The Least Surprising Theory: An Information-Theoretic Approach to the Unification of Physics

1. Introduction

1.1. Physics from Inference

Jaynes (1957) observed that statistical mechanics need not be derived from mechanics at all. Extremizing the Shannon entropy

$$S\left[p\right]=-\sum _i{p}_{i}ln{p}_i\mathrm{subject}\mathrm{to}\sum _i{p}_i{f}_k^{\left(i\right)}=\langle f_k\rangle$$

(1)

returns the Gibbs distribution,

$$p_i\propto exp\left(-\sum _k{\lambda }_k{f}_k^{\left(i\right)}\right),$$

(2)

the multipliers ${\lambda }_k$ being fixed by the constraints. No ergodic hypothesis is required and nothing is assumed about the underlying dynamics; thermodynamics on this reading is a rule of inference rather than a physical law.

The reframing was more ambitious than a rederivation. If one part of physics is inference under constraint, the rest may be as well, and the laws would then not be brute facts but the least biased account of nature available given what is held fixed. We shall call this the inferential programme. The present work is an attempt to complete it; the two subsections that follow set out what has been achieved within it and why it has not been completed before.

Paths.

Jaynes extended the construction from states to histories. In place of a distribution over configurations one seeks a distribution over trajectories, extremizing the path entropy—the caliber—subject to constraints on path functionals $F_k\left[x\right]$. The output has the same form as (2),

$$p\left[x\left(t\right)\right]\propto exp\left(-\sum _k{\lambda }_k{F}_k\left[x\right]\right),$$

(3)

from which follow diffusion, the fluctuation theorems and Onsager reciprocity; Pressé et al. (2013) survey the construction and its applications.

Updates.

A further refinement takes the object of inference to be not a distribution but the revision of one. Given a reference $\rho$, the least biased revision is measured by the relative entropy, and for nearby distributions

$$D\left(\rho +\delta \rho \parallel \rho \right)={\displaystyle \frac{1}{2}}\int {\displaystyle \frac{{\left(\delta \rho \right)}^2}{\rho }}+O\left(\delta {\rho }^3\right),$$

(4)

whose leading term is the Fisher–Rao metric.

Quantum mechanics.

That line reached a landmark in the recovery of the Schrödinger equation (Hall and Reginatto 2002,Reginatto 1998). A classical ensemble is described by a density $\rho$ and a Hamilton–Jacobi action S, with ensemble Hamiltonian

$${\mathcal{H}}_{\mathrm{cl}}[\rho ,S]=\int {\mathrm{d}}^{3}x\rho \left[{\displaystyle \frac{{|\nabla S|}^2}{2m}}+V\right],$$

(5)

whose stationary points are the continuity and Hamilton–Jacobi equations. Adding the Fisher information of the density,

$$\mathcal{H}[\rho ,S]={\mathcal{H}}_{\mathrm{cl}}[\rho ,S]+\lambda \int {\mathrm{d}}^{3}x{\displaystyle \frac{{|\nabla \rho |}^2}{\rho }},\lambda ={\displaystyle \frac{{\hslash }^2}{8m}},$$

(6)

and writing $\psi =\sqrt{\rho }{e}^{iS/\hslash }$, the two stationarity conditions become the real and imaginary parts of

$$i\hslash {\partial }_t\psi =\left(-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+V\right)\psi .$$

(7)

The Fisher term contributes exactly the quantum potential $-\frac{{\hslash }^2}{2m}{\nabla }^2\sqrt{\rho }/\sqrt{\rho }$; the remainder of (7) comes from (5). The Schrödinger equation as a statement about inference rather than a postulate about nature is a genuine success of the programme.

Everything else.

Frieden (1998) carried the idea furthest. In the principle of extreme physical information a measurement is a transfer of information from source to observer, and the equations of physics are the extrema of $I-J$, with I the Fisher information of the observed data and J the bound information of the source. With appropriate choices of J the principle returns the Klein–Gordon, Maxwell and Einstein equations and the Boltzmann distribution.

1.2. Why Each Fell Short

In the guise of completing the inference programme, the sequence above is a sequence of repairs, and each repair leaves something undone.

Maximum entropy assigns a distribution at an instant. It says nothing about evolution, and (2) contains no time. Maximum caliber answers this, replacing distributions over states by distributions over histories.

Maximum caliber therefore delivers dynamics, but (3) is real and positive, and real positive weights do not interfere. What it delivers is stochastic—diffusion and its relatives—and the passage to quantum dynamics still needs $\beta \to i\tau$; that is, it introduces complex amplitudes by fiat. Replacing the entropy by the Fisher information answers this: (7) is quantum, and no rotation was performed to obtain it.

The Fisher route at first appears promising but falls short for two reasons. The first is structural. A divergence vanishes and is stationary where its arguments agree, so (4) has no first-order term and its leading content is the quadratic form; such a principle can supply a metric and nothing else. A metric is a second-order kinetic term, and in (6) that is precisely what it is—the quantum potential, the term of order ${\hslash }^2$—while the kinetic and potential energies are imported with ${\mathcal{H}}_{\mathrm{cl}}$. Second order is moreover the wrong order for fundamental matter. The Dirac Lagrangian $\overline{\psi }i{\gamma }^{\mu }{\partial }_{\mu }\psi$ is linear in the derivative, and it is that linearity which carries the conserved current $\overline{\psi }{\gamma }^{\mu }\psi$ and its positive density; no functional stationary at coincidence produces a term linear in the displacement. Extreme physical information is no counterexample: its extremization returns the Klein–Gordon equation, second order as expected, and Dirac’s equation follows only on factorizing that output—the step Dirac (1928) took in 1928, performed to fix the landed result rather than derived from the principle.

The second reason is methodological. Streater (2007) observes that the bound information J is chosen so that the extremum of $I-J$ is the equation sought, and that as the target theory becomes more structured more of the content sits in J than in the informational term. The breadth of the principle’s reach is bought at the cost of its discriminating power. The result is a program that admits anything at second order, and nothing at first.

1.3. Surprisal

At minimum what is wanted, then, is an informational quantity whose expansion about identical arguments has a first-order term. There is one, and it is the oldest: the surprisal $-ln\rho$ of a single event.

Unlike entropy, surprisal requires no sum, and so needs no ensemble. Where relative maximum entropy delivers the least biased revision given constraints, transition surprisal delivers the least surprising update given a displacement from a reference. The two stand in the same relation to their inputs.

$$\begin{array}{c}\hfill \underset{\mathbf{The}\mathbf{least}\mathbf{biased}\mathbf{revision}}{\underbrace{D\left(\rho +\delta \rho \parallel \rho \right)=\frac{1}{2}\int {\displaystyle \frac{{\left(\delta \rho \right)}^2}{\rho }}+\dots }}\mathrm{vs}.\underset{\mathbf{The}\mathbf{least}\mathbf{surprising}\mathbf{update}}{\underbrace{\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵ\left[\langle \psi \mid D\psi \rangle +{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}^2\right]+\dots }}.\end{array}$$

(8)

The decisive step is to attach surprisal to the normalized complex comparison of a displacement jet and a reference jet. Jets allow the surprisal of both the $\psi \to \psi +iϵD\psi$ and the $0\to \sqrt{ϵ}{\mathbb{D}}^2$ transitions to simultaneously contribute to the same action phase. The result is a complexified transition surprisal. The full construction is given in Definitions 1–3.

Both relative entropy and surprisal measure the informational cost of a change relative to a reference. Relative entropy is real and begins at second order. As for surprisal, the state displacement in the jet comparison has a first-order term, which is imaginary and oriented: it changes sign when the displacement is reversed, as a divergence between probability distributions cannot. Furthermore, its curvature comparison is intrinsically quadratic, but its $\sqrt{ϵ}$ scaling makes it contribute at the same accumulated order, while the signature-dependent complex jet pairing places it in the same action phase. Nothing needs to be imported alongside them: kinetic terms, potentials and curvature terms arise from the transport, reference and invariant pairings of the jets during surprisal accumulation, so there is no analogue of a supplementary source J chosen to reproduce a predetermined equation.

The resulting expression is precisely the global form required of an action. A field history with real action S has the one-parameter transition weight

$${\mathcal{A}}_{ϵ}=e^{iϵS},$$

(9)

where $ϵ$ is the dimensionless deformation parameter of the jets. Its transition surprisal, on the continuous local logarithm branch at $ϵ=0$, is

$$-ln{\mathcal{A}}_{ϵ}=-ln{e}^{iϵS}=-iϵS.$$

(10)

With the carrier pairings understood as global $L^2$ pairings, comparison with (8) therefore identifies

$$\begin{array}{cc}\hfill S& =Re\langle \psi \mid D\psi \rangle +{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}^2\hfill \\ \hfill & ={\int }_X{\mathrm{d}}^{d}x\sqrt{\left|g\right|}\left[Re{\langle \psi \mid D\psi \rangle }_x+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}_x^2\right].\hfill \end{array}$$

(11)

The expression in brackets is the local Lagrangian density,

$$\mathcal{L}=Re{\langle \psi \mid D\psi \rangle }_x+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}_x^2,S={\int }_X{\mathrm{d}}^{d}x\sqrt{\left|g\right|}\mathcal{L}.$$

(12)

The least-surprising update thus has exactly the form of the surprisal of an action amplitude. Accumulation extracts the first-order coefficient of the jet comparison and gives

$$\mathcal{I}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{{\mathcal{K}}_{ϵ}}{ϵ}}=-iS,e^{-\mathcal{I}}=e^{iS}.$$

(13)

Our claim is accordingly that the inferential programme was right but the functionals considered were wrong. Not entropy, relative entropy or Fisher information, but the surprisal of the transition between a displacement jet and a reference jet. In other words, the universe does not evolve according to the least biased revision; it evolves according to the least surprising update.

