four layers of information in the wavefront structure and their geometric roles. Conformal geometry encompasses the first two layers. Directed propagation and metric calibration require additional data.], )") // ============================================================================= // Section 6: Realization of the Abstract Characteristic Signature (Package C) // ============================================================================= define sec_package_c = ArxivSection("Realization of the Abstract Characteristic Signature", "Paper 1 [Atik2026a] introduced the characteristic signature $C(K)$ as an abstract invariant: the zero set of the principal symbol modulo multiplicity and tensor-projector equivalence. Two invariances define the abstraction: - *CHAR-1 (Multiplicity invariance)*: $Z(p^m) = Z(p)$ for $m >= 1$. - *CHAR-2 (Projector forgetting)*: $Z(P dot sigma) = Z(sigma)$ when $P$ is nondegenerate on the physical sector. This paper provides a concrete realization via two independent routes, _neither of which requires a bridge axiom_. === Route 1: Operator-relative (for Green kernels of known operators) Paper 1 defines $C_(\"abstract\")(K) = \"char\"_(\"sig\")(sigma_(\"prin\")(L_K))$. The _nontrivial theorem_ imported here is Duistermaat--Hörmander [DH1972]: for a distinguished parametrix of a real-principal-type operator, the singularity relation of the kernel is precisely the bicharacteristic relation of the principal symbol, so $Sigma_K = \"Char\"(L_K) without 0$. The remaining equalities are definitions and projectivization: $ P Sigma_K = P Z(sigma_(\"prin\")(L_K)). $ The abstract signature retains exactly the zero-set information of $sigma_(\"prin\")$ (by CHAR-1 and CHAR-2). So equal abstract signatures imply equal projective zero loci. The derivation chain: $ C_(\"abstract\")(K_1) = C_(\"abstract\")(K_2) arrow.r.double Z(sigma_1) = Z(sigma_2) arrow.r.double P Z(sigma_1) = P Z(sigma_2) arrow.r.double P Sigma_(K_1) = P Sigma_(K_2). $ This is a _theorem_ for the operator-relative class, not an axiom. === Route 2: Intrinsic (for any algebraically saturated kernel) For an algebraically saturated kernel, the concrete signature is defined directly from $\"WF\"'(K)$: $ C_(\"concrete\")(K) := [I_2(D_K)] tilde.equiv [P Sigma_K]. $ Here $[I_2(D_K)]$ is the one-dimensional quadratic ideal determined by the projective singular directions. No operator is referenced. No comparison target is needed. The concrete signature _is_ the projective quadric, defined from the kernel's own wavefront data. === Invariance checks The two routes relate to Paper 1's abstract invariances differently: On the _intrinsic_ route, $C_(\"concrete\")(K) = [I_2(D_K)]$ is defined from the projective zero locus of wavefront directions. Multiplicity blindness ($Z_P(p^m) = Z_P(p)$) and projector blindness ($Z_P(P dot p) = Z_P(p)$ for nondegenerate $P$) are _structural properties_ of the zero-locus construction --- they hold by definition, not as separate theorems. On the _operator-relative_ route, the same properties must be verified as _compatibility checks_ against Paper 1's CHAR-1 and CHAR-2 axioms. The companion file records: #strong[REA-1 (Multiplicity invariance).] Kernels whose principal symbols differ by multiplicity ($p$ vs $p^m$) have the same concrete signature. This confirms compatibility with CHAR-1. #strong[REA-2 (Projector forgetting).] Kernels whose principal symbols differ by a nondegenerate tensor projector have the same concrete signature. This confirms compatibility with CHAR-2. === Deriving the universality ladder For the operator-relative class, the universality results follow as theorems: #strong[REA-4 (Derive REC-2).] Distinguished parametrices with $C_(\"abstract\")(K_1) = C_(\"abstract\")(K_2)$ have $P Sigma_(K_1) = P Sigma_(K_2)$ (Route 1). By the proportionality lemma (Section 4), $[g_1]_(\"conf\") = [g_2]_(\"conf\")$. #strong[REA-5 (Derive REC-5).] The same derivation applied to the coherence statement. For intrinsic kernels, the derivation goes directly: equal algebraically-determined quadrics imply equal conformal classes by the same proportionality argument. === Relationship between the papers The two papers stand in a _refinement_ relation, not a dependency: #figure( table( columns: 2, [*Paper 1*], [*Paper 2*], [$C(K)$ abstract], [$C(K) tilde.equiv [P Sigma_K]$ concrete], [Zero set up to multiplicity/projector], [Algebraically saturated projective cone field], [Axiomatized: same $C arrow.r.double$ same $[g]$], [Route 1: theorem (D-H); Route 2: definition + proportionality], ), caption: [The abstract characteristic signature (Paper 1) admits two concrete realizations (Paper 2). Paper 1 is not modified; Paper 2 refines it. No bridge