1.4. The Main Result

Ordinarily, a carrier is chosen to realize a Lagrangian already given: the theory determines which fields, representations, and internal spaces must be supplied. Accumulated surprisal permits the reverse direction. Given a carrier and its transition jets, surprisal accumulation determines the corresponding action. Fundamental theory construction is thereby reduced to transition-jet construction.

This simplifies theory construction sufficiently that a power-counting-renormalizable unified matter–gravity theory can be attempted from first principles. This constitutes our main result.

Before stating the central construction, it is useful to identify what is absent from the least-surprising action of $\psi \left(x\right)$, given the usual displacement and reference jets:

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi +iϵD_x\psi ,\sqrt{ϵ}{\mathbb{D}}_x^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & =\left(\psi ,\sqrt{ϵ}{\mathbb{F}}_{0,x}\right),\hfill \end{array}$$

(14)

where ${\mathbb{F}}_{0,x}$ is the curvature configuration declared unsurprising. For a suitable constant-curvature reference (Equation 104), accumulated surprisal gives the Dirac action coupled to quadratic gravity, as shown in Section 2.4. In this sense, quadratic gravity with Dirac matter is therefore the least-surprising theory that a wavefunction in spacetime furnishes of itself: the jets (14) retain the transitions available to an x-parametrized amplitude, but nothing else. The action contains gravity and Dirac matter, but neither a Standard Model particle carrier nor family multiplicity.

The central construction remedies this absence. The physical theory will instead be defined using the canonical deformation geometry associated with the Dirac–Hestenes amplitude

$$\psi \left(x\right)\in {Cl}^{+}(T_{x}X,g_x).$$

(15)

Matter deforms spacetime. It is therefore natural yet remarkable that closing the Dirac–Hestenes amplitude $\psi \left(x\right)$ under deformations of the geometric structures required for its own definition lands on its canonical particle sector. Namely that the metric and right-Hermitian deformation geometries furnish, respectively, the particle content of one Standard Model generation and its three-family multiplicity. The construction is endogenous: rather than adjoining an independently postulated internal space to the wavefunction, it derives its candidate internal carrier from deformations of structures already required to define the wavefunction itself. The theory is thus entirely self-contained, and its implications, canonical. By contrast, GUT constructions postulate their internal spaces, hence are less natural.

We are thus lead to conclude that, up to regular reductions, the three-generation Standard Model matter coupled to power-counting-renormalizable gravity is the least-surprising theory that a deformable wavefunction in spacetime furnishes of itself.

This section gives a sketch. The result will be shown in full in Section 3.

Left and right defining forms.

At each $x\in X$, the Dirac–Hestenes amplitude admits commuting left spacetime-spin and right current-preserving actions,

$$\psi \left(x\right)⟼S\psi \left(x\right)U,S\in Spin(3,1),U\in U{\left(2\right)}_R.$$

(16)

The left action preserves a Lorentzian form $g_x$ on $T_{x}X$, while the right action preserves a positive Hermitian form $q_x$ on the complex rank-two right module $W_{x,g}$. We allow both $g_x$ and $q_x$ to vary, thereby retaining the corresponding deformations of $\psi \left(x\right)$.

The defining forms $(g_x,q_x)$ are promoted to configuration variables of the extended amplitude $\Psi$. Their local configuration spaces are

$$\begin{array}{cccc}\hfill {Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)& \cong {\displaystyle \frac{{\widetilde{GL}}^{+}(4,\mathbb{R})}{Spin(3,1)}},\hfill & \hfill {Herm}^{+}\left(W_{x,g}\right)& \cong {\displaystyle \frac{GL(2,\mathbb{C})}{U\left(2\right)}}.\hfill \end{array}$$

(17)

These spaces assemble into the configuration bundle ${\pi }_{\mathcal{C}}:{\mathcal{C}}_X⟶X,$ whose local points are $z=(x,g_x,q_x).$ Its configuration fibre has real dimension

$${dim}_{\mathbb{R}}{Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)+{dim}_{\mathbb{R}}{Herm}^{+}\left(W_{x,g}\right)=10+4=14,$$

(18)

and therefore

$${dim}_{\mathbb{R}}{\mathcal{C}}_X=4+10+4=18.$$

(19)

The additional fourteen directions are configuration directions, not additional spatial or temporal dimensions.

Deformation geometries.

At $z=(x,g_x,q_x)\in {\mathcal{C}}_X$, the corresponding vertical tangent spaces are

$$\begin{array}{cc}\hfill V_z^g& :=T_{g_x}{Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)\cong {Sym}^2\left(T_x^{*}X\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill V_z^q& :=T_{q_x}{Herm}^{+}\left(W_{x,g}\right)\cong Herm\left(W_{x,g}\right).\hfill \end{array}$$

(21)

The first consists of infinitesimal Lorentzian-metric deformations, while the second consists of infinitesimal deformations of the right-module Hermitian form.

The metric-deformation space carries the DeWitt form

$$G_{g_x}(h,k)={tr}_{g_x}\left(hk\right)-\lambda {tr}_{g_x}\left(h\right){tr}_{g_x}\left(k\right),h,k\in V_z^g,$$

(22)

while the right-Hermitian deformation space carries the invariant positive form

$$K_{q_x}(a,b)=tr\left(q_x^{-1}a{q}_x^{-1}b\right),a,b\in V_z^q.$$

(23)

Complementary physical roles are assigned to these native deformation geometries. The metric-deformation geometry is represented by its chiral half-spin module and supplies particle identity, whereas the right-Hermitian deformation geometry is represented by its tangent module and supplies multiplicity. The corresponding canonical deformation carrier is

$$\begin{array}{|c|}\hline \begin{array}{ccc}\hfill {\left(\mathcal{E}\right)}_z:=& {Cl}^{+}(T_{x}X,g_x){\otimes }_{\mathbb{R}}{\Sigma }^{\epsilon }\left(V_z^g,G_{g_x}\right){\otimes }_{\mathbb{R}}{V}_z^q,\hfill & \hfill \epsilon \in \{+,-\}.\end{array}\\ \hline\end{array}$$

(24)

The associated deformation amplitude is a section

$$\Psi \in \Gamma \left({\mathcal{C}}_X,\mathcal{E}\right),\Psi =\Psi (x,g_x,q_x).$$

(25)

Thus $\psi \left(x\right)$ and $\Psi (x,g_x,q_x)$ belong to distinct but related carriers: $\psi$ is the original spacetime-algebra amplitude, while $\Psi$ is the amplitude associated with its deformation geometry.

Universal transport and jets.

The generalized covariant transport on the deformation carrier decomposes according to the three types of directions in ${\mathcal{C}}_X$:

$${\mathbb{D}}_{\mathrm{tot}}={\mathbb{D}}_x+{\mathbb{D}}_g+{\mathbb{D}}_q+\Phi .$$

(26)

Here ${\mathbb{D}}_x$ denotes horizontal spacetime transport, ${\mathbb{D}}_g$ vertical transport through Lorentzian-metric configurations, ${\mathbb{D}}_q$ vertical transport through right-Hermitian configurations, and $\Phi$ odd scalar transport. Let $D_{\mathrm{tot}}$ denote the corresponding formally Hermitian first-order operator on the state carrier.

Its curvature decomposes schematically as

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mathrm{tot}}^2& =F_{xx}+F_{xg}+F_{xq}+F_{gg}+F_{gq}+F_{qq}\hfill \\ \hfill & +[{\mathbb{D}}_x,\Phi ]+[{\mathbb{D}}_g,\Phi ]+[{\mathbb{D}}_q,\Phi ]+{\Phi }^2.\hfill \end{array}$$

(27)

The diagonal terms are the curvatures of the three transport sectors, while the off-diagonal terms are their mixed curvatures.

The transition jets of $\Psi$ are the deformation jets of $\psi$:

$$\begin{array}{cc}\hfill {\mathfrak{J}}_{ϵ}\Psi & :=\left(\Psi +iϵD_{\mathrm{tot}}\Psi ,\sqrt{ϵ}{\mathbb{D}}_{\mathrm{tot}}^2\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\mathfrak{J}}_r\Psi & :=\left(\Psi ,\sqrt{ϵ}{\mathbb{F}}_{0,\mathrm{tot}}\right).\hfill \end{array}$$

(29)

By Proposition 1,

$$\begin{array}{c}\hfill \mathcal{K}\left({\mathfrak{J}}_{ϵ}\Psi ,{\mathfrak{J}}_r\Psi \right)=-iϵ\left[Re〈\Psi |D_{\mathrm{tot}}\Psi 〉+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}_{\mathrm{tot}}^2-{\mathbb{F}}_{0,\mathrm{tot}}∥}^2\right]+O\left({ϵ}^2\right).\end{array}$$

(30)

Accumulated surprisal of these jets determines the action of $\Psi$, thereby furnishing the deformation action associated with the original amplitude $\psi$.

Physical pullback and reductions.

A physical realization is a section

$$s=(g,q):X⟶{\mathcal{C}}_X,s\left(x\right)=\left(x,g\left(x\right),q\left(x\right)\right).$$

(31)

The physical matter field and jet data are obtained by pullback along s. Positivity, complex structure, and unimodularity are then imposed on the pulled-back carrier and transport.

As shown in Section 3.3, the metric-deformation space has signature $(6,4)$ and ambient spin-frame symmetry $Spin(6,4)$. Positive polarization $R_x$, compatible complex structure $J_x$, and unimodularity give

$$\begin{array}{|c|}\hline Spin(6,4)\stackrel{R_x}{\to }Spin\left(6\right)\times Spin\left(4\right)\stackrel{J_x}{\to }U\left(3\right)\times U\left(2\right)\stackrel{\mathrm{unimodularity}}{\to }{G}_{\mathrm{SM}}.\\ \hline\end{array}$$

(32)

Its selected half-spin module supplies the particle content of one Standard Model generation.

The right-Hermitian deformation pairing $K_q$ is already positive, while complex linearity of the complete matter carrier is inherited from the complex particle factor. The remaining nontrivial condition is right-sector unimodularity, $det{q}_x=1.$ Its infinitesimal form is $tr\left(q_x^{-1}\delta q_x\right)=0,$ which removes the trace deformation and leaves

$$V_z^{q,1}:=T_{q_x}{Herm}_1^{+}\left(W_{x,g}\right)\cong {Herm}_{0,q_x}\left(W_{x,g}\right),{dim}_{\mathbb{R}}{V}_z^{q,1}=3.$$

(33)

If $E_{\mathrm{part}}$ denotes the resulting complex one-generation particle carrier, then

$$\begin{array}{|c|}\hline E_{\mathrm{part}}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\cong E_{\mathrm{part}}{\otimes }_{\mathbb{C}}\left(\mathbb{C}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\right)\cong E_{\mathrm{part}}{\otimes }_{\mathbb{C}}{\mathbb{C}}^3.\\ \hline\end{array}$$

(34)

The right-Hermitian deformation sector therefore supplies three gauge-identical family components. Scalar transport on this multiplicity factor admits general complex matrices

$$Y_u,Y_d,Y_e,Y_{\nu },M_R\in {Mat}_3\left(\mathbb{C}\right),M_R=M_R^T.$$

(35)

The reduced action.

Accumulated surprisal of the pulled-back and reduced universal deformation jets gives

$$\begin{array}{cc}\hfill S_{\mathrm{SM}+\mathrm{QG}}^{\left(3\right)}& ={\int }_X{\mathrm{d}}^{4}x\sqrt{-g}[a{C}_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+b{R}^2+c{E}_4+{\displaystyle \frac{M_{\mathrm{P}}^2}{2}}R-\Lambda \hfill \\ \hfill & -{\displaystyle \frac{1}{4g_3^2}}{G}_{\mu \nu }^A{G}^{A\mu \nu }-{\displaystyle \frac{1}{4g_2^2}}{W}_{\mu \nu }^I{W}^{I\mu \nu }-{\displaystyle \frac{1}{4g_1^2}}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ \hfill & +{\displaystyle \sum _{n=1}^3}\sum _f{\overline{\psi }}_{f,n}i{\gamma }^{\mu }{D}_{\mu }^{\mathrm{SM}}{\psi }_{f,n}-{\left(D_{\mu }^{\mathrm{SM}}H\right)}^{†}{D}^{\mu ,\mathrm{SM}}H-\xi R{H}^{†}H\hfill \\ \hfill & -m_H^2{H}^{†}H-{\lambda }_H{\left(H^{†}H\right)}^2\hfill \\ \hfill & -\left({\overline{Q}}_L{Y}_u\tilde{H}{u}_R+{\overline{Q}}_L{Y}_{d}H{d}_R+{\overline{L}}_L{Y}_{e}H{e}_R+{\overline{L}}_L{Y}_{\nu }\tilde{H}{\nu }_R+\mathrm{h}.\mathrm{c}.\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left(\overline{{\nu }_R^c}{M}_R{\nu }_R+\mathrm{h}.\mathrm{c}.\right)],\hfill \end{array}$$

(36)

where $\tilde{H}=i{\sigma }_2{H}^{*}$. This is the three-generation Standard Model coupled to local, power-counting-renormalizable quadratic gravity, here derived canonically from the deformation space of $\psi \left(x\right)$ as its least-surprising theory, under usual group reductions.

This subsection states the destination of the construction. The configuration bundle, deformation carrier and transport, physical pullback and reductions, and corresponding operator sectors are developed in Section 3.

1.5. An Observer Internal to the Universe

Accumulated surprisal produces an action quadratic in the curvature: the curvature slot of the jet comparison contributes through the square ${\parallel {\mathbb{D}}^2-{\mathbb{F}}_0\parallel }^2$ (8). For a gauge connection this is precisely the Yang–Mills term, and its quantization is unproblematic. However, the spin connection enjoys no such protection: its curvature ${\mathbb{D}}^2=R_g$ already contains two derivatives of the metric, so its square is a four-derivative action—quadratic gravity, containing $C_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }$ and $R^2$ alongside the Einstein–Hilbert term R and the cosmological term $\Lambda$. Since the procedure cannot avoid this outcome, the standing objections to quadratic gravity must be addressed lest the procedure must be discarded. There are two. The dynamical objection—stability—is taken up in Section 1.6. The kinematical objection, addressed here, is that retaining the massive opposite-norm spin-two mode (Stelle 1977) places the state space in a Krein space rather than a Hilbert space: the kinematical carrier of (36) is the Hilbert–Krein space

$$K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}},{\eta }_{universe}={\mathbb{1}}_{\mathrm{SM}}^{\left(3\right)}\otimes {\eta }_{\mathrm{QG}}.$$

(37)

A Krein space K carries a positive Hilbert topology $(·,·)$ together with a self-adjoint involution

$${\eta }^{†}=\eta ,{\eta }^2=\mathbb{1},$$

(38)

which defines the conserved indefinite form

$$[\varphi ,\psi ]:=(\varphi ,\eta \psi ).$$

(39)

The decomposition

$$K=K_{+}\oplus K_{-}{,\eta |}_{K_{\pm }}=\pm \mathbb{1},$$

(40)

admits nonzero states of positive, negative, or vanishing signed norm. Consequently, the conserved form cannot provide a universal positive normalization

$$[\psi ,\psi ]=1$$

(41)

for all physical states. If quantum measurement is assumed to require such a positively normalized state, the opposite-norm sector appears to have no measurement interpretation. This is the conventional negative-norm objection.

The objection mistakes the cure for the disease.

First, why it’s not a disease.

The objection conflates physical measurement with representation by a positively normalized probability measure. However, a detector need only return nonnegative, countably additive intensities, thus satisfying two of the three Kolmogorov axioms. Let

$$P_{\pm }={\displaystyle \frac{1}{2}}(\mathbb{1}\pm \eta ),A_{\pm }=\eta P_{\pm }.$$

(42)

Both channels are intensity-positive, since

$$I_{\pm }\left(\psi \right):=[\psi ,A_{\pm }\psi ]=(\psi ,P_{\pm }\psi )\ge 0.$$

(43)

Their sum gives the total positive intensity,

$$I_{+}\left(\psi \right)+I_{-}\left(\psi \right)=(\psi ,\psi ),$$

(44)

whereas their difference reconstructs the conserved signed Krein charge,

$$I_{+}\left(\psi \right)-I_{-}\left(\psi \right)=[\psi ,\psi ].$$

(45)

A negative conserved value is therefore not a negative detector event: it is a difference between two nonnegative detector records.

Intensities are standard in quantum optics, hence the present structure is already familiar. In balanced photodetection, the two photodetectors return nonnegative photon-count intensities, while their difference defines a signed photocurrent from which field quadratures are reconstructed. A negative quadrature or difference current is not a negative photon count; it records which positive channel has the greater intensity. Likewise, a signed Krein observable is reconstructed from differences of nonnegative intensities rather than measured as a negative event.

More generally, for a Krein-self-adjoint channel A,

$$I_A\left(\psi \right):=[\psi ,A\psi ]=(\psi ,\eta A\psi )$$

(46)

is nonnegative precisely when $\eta A\ge 0$. The order isomorphism

$$A⟼E:=\eta A$$

(47)

therefore identifies intensity-positive Krein channels with ordinary positive Hilbert-space effects. Since the intensity cone generates the full observable space, differences of nonnegative records recover signed observables and provide complete tomography. Section 4 develops this construction in full.

Second, why it is a cure.

An observer without a conserved global normalizer cannot prove that a quantum system has been exhaustively measured. We qualify such an observer as internal. By contrast, an external observer can. Internal observers are naturally suited to quantum cosmology because they resolve its longest standing philosophical objection: the need for an observer external to the universe. Intensities are precisely the records available to an internal observer. Such an observer accesses a finite, nonexhaustive collection of local channels, not every possible outcome of the system as a whole. The carrier

$$K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}}$$

(48)

positions the observer externally to the particle physics sector (i.e., ${\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}$ is a Hilbert space) while internally to the universe (i.e., $K_{QG}$ and by extension $K_{universe}$ are Krein spaces).

The Krein structure is therefore not a defect of the quantum theory. Krein spaces are to internal observers, what Hilbert spaces are to external observers.

1.6. The Stability Objection

The second objection to quadratic gravity concerns its dynamics rather than its kinematics. The massive spin-two mode couples universally to the stress tensor, and interactions generically displace its pole into a complex-conjugate pair $E,\overline{E}$: the positive sector leaks into the opposite-norm sector.