axiom is required.], )") // ============================================================================= // Section 7: Epistemic Accounting // ============================================================================= define sec_epistemic = ArxivSection("Epistemic Accounting", "=== Verified results The companion theory file `pot_causal_reconstruction.kleis` contains 20 Z3-verified results: #figure( table( columns: 4, [*Label*], [*Result*], [*Type*], [*Status*], [REC-1], [Rescaled symbol preserves characteristic cone], [Theorem], [Axiomatized], [REC-2], [Same char cone $arrow.r.double$ same conformal class], [Theorem], [Axiomatized; grounded by Package A], [REC-3], [Conformal class independent of time orientation], [Theorem], [Axiomatized], [REC-4], [C1+C2+C3 gives conformal class], [Theorem], [Compositional: definitions + proportionality lemma], [REC-5], [Characteristic coherence $arrow.r.double$ conformal coherence], [Theorem], [Axiomatized; grounded by Package A], [CM-1], [Complement-components test fails for index 2], [Countermodel], [Rejection of early C2 candidate], [CM-2], [Quadric topology distinguishes index 1 vs 2 (dim 4)], [Countermodel], [Auxiliary], [GRD-1], [Same projective quadric $arrow.r.double$ same conformal class], [Grounding], [From proportionality lemma], [GRD-3], [Derive REC-2 via proportionality], [Derivation], [Compositional], [GRD-4], [Derive REC-5 via proportionality], [Derivation], [Compositional], [MIN-1], [Algebraic saturation suffices], [Theorem], [From saturation + proportionality], [MIN-2], [$[g]_K arrow.r.double.not R_K^(arrow.r)$ (directed propagation)], [Countermodel], [Retarded/advanced witness], [MIN-3], [Under-determined directions $arrow.r.double$ wrong $[g]$], [Z3 consistency model], [Irredundancy of C1], [MIN-4], [Witt index $> 1$ $arrow.r.double$ non-Lorentzian class], [Z3 consistency model], [Irredundancy of C2], [MIN-5], [No smooth variation $arrow.r.double$ no global $[g]$], [Z3 consistency model], [Irredundancy of C3], [REA-1], [Concrete realization: multiplicity invariance], [Derivation], [From invariance axioms], [REA-2], [Concrete realization: projector forgetting], [Derivation], [From invariance axioms], [REA-3], [Operator-grounded char equality $arrow.r.double$ same quadric], [Theorem], [Route 1 (D-H)], [REA-4], [Derive REC-2 for operator-grounded kernels], [Derivation], [Compositional], [REA-5], [Derive REC-5 for operator-grounded kernels], [Derivation], [Compositional], ), caption: [All 20 Z3-verified results with their epistemic status. _Axiomatized_ means the mathematical content is established but encoded as an axiom rather than derived from Kleis primitives. _Grounded_ means the axiom is supported by an explicit derivation path. _Compositional_ means the result follows from combining other verified results.], ) === Rejected formulations Three early candidate conditions were investigated and rejected: - *C1 candidate 1: $C_K$ is locally a graph of a canonical transformation.* Rejected because the bicharacteristic relation has a one-dimensional flow fiber and is generally not locally a graph. Replaced by the characteristic-leaf formulation via $Xi_K = ker(omega|_(T Sigma_K))$. - *C1 candidate 2: cone completeness ($Sigma_K = pi_1(C_K)$).* Rejected as tautological: $Sigma_K$ was defined from the same wavefront data it was required to 'complete.' Replaced by algebraic saturation ($dim I_2(D_(K,x)) = 1$), which is a nontrivial algebraic condition on the extracted directions. - *C2 candidate: complement of projective quadric has two connected components.* Rejected because this is trivially true for _all_ indefinite signatures, not just Lorentzian (CM-1). Replaced by the algebraic condition: Witt index one. The rejected formulations and their countermodels are retained in the companion file as part of the scientific record. === Bridge axiom elimination A previous version used an axiom `abstract_determines_concrete` to connect Paper 1's abstract $C(K)$ with Paper 2's concrete $P Sigma_K$. This was identified as a renamed bridge (the referee correctly noted it assumed the conclusion it claimed to derive). The revised Package C eliminates all bridge axioms by providing two independent routes (Section 6). No connecting assumption remains on any active proof path. === What the formalization contributes The value of the Z3 verification is not the number of passing results. It is: - *Dependency tracking*: each result records which axioms it requires, preventing circular reasoning. - *Consistency verification*: the rejected formulations (CM-1, CM-2) and the irredundancy models (MIN-3 through MIN-5) are machine-checked for internal consistency. These are Z3 consistency models establishing