In the received view this is fatal, because measurement was held to require a positive-definite carrier: a populated opposite-norm sector admits no Born rule, and with it the theory was thought to lose its probability interpretation altogether. The response of the literature has accordingly been to seek a unitary sector: a positive-norm subspace of real spectrum, containing ordinary matter and dynamically preserved by the interactions, whose opposite-norm complement is never populated from matter within it and can be excised in a Lorentz-invariant manner—completed by the assumption that the initial conditions of the universe lie within the retained sector. Each clause of this program remains unproven, its treatments of the complex-pair sector remain in explicit disagreement (Donoghue and Menezes 2019,Kubo and Kugo 2024), and the program as a whole is now in its sixth decade (Cutkosky et al. 1969,Lee and Wick 1969).

The preceding subsection removes the premise. Intensities furnish a measurement theory on the full Krein carrier directly: no protected subspace is required for detector records to be nonnegative, additive, and tomographically complete. A leak is therefore not an interpretational catastrophe but a physical process—a transfer between signed intensity channels. We accept leaks.

What the framework does require is weaker but not empty. Detector records are intensities, so predicted records must remain finite. This is nontrivial precisely on the complex-pair states: their eigenvectors are Krein-null, $[\psi ,\psi ]=0$, the signed charge $[{\psi }_t,{\psi }_t]$ is exactly conserved, and yet the total intensity

$$I_{+}\left({\psi }_t\right)+I_{-}\left({\psi }_t\right)=({\psi }_t,{\psi }_t)$$

(49)

may grow without bound under Krein-unitary evolution. The stability requirement is thus intensity-boundedness of the full Krein-unitary dynamics, in place of reduction to a no-leak unitary sector.

The evidence for boundedness is as follows. Classically, quadratic gravity exhibits no runaways below computable energy thresholds (Salvio 2019). In the quantum theory, ghost pairs are produced from ordinary matter only above a definite threshold, and the populated sector is anti-unstable: conjugate pairs carry real total energy and persist without amplification (Kubo and Kugo 2024). We adopt intensity-boundedness as a conjecture supported where the dynamics has been solved exactly and denied nowhere; its residual conditions—the extension of anti-instability to the multi-ghost sector and the frame consistency of the complex-pair kinematics—remain open questions of the operator formalism (Kubo and Kugo 2024). The stability objection is thus not an axiomatic violation but an unfinished boundedness question.

1.7. Conventions and Definitions

Throughout, $\hslash =1$. Unless stated otherwise, the signature is $(-,+,+,+)$, and the state pairing $\langle ·\mid ·\rangle$ is antilinear in its first argument.

To construct a Lagrangian from surprisal, we require a local complex comparison between a displaced amplitude and a reference. The parametrization of the amplitude supplies two forms of local data: its first-order transport and the curvature of that transport. We collect these data into jets and define transition surprisal directly from their normalized complex comparison.

Two operators built from the same connection must therefore be distinguished. The covariant exterior derivative $\mathbb{D}$ has curvature ${\mathbb{D}}^2$, while its contraction D is a first-order operator on the carrier, typically Clifford. They are not independent: D and ${\mathbb{D}}^2$ are respectively the contracted transport and curvature supplied by the same connection.

Definition 1

(Jets). Let $\psi \in \mathcal{H}$, let $\mathbb{D}$ be a covariant transport with curvature ${\mathbb{D}}^2$ and contraction D, let $ϵ>0$, and let ${\mathbb{F}}_0$ be a curvature-valued reference. The displacement and referenc jets of ψ are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & :=\left(\psi +iϵD\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & :=\left(\psi ,\sqrt{ϵ}{\mathbb{F}}_0\right),\hfill \end{array}$$

(50)

the first slot of the displacement jet being the infinitesimal unitary transport generated by D, and the reference jet supplying the state against which the displaced jet is compared.

Definition 2

(Complex jet pairing). For $U=(u_1,u_2)$ and $V=(v_1,v_2)$ in the jet carrier ${\mathcal{H}}_{\mathfrak{J}}=\mathcal{H}\oplus {\mathcal{H}}_F$, define the complex jet pairing

$${\langle U\mid V\rangle }_{\mathfrak{J}}:=w_1\langle u_1\mid v_1\rangle +w_2\langle u_2\mid v_2\rangle ,{\displaystyle \frac{w_2}{w_1}}=-i{s}_F,s_F=\left\{\begin{array}{cc}-1\hfill & \mathrm{Lorentzian},\hfill \\ +1\hfill & \mathrm{Euclidean},\hfill \end{array}\right.$$

(51)

the slots being mutually orthogonal. Because their relative weight is imaginary, the resulting jet pairing is generally a complex comparison form rather than a Hermitian inner product. The form of the complex jet pairing places the surprisal of both arguments of the jets into the same action phase.

Definition 3

(Transition surprisal). Let $U,V\in {\mathcal{H}}_{\mathfrak{J}}$ be jets for which the following comparison is nonzero and continuously connected to coincidence. The transition surprisa from V to U is the negative local logarithm of their normalized complex jet comparison,

$$\mathcal{K}(U,V):=-ln{\displaystyle \frac{{\langle V\mid U\rangle }_{\mathfrak{J}}}{\sqrt{{\langle V\mid V\rangle }_{\mathfrak{J}}{\langle U\mid U\rangle }_{\mathfrak{J}}}}},$$

(52)

where the square root and logarithm are chosen by continuous local continuation from coincidence, with $\mathcal{K}(V,V)=0$.

Proposition 1

(Expansion of the jet surprisal). Let the jets be those of Definition 1, with normalized amplitude $\langle \psi \mid \psi \rangle =1$, and let the relative curvature-slot weight be

$$\sigma =-i{s}_F,s_F\in \{+1,-1\}.$$

(53)

If D is formally Hermitian and the curvature pairing is real and symmetric, then

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵ\left[\langle \psi \mid D\psi \rangle +{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}^2\right]+O\left({ϵ}^2\right),$$

(54)

and the coefficient in brackets is real.

Proof. 

Write

$$\begin{array}{cccc}\hfill a& :=\langle \psi \mid D\psi \rangle \in \mathbb{R},\hfill & \hfill b& :={\parallel D\psi \parallel }^2,\hfill \end{array}$$

(55)

$$\begin{array}{cccccc}\hfill c& :=〈{\mathbb{F}}_0|{\mathbb{D}}^2〉\in \mathbb{R},\hfill & \hfill d& :={\parallel {\mathbb{D}}^2\parallel }^2,\hfill & \hfill f& :={\parallel {\mathbb{F}}_0\parallel }^2.\hfill \end{array}$$

(56)

The three pairings entering the normalized jet comparison are computed separately. First, the cross pairing is

$$\begin{array}{cc}\hfill {〈{\mathfrak{J}}_r\psi |{\mathfrak{J}}_{ϵ}\psi 〉}_{\mathfrak{J}}& =w_1\left[\langle \psi \mid \psi +iϵD\psi \rangle +\sigma ϵ〈{\mathbb{F}}_0|{\mathbb{D}}^2〉\right]\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill & =w_1\left[1+iϵa+\sigma ϵc\right].\hfill \end{array}$$

(58)

The reference self-pairing is

$$\begin{array}{cc}\hfill {〈{\mathfrak{J}}_r\psi |{\mathfrak{J}}_r\psi 〉}_{\mathfrak{J}}& =w_1\left[\langle \psi \mid \psi \rangle +\sigma ϵ{\parallel {\mathbb{F}}_0\parallel }^2\right]\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & =w_1\left[1+\sigma ϵf\right].\hfill \end{array}$$

(60)

Finally, the displaced self-pairing is

$$\begin{array}{cc}\hfill {〈{\mathfrak{J}}_{ϵ}\psi |{\mathfrak{J}}_{ϵ}\psi 〉}_{\mathfrak{J}}& =w_1\left[\langle \psi +iϵD\psi \mid \psi +iϵD\psi \rangle +\sigma ϵ{\parallel {\mathbb{D}}^2\parallel }^2\right]\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & =w_1\left[1+iϵ\langle \psi \mid D\psi \rangle -iϵ\langle D\psi \mid \psi \rangle +{ϵ}^2{\parallel D\psi \parallel }^2+\sigma ϵd\right]\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & =w_1\left[1+{ϵ}^{2}b+\sigma ϵd\right],\hfill \end{array}$$

(63)

where the two terms linear in $ϵ$ cancel because D is formally Hermitian and hence

$$\langle D\psi \mid \psi \rangle =\langle \psi \mid D\psi \rangle =a.$$

(64)

By Definition 3, the surprisal of the normalized jet comparison is

$$\begin{array}{cc}\hfill \mathcal{K}& =-ln{〈{\mathfrak{J}}_r\psi |{\mathfrak{J}}_{ϵ}\psi 〉}_{\mathfrak{J}}\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{2}}ln{〈{\mathfrak{J}}_r\psi |{\mathfrak{J}}_r\psi 〉}_{\mathfrak{J}}+{\displaystyle \frac{1}{2}}ln{〈{\mathfrak{J}}_{ϵ}\psi |{\mathfrak{J}}_{ϵ}\psi 〉}_{\mathfrak{J}}.\hfill \end{array}$$

(66)

Substituting (58)–(63), the constant normalization cancels:

$$-ln{w}_1+{\displaystyle \frac{1}{2}}ln{w}_1+{\displaystyle \frac{1}{2}}ln{w}_1=0.$$

(67)

Using

$$ln(1+z)=z-{\displaystyle \frac{1}{2}}{z}^2+O\left(z^3\right),$$

(68)

the three remaining logarithms give

$$\begin{array}{cc}\hfill -ln\left(1+iϵa+\sigma ϵc\right)& =-iϵa-\sigma ϵc+{\displaystyle \frac{{ϵ}^2}{2}}{(ia+\sigma c)}^2+O\left({ϵ}^3\right),\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}ln\left(1+\sigma ϵf\right)& ={\displaystyle \frac{\sigma ϵ}{2}}f-{\displaystyle \frac{{\sigma }^2{ϵ}^2}{4}}{f}^2+O\left({ϵ}^3\right),\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}ln\left(1+\sigma ϵd+{ϵ}^{2}b\right)& ={\displaystyle \frac{\sigma ϵ}{2}}d+{\displaystyle \frac{{ϵ}^2}{2}}b-{\displaystyle \frac{{\sigma }^2{ϵ}^2}{4}}{d}^2+O\left({ϵ}^3\right).\hfill \end{array}$$

(71)

Collecting the first-order terms yields

$$\begin{array}{cc}\hfill \mathcal{K}& =-iϵa+\sigma ϵ\left({\displaystyle \frac{1}{2}}d-c+{\displaystyle \frac{1}{2}}f\right)+O\left({ϵ}^2\right)\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill & =-iϵa+{\displaystyle \frac{\sigma ϵ}{2}}\left(d+f-2c\right)+O\left({ϵ}^2\right).\hfill \end{array}$$

(73)

Since the curvature pairing is real and symmetric,

$$d+f-2c={∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}^2,$$

(74)

and therefore

$$\mathcal{K}=-iϵa+{\displaystyle \frac{\sigma ϵ}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}^2+O\left({ϵ}^2\right).$$

(75)

Substituting $\sigma =-i{s}_F$ and restoring $a=\langle \psi \mid D\psi \rangle$ gives (54).

Both terms in brackets are real. The displacement term is necessarily proportional to $-i$, so the curvature contribution shares the same phase precisely when its relative weight is imaginary. Its real magnitude is absorbed into the invariant curvature pairing, while the signature convention gives $\sigma =-i{s}_F$ in the normalization used here. □

Definition 4

(Surprisal accumulation). Using the global $L^2$ pairings of the carrier, define the accumulated surprisal by

$$\mathcal{I}:=\underset{ϵ\to 0}{lim}{\displaystyle \frac{1}{ϵ}}\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iS,$$

(76)

where

$$S={\int }_X{\mathrm{d}}^{d}x\sqrt{\left|g\right|}\left[Re{\langle \psi \mid D\psi \rangle }_x+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}_x^2\right].$$

(77)

The parameter ϵ is a dimensionless deformation parameter: division by ϵ extracts the first variation of the global jet comparison at coincidence.

Remark 1

(The exponentiated accumulated surprisal). Because the carrier pairing is the global $L^2$ pairing, the first-order coefficient of the jet comparison is already the complete action:

$${\mathcal{K}}_{ϵ}=-iϵS+O\left({ϵ}^2\right).$$

(78)

Consequently,

$$\mathcal{I}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{{\mathcal{K}}_{ϵ}}{ϵ}}=-iS,e^{-\mathcal{I}}=e^{iS}.$$

(79)

Thus exponentiating the accumulated surprisal gives the usual Lorentzian history weight. Construction of the full propagator then proceeds, as usual, by integrating $e^{iS}$ over field histories with the appropriate measure, gauge fixing, and boundary conditions.

2. Calibration

A procedure that attempts to output a unified matter-gravity theory from first principles must first be shown to output the laws already known. The five subsections that follow are therefore not illustrations but calibration: in each, the same kernel is applied to the same pair of jets, only the reference changing, and the output is compared with a Lagrangian established independently by other means. The success of this calibration retrodictively justifies definitions 1-4.

Because surprisal carries no absolute value, a theory built on it is a declaration of what counts as un surprising, and the declaration is made by the reference. The Dirac equation declares a non-displaced state, vacuum declares flatness; the instanton declares self-duality; gravity declares constant curvature; the Higgs declares a nonvanishing vacuum value. And of what it counts as surprising: namely, an infinitesimal displacement and a curvature. Fixing these fixes a theory, in the strong sense that its Lagrangian is outputted rather than posited.

2.1. Dirac

Let X be flat, let $\psi \in \mathcal{H}$ be an amplitude, let $D=i{\gamma }^{\mu }{\partial }_{\mu }-m$ be the distinguished transport operator on $\mathcal{H}$, formally Hermitian, and let $ϵ>0$. Since X is flat and no gauge field is present, ${\mathbb{D}}^2=0$, and the curvature slot is empty. The relevant jets are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi +iϵD\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & =\left(\psi ,0\right).\hfill \end{array}$$

(80)

By Proposition 1, with ${\mathbb{D}}^2={\mathbb{F}}_0=0$,

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵRe\langle \psi \mid D\psi \rangle +O\left({ϵ}^2\right),$$

(81)

so that, on accumulating we find

$$\begin{array}{cccc}\hfill \mathcal{I}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)& =-iS,\hfill & \hfill S& =Re\langle \psi \mid D\psi \rangle =\langle \psi \mid D\psi \rangle \hfill \end{array}$$

(82)

$$\begin{array}{cccc}\hfill & \hfill & \hfill & ={\int }_X{\mathrm{d}}^{4}x\sqrt{\left|g\right|}\overline{\psi }\left(i{\gamma }^{\mu }{\partial }_{\mu }-m\right)\psi ,\hfill \end{array}$$

(83)

the Dirac action.

2.2. Dirac–Yang–Mills

Let X be flat, let $\psi \in \mathcal{H}$ be an amplitude, and let

$$\mathbb{D}=d+A,{\mathbb{D}}_{\mu }={\partial }_{\mu }-ig{A}_{\mu },{\mathbb{D}}^2=F=dA+A\wedge A$$

(84)

be a covariant exterior derivative with Hermitian generators normalized by $tr\left(T^a{T}^b\right)={\delta }^{ab}$. Let $D=i{\gamma }^{\mu }{\mathbb{D}}_{\mu }-m$ be its Clifford contraction and the distinguished transport operator on $\mathcal{H}$. Unlike the previous example, the curvature slot is now occupied, ${\mathbb{D}}^2=F\ne 0$, while the reference remains flat. The relevant jets are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi +iϵD\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & =\left(\psi ,0\right).\hfill \end{array}$$

(85)

By Proposition 1, with ${\mathbb{F}}_0=0$, $s_F=-1$, and the two-form norm

$${\langle F\mid F\rangle }_x={\displaystyle \frac{1}{2}}tr\left(F_{\mu \nu }{F}^{\mu \nu }\right),$$

(86)

one has

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵ\left[Re\langle \psi \mid D\psi \rangle -{\displaystyle \frac{1}{4}}tr\left(F_{\mu \nu }{F}^{\mu \nu }\right)\right]+O\left({ϵ}^2\right).$$

(87)

Both terms are of order $ϵ$, which is the purpose of the $\sqrt{ϵ}$ weighting: a single accumulation serves both slots. Accumulating yields

$$S={\int }_X{\mathrm{d}}^{4}x\sqrt{\left|g\right|}\left[\overline{\psi }\left(i{\gamma }^{\mu }{\mathbb{D}}_{\mu }-m\right)\psi -{\displaystyle \frac{1}{4}}tr\left(F_{\mu \nu }{F}^{\mu \nu }\right)\right],$$

(88)

the Dirac–Yang–Mills action.

2.3. Instantons

Nothing requires the curvature slot to be compared against flatness. Let X be a Euclidean four-manifold, so that ${\star }^2=+1$ on two-forms and consequently $\parallel \star F\parallel =\parallel F\parallel$. Let the curvature slot carry the Euclidean weight $s_F=+1$ rather than the Lorentzian $-1$, so that the accumulated surprisal carries the Euclidean action. Let $\mathbb{D}=d+A$ with ${\mathbb{D}}^2=F$ as before. Comparing the curvature against its own Hodge dual, the relevant jets are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_{\star }\psi & =\left(\psi ,\sqrt{ϵ}\star {\mathbb{D}}^2\right).\hfill \end{array}$$

(89)

The displacement slot is common to both jets and therefore inert; this calibration concerns the geometry alone. By Proposition 1, with $D=0$ and ${\mathbb{F}}_0=\star F$,

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_{\star }\psi \right)=-iϵ{\displaystyle \frac{1}{2}}{∥F-\star F∥}^2+O\left({ϵ}^2\right),$$

(90)

and, on accumulating,

$$\begin{array}{cc}\hfill S_{\star }& ={\displaystyle \frac{1}{2}}{∥F-\star F∥}^2={\parallel F\parallel }^2-〈\star F|F〉\ge 0\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill & =2S_{\mathrm{YM}}-〈\star F|F〉,S_{\mathrm{YM}}:={\displaystyle \frac{1}{2}}{\parallel F\parallel }^2={\displaystyle \frac{1}{4}}{\int }_X{\mathrm{d}}^{4}xtr\left(F_{\mu \nu }{F}^{\mu \nu }\right).\hfill \end{array}$$

(92)

Here $S_{\mathrm{YM}}$ is the action accumulated by the same curvature against the flat reference of the previous example. Nonnegativity is unconditional because the setting is Euclidean and the slot arguments are Lie-algebra-valued forms with a positive Hilbert–Schmidt norm. The cross term is topological: with the coupling absorbed into the normalization of the pairing,

$${\displaystyle \frac{1}{2}}〈\star F|F〉={\displaystyle \frac{8{\pi }^{2}k}{g^2}},k\in \mathbb{Z},$$

(93)

where k is the instanton number. Nonnegativity of the surprisal is therefore the Bogomolny bound,

$$S_{\mathrm{YM}}\ge {\displaystyle \frac{8{\pi }^{2}k}{g^2}},S_{\star }=0\iff F=\star F.$$

(94)

Replacing the reference by $-\star {\mathbb{D}}^2$ reverses the orientation and gives the anti-self-dual branch,

$$S_{\mathrm{YM}}\ge -{\displaystyle \frac{8{\pi }^{2}k}{g^2}},F=-\star F$$

(95)

at equality; together the two references yield

$$S_{\mathrm{YM}}\ge {\displaystyle \frac{8{\pi }^2\left|k\right|}{g^2}}.$$

(96)

Two features are worth mentioning. First, the topological charge is the irreducible surprisal of a sector: no field in the class k can be less surprising than $8{\pi }^2\left|k\right|/g^2$ relative to flatness, and the instanton saturates the bound. Second, the same curvature compared against two different references returns two different theories: against 0 the Yang–Mills action, against $\star F$ the Bogomolny functional—therefore illustrating quite clearly that a theory is, in part, a declaration of what counts as unsurprising, and the declaration is made by the reference.