logical irredundancy, not exhibited distributions. - *Epistemic boundary enforcement*: axiomatized results are explicitly labeled; no downstream result claims derivation from unlabeled assumptions. - *Compositional closure*: the grounding and realization chains (GRD, REA) are verified to compose correctly. - *Bridge elimination*: the progression from axiomatized bridge to theorem/definition is tracked and machine-verified.") // ============================================================================= // Section 8: Discussion — Shape, Propagation, Calibration // ============================================================================= define sec_discussion = ArxivSection("Discussion: Shape, Propagation, Calibration", "=== The reconstruction chain and the conformal fork This paper's results yield a reconstruction chain terminating at the conformal class: #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt)[ $ K arrow.r \"WF\"'(K) arrow.r D_K arrow.r chevron.l q chevron.r arrow P Sigma_K arrow [g]_K arrow R_K^(\"leaf\") $ ]] Conformal reconstruction is not an intermediate approximation to full geometry. It is the natural endpoint of the individual kernel's projective geometry-bearing information. Two distinct enrichments remain, branching from the conformal class rather than forming successive stages: #align(center)[#box(stroke: 0.5pt, inset: 12pt, radius: 4pt)[ $ [g]_K cases(limits(+O(K))^() arrow.r R_K^(arrow.r) quad &\"(directed propagation)\", limits(+S(F))^() arrow.r g_F quad &\"(metric calibration)\") $ ]] Here $O(K)$ denotes schematically whatever orientation or boundary-condition information selects directed propagation from the conformal/leaf structure, and $S(F)$ is the family-level scale invariant introduced in [Atik2026b]. Neither enrichment is supplied by the projective characteristic cone $P Sigma_K$ itself, and neither is obviously reducible to the other: retarded and advanced kernels share $[g]$ and $R^(\"leaf\")$ while differing in $R^(arrow.r)$, yet both must live on the same calibrated metric if they belong to the same physical universe. === What 'geometry is emergent' means precisely At this stage, the claim has a precise and limited content: - *Primitive input*: the smooth differentiable carrier $(M, C^infinity)$. - *Emergent output*: the Lorentzian conformal class $[g]_K$, reconstructed from $K$. - *Not emergent*: the smooth manifold $M$ itself, its dimension, or its topology. - *Not yet emergent*: the full metric $g$ (requires collective scale $S(F)$). The differentiable carrier remains prior input. Conformal metric structure is projected from the kernel's singularity data. This is the exact boundary of the current POT construction. === The family-level calibration problem Selecting a metric representative $g$ from the conformal class $[g]_K$ requires information not available from any individual kernel. The metric-coherence formalism [Atik2026b] introduces a _family scale_ $S(F)$ for a coherent family $F = {K_i}$ of physical kernels: $ ([g]_(\"conf\"), S(F)) arrow.r g^*. $ Paper 2 constrains what $S(F)$ must be: - *Family-level*: individual-kernel admissibility does not imply metric coherence (COH-5 in [Atik2026b]). - *Sufficient to resolve $g tilde Omega^2 g$*: $S(F)$ must supply precisely the information quotiented out by the projective construction $P Sigma_K$. - *Independent of the retarded/advanced choice*: if a proposed $S(F)$ changes merely because one replaces a retarded kernel by the corresponding advanced kernel while leaving the physical family and conformal geometry unchanged, it is not the metric-calibration invariant. The physical identity of $S(F)$ is deliberately left uninterpreted. Paper 1 suggested that it may originate in the massive sector of QED, where the mass-shell condition $g_(mu nu) p^mu p^nu = m^2$ is not conformally invariant. The metric-coherence formalism establishes that individual field-specific masses cannot serve as the universal scale: different fields in the same coherent family may have different masses while reconstructing the same metric. === Conformal gravity revisited The structural connection to Mannheim's conformal gravity [Man2012] identified in Paper 1 is sharpened by the present results. Paper 1 showed that the geometric boundary of the POT construction is conformal. This paper shows _why_: the geometry-bearing invariant of a kernel is the projective cone field, which is inherently a conformal object (both covector rescaling and metric conformal rescaling drop out at the projective level). The conformal endpoint is therefore not necessarily a failure to reconstruct the metric. It may identify the natural boundary of geometry available at the individual-kernel