2.4. Dirac–Einstein–Stelle

The first two examples compared curvature against flatness. Like the third example, here the reference is nonzero, and this supplies the dimensionful content of gravity: one quadratic comparison contributes at three orders in the reference scale at once and returns the Lagrangian of Stelle (Salvio 2018,Stelle 1977)—quadratic gravity with an Einstein–Hilbert term, a cosmological term and Dirac matter.

Two load-bearing features are visible in advance. First, the first-order coefficient is quadratic in the curvature slot, and curvature has mass dimension two, so a comparison of curvatures produces operators of mass dimension four—which in four dimensions is precisely the power-counting renormalizable ceiling. The procedure cannot emit a dimension-six operator, and it cannot emit the Einstein–Hilbert term alone: unrenormalizable gravity is outside its range. Second, a nonvanishing reference reaches below dimension four, to the dimension-two and dimension-zero terms $M_P^{2}R$ and $\Lambda$, and this is where the dimensionful content of physics resides.

Carrier and transport.

Let $\psi \in \mathcal{H}$ be an amplitude and let $\omega$ be the Levi-Civita spin connection of g, so that the construction is second-order and torsion-free. Set

$$\mathbb{D}=d+\omega ,{\mathbb{D}}^2=:\mathbb{F},{{\mathbb{F}}_{\mu \nu }}^{ab}={R_{\mu \nu }}^{ab},$$

(97)

so that the curvature slot carries the Riemann curvature two-form. Let

$$D=i{\gamma }^{\mu }{\nabla }_{\mu }-m$$

(98)

be the Clifford contraction of the same connection.

The curvature-slot pairing.

The curvature slot takes values in the space of algebraic curvature tensors, which in four dimensions is twenty-dimensional and splits into three irreducible sectors: Weyl $\left(10\right)$, traceless Ricci $\left(9\right)$ and scalar $\left(1\right)$. For a curvature two-form $\mathbb{K}$, write $C\left[\mathbb{K}\right]$, $S\left[\mathbb{K}\right]$ and $R\left[\mathbb{K}\right]$ for its Weyl, traceless-Ricci and scalar parts. The general invariant real pairing carries one weight per sector:

$${∥\mathbb{K}∥}^2:={\displaystyle \frac{1}{2}}\left[{\kappa }_{C}C\left[\mathbb{K}\right]·C\left[\mathbb{K}\right]+2{\kappa }_{S}S\left[\mathbb{K}\right]·S\left[\mathbb{K}\right]+{\displaystyle \frac{{\kappa }_R}{6}}R{\left[\mathbb{K}\right]}^2\right].$$

(99)

For the Riemann two-form itself,

$$S_{\mu \nu }:=R_{\mu \nu }-{\displaystyle \frac{1}{4}}R{g}_{\mu \nu },$$

(100)

and the identities used below are

$$\begin{array}{cc}\hfill R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }& =C_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+2S_{\mu \nu }{S}^{\mu \nu }+{\displaystyle \frac{1}{6}}{R}^2,\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill E_4& =C_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }-2S_{\mu \nu }{S}^{\mu \nu }+{\displaystyle \frac{1}{6}}{R}^2,\hfill \end{array}$$

(102)

where $E_4$ is the Euler density. The choice

$$({\kappa }_C,{\kappa }_S,{\kappa }_R)=(3,-2,1)$$

(103)

is carried through this calibration for concreteness.

The reference.

The metric supplies a canonical nonzero element of the same slot: the constant-curvature two-form of radius ℓ,

$${\mathbb{F}}_0:={\displaystyle \frac{1}{2{\ell }^2}}{g}_x\overline{)\wedge }{g}_x,{{\mathbb{F}}_{0\mu \nu }}^{ab}={\displaystyle \frac{1}{{\ell }^2}}\left(e_{\mu }^a{e}_{\nu }^b-e_{\nu }^a{e}_{\mu }^b\right),$$

(104)

with

$${\left(h\overline{)\wedge }k\right)}_{\mu \nu \rho \sigma }=h_{\mu \rho }{k}_{\nu \sigma }+h_{\nu \sigma }{k}_{\mu \rho }-h_{\mu \sigma }{k}_{\nu \rho }-h_{\nu \rho }{k}_{\mu \sigma }.$$

(105)

Its sector decomposition is

$$C\left[{\mathbb{F}}_0\right]=0,S\left[{\mathbb{F}}_0\right]=0,R\left[{\mathbb{F}}_0\right]=R_0:={\displaystyle \frac{12}{{\ell }^2}},$$

(106)

and therefore

$${∥{\mathbb{F}}_0∥}^2={\displaystyle \frac{{\kappa }_R{R}_0^2}{12}},〈{\mathbb{F}}_0|\mathbb{F}〉={\displaystyle \frac{{\kappa }_R{R}_{0}R}{12}}.$$

(107)

Jets and surprisal.

The jets are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi +iϵD\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & =\left(\psi ,\sqrt{ϵ}{\mathbb{F}}_0\right).\hfill \end{array}$$

(108)

By Proposition 1, with $s_F=-1$,

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵ\left[Re\langle \psi \mid D\psi \rangle -{\displaystyle \frac{1}{2}}{∥\mathbb{F}-{\mathbb{F}}_0∥}^2\right]+O\left({ϵ}^2\right).$$

(109)

Expanding the square,

$${\displaystyle \frac{1}{2}}{∥\mathbb{F}-{\mathbb{F}}_0∥}^2=\underset{\mathrm{quadratic}}{\underbrace{{\displaystyle \frac{1}{2}}{\parallel \mathbb{F}\parallel }^2}}-\underset{\mathrm{linear}}{\underbrace{〈{\mathbb{F}}_0|\mathbb{F}〉}}+\underset{\mathrm{constant}}{\underbrace{{\displaystyle \frac{1}{2}}{\parallel {\mathbb{F}}_0\parallel }^2}}.$$

(110)

The comparison thus contributes at three orders in the reference: quadratic curvature, the Einstein–Hilbert term and the cosmological term. Substitution of (99) and (107) gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{∥\mathbb{F}-{\mathbb{F}}_0∥}^2& ={\displaystyle \frac{1}{4}}\left[{\kappa }_C{C}_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+2{\kappa }_S{S}_{\mu \nu }{S}^{\mu \nu }+{\displaystyle \frac{{\kappa }_R}{6}}{R}^2\right]\hfill \\ \hfill & -{\displaystyle \frac{{\kappa }_R{R}_{0}R}{12}}+{\displaystyle \frac{{\kappa }_R{R}_0^2}{24}}.\hfill \end{array}$$

(111)

Eliminating $S_{\mu \nu }{S}^{\mu \nu }$ with (101) and (102),

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{∥\mathbb{F}-{\mathbb{F}}_0∥}^2& ={\displaystyle \frac{1}{4}}\left[({\kappa }_C+{\kappa }_S)C^2+{\displaystyle \frac{{\kappa }_S+{\kappa }_R}{6}}{R}^2-{\kappa }_S{E}_4\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{{\kappa }_R{R}_0}{3}}R+{\displaystyle \frac{{\kappa }_R{R}_0^2}{6}}\right].\hfill \end{array}$$

(112)

Accumulation.

Accumulating yields

$$\begin{array}{c}\hfill S={\int }_X{\mathrm{d}}^{4}x\sqrt{\left|g\right|}\left[\overline{\psi }\left(i{\gamma }^{\mu }{\nabla }_{\mu }-m\right)\psi +a{C}_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+b{R}^2+c{E}_4+{\displaystyle \frac{M_{\mathrm{P}}^2}{2}}R-\Lambda \right],\end{array}$$

(113)

with

$$\begin{array}{cccccccccc}\hfill a& =-{\displaystyle \frac{{\kappa }_C+{\kappa }_S}{4}},\hfill & \hfill b& =-{\displaystyle \frac{{\kappa }_S+{\kappa }_R}{24}},\hfill & \hfill c& ={\displaystyle \frac{{\kappa }_S}{4}},\hfill & \hfill M_{\mathrm{P}}^2& ={\displaystyle \frac{2{\kappa }_R}{{\ell }^2}},\hfill & \hfill \Lambda & ={\displaystyle \frac{6{\kappa }_R}{{\ell }^4}}.\hfill \end{array}$$

(114)

At (103),

$$\begin{array}{cccccccccc}\hfill a& =-{\displaystyle \frac{1}{4}},\hfill & \hfill b& ={\displaystyle \frac{1}{24}},\hfill & \hfill c& =-{\displaystyle \frac{1}{2}},\hfill & \hfill M_{\mathrm{P}}^2& ={\displaystyle \frac{2}{{\ell }^2}},\hfill & \hfill \Lambda & ={\displaystyle \frac{6}{{\ell }^4}}.\hfill \end{array}$$

(115)

This is quadratic gravity with an Einstein–Hilbert and a cosmological term. It is local, generally covariant and perturbatively renormalizable in four dimensions (Stelle 1977), with known asymptotically free trajectories in its coupling space (Avramidi and Barvinsky 1985,Fradkin and Tseytlin 1982,Salvio 2018). Its linearized spectrum contains the massless graviton, a massive spin-two mode with parameter

$$m_2^2=-{\displaystyle \frac{M_{\mathrm{P}}^2}{4a}},$$

(116)

and a scalaron with parameter

$$m_0^2={\displaystyle \frac{M_{\mathrm{P}}^2}{12b}},$$

(117)

both positive at (103). Variation with respect to $\overline{\psi }$ gives the Dirac equation, while variation with respect to the metric gives the fourth-order gravitational equation.

The zero-surprisal vacuum.

In vacuum, the configuration $\mathbb{F}={\mathbb{F}}_0$ has vanishing surprisal and solves that equation. It is de Sitter space of radius ℓ:

$$R_{\mu \nu \rho \sigma }={\displaystyle \frac{1}{{\ell }^2}}\left(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho }\right),R=R_0=4{\Lambda }_{\mathrm{geo}},$$

(118)

where

$${\Lambda }_{\mathrm{geo}}:={\displaystyle \frac{\Lambda }{M_{\mathrm{P}}^2}}={\displaystyle \frac{3}{{\ell }^2}}.$$

(119)

Being conformally flat, it annihilates the Bach tensor; being Einstein with constant R, it also annihilates the variation of $R^2$. It therefore solves the full fourth-order equation, not merely its Einstein–Hilbert truncation.

2.5. Dirac–Einstein–Stelle–Standard Model

The preceding calibrations may be assembled into the established matter operator basis coupled to quadratic gravity. Let the fermion amplitude take values in the conventional carrier

$$\psi \left(x\right)\in S(T_{x}X,g_x)\otimes E_{\mathrm{SM}},$$

(120)

where $E_{\mathrm{SM}}$ contains one or more Standard Model generations. More explicitly, we may write $\psi$ as:

$$\psi \left(x\right)\in \left[S_L(T_{x}X,g_x)\otimes \left({(\mathbf{3},\mathbf{2})}_{1/6}\oplus {(\mathbf{1},\mathbf{2})}_{-1/2}\right)\right]\oplus \left[S_R(T_{x}X,g_x)\otimes \left({(\mathbf{3},\mathbf{1})}_{2/3}\oplus {(\mathbf{3},\mathbf{1})}_{-1/3}\oplus {(\mathbf{1},\mathbf{1})}_{-1}\oplus {(\mathbf{1},\mathbf{1})}_0\right)\right],$$

(121)

Let

$$\mathbb{D}={\nabla }_g^{\mathrm{spin}}+{\nabla }_A^{\mathrm{SM}}+{\Phi }_H,D=i{\gamma }^{\mu }\left({\nabla }_{g,\mu }^{\mathrm{spin}}+{\nabla }_{A,\mu }^{\mathrm{SM}}\right)-{\Phi }_H,$$

(122)

where ${\Phi }_H$ is the odd zero-form containing the Higgs and Yukawa transport. Its curvature is

$${\mathbb{D}}^2=R_g+F_A^{\mathrm{SM}}+D_{g,A}{\Phi }_H+{\Phi }_H^2.$$

(123)

Take the reference

$${\mathbb{F}}_0={\displaystyle \frac{1}{2{\ell }^2}}g\overline{)\wedge }g+C_H,$$

(124)

where the first term is the constant-curvature gravitational reference and $C_H$ is the scalar zero-form reference. The jets are

$$\begin{array}{cccc}\hfill {\mathfrak{J}}_{ϵ}\psi & =\left(\psi +iϵD\psi ,\sqrt{ϵ}{\mathbb{D}}^2\right),\hfill & \hfill {\mathfrak{J}}_r\psi & =\left(\psi ,\sqrt{ϵ}{\mathbb{F}}_0\right).\hfill \end{array}$$

(125)

Choose the invariant curvature pairing sectorwise on the gravitational, gauge, Higgs-derivative, and scalar endomorphism components, including the allowed mixed invariant $\xi R{H}^{†}H$. Proposition 1 then gives

$$\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iϵS_{\mathrm{ESM}}+O\left({ϵ}^2\right),$$

(126)

and accumulating gives

$$\mathcal{I}=-i{S}_{\mathrm{ESM}},S_{\mathrm{ESM}}={\int }_X{\mathrm{d}}^{4}x\sqrt{-g}{\mathcal{L}}_{\mathrm{ESM}},$$

(127)

where

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{ESM}}& =a{C}_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+b{R}^2+c{E}_4+{\displaystyle \frac{M_{\mathrm{P}}^2}{2}}R-\Lambda \hfill \\ \hfill & -{\displaystyle \frac{1}{4g_3^2}}{G}_{\mu \nu }^A{G}^{A\mu \nu }-{\displaystyle \frac{1}{4g_2^2}}{W}_{\mu \nu }^I{W}^{I\mu \nu }-{\displaystyle \frac{1}{4g_1^2}}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ \hfill & +\sum _f{\overline{\psi }}_{f}i{\gamma }^{\mu }{D}_{\mu }^{\mathrm{SM}}{\psi }_f-{\left(D_{\mu }^{\mathrm{SM}}H\right)}^{†}{D}^{\mu ,\mathrm{SM}}H-\xi R{H}^{†}H\hfill \\ \hfill & -m_H^2{H}^{†}H-{\lambda }_H{\left(H^{†}H\right)}^2\hfill \\ \hfill & -\left({\overline{Q}}_L{Y}_u\tilde{H}{u}_R+{\overline{Q}}_L{Y}_{d}H{d}_R+{\overline{L}}_L{Y}_{e}H{e}_R+{\overline{L}}_L{Y}_{\nu }\tilde{H}{\nu }_R+\mathrm{h}.\mathrm{c}.\right),\hfill \end{array}$$

(128)

with $\tilde{H}=i{\sigma }_2{H}^{*}$ — the Standard Model matter action coupled to Einstein–Stelle quadratic gravity.

3. Main Result

Having calibrated least surprisal against established Lagrangians, we now construct the deformations jets of $\psi \left(x\right)$, and determine the theory produced by their accumulated surprisal. Insofar as the jets describe its canonical deformations, the resulting theory will be the least-surprising theory a deformable $\psi \left(x\right)$ furnishes of itself.

3.1. The Arena

The dimensionality of surprisal fixes the arena and its signature.

Theorem 1

(Dimensionality of surprisal). Let Ψ be a canonically normalized Dirac field in d spacetime dimensions, let the contracted transport have mass dimension one, and let its curvature and reference have mass dimension two:

$$[\Psi ]={\displaystyle \frac{d-1}{2}},\left[D\right]=1,\left[{\mathbb{D}}^2\right]=\left[{\mathbb{F}}_0\right]=2.$$

(129)

The local coefficients furnished by the matter and curvature slots then have dimensions

$$\begin{array}{cccc}\hfill \left[Re{\langle \Psi \mid D\Psi \rangle }_x\right]& =d,\hfill & \hfill \left[Q\left({\mathbb{D}}^2-{\mathbb{F}}_0,{\mathbb{D}}^2-{\mathbb{F}}_0\right)\right]& =4.\hfill \end{array}$$

(130)

Consequently, the two slots contribute to a common local Lagrangian with a dimensionless relative slot weight precisely when

$$d=4.$$

(131)

Equivalently, only in four dimensions does the given transition-jet pairing combine canonically normalized Dirac transport and curvature-square dynamics without an additional dimensionful relative coefficient.

Having selected $d=4$, the remaining metric signatures are

$$(4,0),(3,1),(2,2),(1,3),(0,4).$$

(132)

The matter slot furnishes the Dirac equation and its conserved current

$$J^{\mu }=\overline{\Psi }{\gamma }^{\mu }\Psi .$$

(133)

For this current to represent the locally measurable density of the amplitude, we require the existence of a unit covector n whose contraction with the current,

$$n_{\mu }{J}^{\mu },$$

(134)

is positive-definite in the orientation defined by n: nonvanishing for every nonzero physical state, with a common sign fixed to be positive by the choice of orientation. Any such n is necessarily timelike, and its existence defines the temporal orientation of the arena. The intrinsic Clifford-compatible Dirac adjoint admits such a positive-definite density in the one-time Lorentzian signatures

$$(3,1),(1,3),$$

(135)

which differ only by the overall metric-sign convention. In Euclidean signature the contraction is indefinite for every unit vector: $n·\gamma$ squares to the identity and is traceless, so its spectrum is symmetric about zero. In split signature the same failure occurs on the unrestricted Dirac carrier for every choice of n, timelike in either of the two temporal planes or otherwise.