level. Metric scale would then arise only from coherence across a physical family of kernels. === Relation to existing microlocal and causal-reconstruction literature The result that null cones determine conformal geometry is classical. It underlies the Malament--Hawking--King--McCarthy theorem [Malament1977, HKM1976] and is used constructively in Lorentzian inverse problems [KLU2018]. The Kashiwara--Schapira framework [KS1990] provides a purely sheaf-theoretic, metric-free formulation of causal cone structure. Radzikowski's microlocal spectrum condition [Radzikowski1996] characterizes Hadamard two-point functions via wavefront sets. The present paper does not claim to have newly discovered any of these results. Its contribution is different: _The novelty is not that null cones determine conformal geometry. It is the identification of the minimal kernel-extracted projective signature, its algebraic-saturation formulation, and the separation between conformal geometry and directed propagation._ Specifically: (i) the projective singular directions $D_(K,x)$ extracted from $\"WF\"'(K)$ are identified as the minimal geometry-bearing data; (ii) the algebraic-saturation condition ($dim I_2(D_(K,x)) = 1$) gives an intrinsic, operator-free criterion for conformal reconstruction; (iii) the irredundancy results (MIN-3 through MIN-5) establish that the reconstruction signature cannot be weakened; (iv) the geometry/dynamics separation (MIN-2) shows that directed propagation is provably extra information. === Conclusion The conformal geometry encoded by a projection kernel is determined by the unique nondegenerate quadratic line extracted from its projective singular directions at each fiber. The conditions --- algebraic saturation, Witt index one, smooth variation --- form an irredundant sufficient signature for Lorentzian conformal reconstruction within this architecture. Green kernels and distinguished parametrices of suitable real-principal-type operators provide an important sufficient class realizing the intrinsic construction. Two distinct enrichments branch from the conformal class: directed propagation (requiring orientation or boundary-condition data) and metric calibration (requiring family-level scale information). Neither is supplied by the projective characteristic cone, and neither reduces to the other. The result $K arrow.r.not g$ is therefore not a negative finding but a structural boundary: it converts the question _how does POT produce a metric?_ into the sharper question _what collective structure of a physical kernel family selects $g in [g]$?_") // ============================================================================= // References // ============================================================================= define ref_atik_paper1 = ArxivReference("Atik2026a", "Atik, E. Emergent Conformal Geometry from Projection Kernels: A Projected Ontology Account of QED and Gravitation. Preprint (2026). Formal companion: theories/pot_characteristic_topology.kleis.") define ref_atik_coherence = ArxivReference("Atik2026b", "Atik, E. Metric Coherence for Projection Kernel Families. Kleis theory file: theories/pot_metric_coherence.kleis (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). DOI: 10.1063/1.523436.") 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). DOI: 10.1063/1.522874.") define ref_dh = ArxivReference("DH1972", "Duistermaat, J. J. and Hörmander, L. Fourier integral operators. II. Acta Math. 128, 183--269 (1972).") define ref_radzikowski = ArxivReference("Radzikowski1996", "Radzikowski, M. J. Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529--553 (1996).") define ref_fewster = ArxivReference("Fewster2026", "Fewster, C. J. Hadamard states for decomposable Green-hyperbolic operators. Commun. Math. Phys. 407, Paper 14 (2026). arXiv:2503.12537.") define ref_klu = ArxivReference("KLU2018", "Kurylev, Y., Lassas, M. and Uhlmann, G. Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations. Invent. Math. 212, 781--857 (2018).") define ref_ks = ArxivReference("KS1990", "Kashiwara, M. and Schapira, P. Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften 292, Springer (1990).") 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_hoermander = ArxivReference("Hoermander1971", "Hörmander, L. Fourier integral operators. I. Acta Math. 127, 79--183 (1971).") 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_singularity, sec_theorem, sec_package_a, sec_package_b, sec_package_c, sec_epistemic, sec_discussion, ref_atik_paper1, ref_atik_coherence, ref_malament, ref_hkm, ref_dh, ref_radzikowski, ref_fewster, ref_klu, ref_ks, ref_mannheim, ref_hoermander, 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)) }