The transition jets therefore select Lorentzian $3+1$ spacetime: four dimensions are required for dimensionless compatibility of their matter and curvature slots, and Lorentzian signature is required for the positive conserved density of the Dirac amplitude. The selected dimension is additionally the marginal dimension of curvature-square gravity and hence the dimension in which the resulting gravitational theory is power-counting renormalizable with dimensionless higher-curvature couplings.

3.2. The Deformation Carrier

Section 1.4 motivated closing $\psi \left(x\right)$ under the intrinsic deformations of the defining forms exposed by the commuting left and right actions of the Dirac–Hestenes amplitude. We now construct the resulting configuration spaces, deformation geometries, and carrier.

As established earlier, the least-surprising theory furnished by an amplitude parametrized only by spacetime contains Dirac matter coupled to quadratic gravity, but supplies neither a Standard Model particle carrier nor family multiplicity. Matter, however, deforms spacetime. It should therefore not be entirely surprising that closing $\psi \left(x\right)$ under deformations of its defining geometry furnishes a particle sector. What is nontrivial and remarkable is the specific sector thereby obtained: the metric-deformation geometry furnishes the particle content of one Standard Model generation, while the right-Hermitian deformation geometry furnishes three-family multiplicity.

Definition 5

(Dirac–Hestenes amplitude and defining forms). Let X be a four-dimensional, oriented, time-oriented Lorentzian manifold. At $x\in X$, let $\psi \left(x\right)\in {Cl}^{+}(T_{x}X,g_x)$ be a Dirac–Hestenes amplitude. The even spacetime algebra satisfies ${Cl}^{+}(T_{x}X,g_x)\cong {Mat}_2\left(\mathbb{C}\right)$ as a real algebra and admits commuting left spacetime-spin and right current-preserving actions,

$$\begin{array}{c}\hfill \psi \left(x\right)⟼S\psi \left(x\right)U,S\in Spin(3,1),U\in U{\left(2\right)}_R.\end{array}$$

(136)

In the standard Dirac–Hestenes construction, invariance of the current $J\left(\psi \right)=\psi {\gamma }_0\tilde{\psi }$ under right multiplication requires $U{\gamma }_0\tilde{U}={\gamma }_0,$ whose connected right-action group is

$$U{\left(2\right)}_R\cong {\displaystyle \frac{SU\left(2\right)\times U\left(1\right)}{{\mathbb{Z}}_2}}.$$

(137)

Equivalently, $U{\left(2\right)}_R=U(W_{x,g},q_x)$ for the positive Hermitian form $q_x$ on the complex rank-two right module $W_{x,g}\cong {\mathbb{C}}^2$.

The left action preserves the Lorentzian form $g_x$ on $T_{x}X$. The right action preserves the Hermitian form $q_x$ on $W_{x,g}$.

Allowing $(g_x,q_x)$ to vary promotes them to configuration variables of the universal extension $\Psi$. Their possible values define the corresponding configuration spaces, which we now define.

Definition 6

(Configuration spaces). Let ${\widetilde{GL}}^{+}\left(T_{x}X\right)$ denote the connected double cover of $G{L}^{+}\left(T_{x}X\right)$. The spin-metric configuration space associated with the left action of the Dirac–Hestenes amplitude is

$${Met}_{1,3,x}^{\mathrm{spin}}\left(X\right):={\displaystyle \frac{{\widetilde{GL}}^{+}\left(T_{x}X\right)}{Spin(3,1)}}\cong {\displaystyle \frac{{\widetilde{GL}}^{+}(4,\mathbb{R})}{Spin(3,1)}}.$$

(138)

It has real dimension ten.

The Hermitian-metric configuration space associated with the right action is

$${Herm}^{+}\left(W_{x,g}\right)\cong {\displaystyle \frac{GL\left(W_{x,g}\right)}{U(W_{x,g},q_x)}}\cong {\displaystyle \frac{GL(2,\mathbb{C})}{U\left(2\right)}},$$

(139)

and has real dimension four.

The configuration bundle ${\pi }_{\mathcal{C}}:{\mathcal{C}}_X⟶X$ has local points $z=(x,g_x,q_x)$ and local fibre

$${\left({\mathcal{C}}_X\right)}_x\cong {Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)\times {Herm}^{+}\left(W_{x,g}\right).$$

(140)

The configuration fibre has real dimension

$${dim}_{\mathbb{R}}{\left({\mathcal{C}}_X\right)}_x=10+4=14,$$

(141)

and hence

$${dim}_{\mathbb{R}}{\mathcal{C}}_X=4+14=18.$$

(142)

Infinitesimal deformations lie in the tangent spaces to the configuration fibres.

Definition 7

(Deformation spaces). At a configuration $z=(x,g_x,q_x)\in {\mathcal{C}}_X,$ define the metric and right-Hermitian deformation spaces by

$$\begin{array}{cc}\hfill V_z^g& :=T_{g_x}{Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)\cong {Sym}^2\left(T_x^{*}X\right),\hfill \end{array}$$

(143)

$$\begin{array}{cc}\hfill V_z^q& :=T_{q_x}{Herm}^{+}\left(W_{x,g}\right)\cong Herm\left(W_{x,g}\right).\hfill \end{array}$$

(144)

Under the local product identification, their direct sum $V_z{\mathcal{C}}_X=V_z^g\oplus V_z^q$ is the fourteen-dimensional vertical deformation space at z.

The metric-deformation space carries the DeWitt form

$$G_{g_x}(h,k)={tr}_{g_x}\left(hk\right)-\lambda {tr}_{g_x}\left(h\right){tr}_{g_x}\left(k\right),h,k\in V_z^g.$$

(145)

For $\lambda >\frac{1}{4}$, its signature is $(6,4)$, as established in Lemma 1. It therefore supplies the complex chiral half-spin modules

$${\Sigma }^{\epsilon }(V_z^g,G_{g_x}),\epsilon \in \{+,-\}.$$

(146)

The right-Hermitian deformation space carries the invariant positive form

$$K_{q_x}(a,b):=tr\left(q_x^{-1}a{q}_x^{-1}b\right),a,b\in V_z^q.$$

(147)

Thus the two native deformation geometries are

$$(V_z^g,G_{g_x})\mathrm{and}(V_z^q,K_{q_x}).$$

(148)

Definition 8

(Deformation carrier). The deformation carrie of $\psi \left(x\right)$ over ${\mathcal{C}}_X$ is the vector bundle $\mathcal{E}⟶{\mathcal{C}}_X$ whose fibre at $z=(x,g_x,q_x)$ is

$$\begin{array}{|c|}\hline \begin{array}{ccc}\hfill {\left(\mathcal{E}\right)}_z:=& {Cl}^{+}(T_{x}X,g_x){\otimes }_{\mathbb{R}}{\Sigma }^{\epsilon }\left(V_z^g,G_{g_x}\right){\otimes }_{\mathbb{R}}{V}_z^q,\hfill & \hfill \epsilon \in \{+,-\}.\end{array}\\ \hline\end{array}$$

(149)

A deformation amplitud is a section

$$\Psi \in \Gamma \left({\mathcal{C}}_X,\mathcal{E}\right),\Psi =\Psi (x,g_x,q_x).$$

(150)

The metric-deformation factor is represented by its chiral half-spin module and will supply particle identity. The right-Hermitian deformation factor is represented by its tangent module and will supply multiplicity. These representation assignments are those introduced in Section 1.4.

The seed amplitude $\psi \left(x\right)$ and the deformation amplitude $\Psi (x,g_x,q_x)$ are therefore distinct but related objects. The former takes values in the original spacetime-algebra carrier; the latter takes values in the carrier constructed from its left and right deformation geometries.

Definition 9

(Physical pullback). A physical realization of the defining forms is a section

$$s=(g,q):X⟶{\mathcal{C}}_X,s\left(x\right)=\left(x,g\left(x\right),q\left(x\right)\right).$$

(151)

The corresponding physical carrier and amplitude are

$$\begin{array}{cc}\hfill {\mathcal{E}}_s& :=s^{*}\mathcal{E}⟶X,\hfill \end{array}$$

(152)

$$\begin{array}{cc}\hfill {\Psi }_s& :=s^{*}\Psi \in \Gamma (X,{\mathcal{E}}_s).\hfill \end{array}$$

(153)

Pointwise,

$${\Psi }_s\left(x\right)=\Psi \left(x,g\left(x\right),q\left(x\right)\right).$$

(154)

The notation ${\Psi }_s$ is deliberately distinguished from the seed amplitude $\psi$. Pullback changes the domain of the universal amplitude but retains its deformation-derived particle and multiplicity factors. Their physical reductions are developed in Section 3.3 and Section 3.4.

The horizontal–vertical decomposition and the corresponding generalized transport on this carrier are constructed in Section 3.5.

3.3. The Particle Sector

By Definition 7, the metric-deformation space in the selected arena is

$$V_z^g=T_{g_x}{Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)\cong {Sym}^2\left(T_x^{*}X\right),{dim}_{\mathbb{R}}{V}_z^g=10.$$

(155)

Its DeWitt form determines the spin-frame symmetry of the particle carrier.

Lemma 1

(DeWitt signature). Let

$$G_g(h,k)=h_{\mu \nu }{k}_{\rho \sigma }\left(g^{\mu \rho }{g}^{\nu \sigma }-\lambda g^{\mu \nu }{g}^{\rho \sigma }\right)$$

(156)

on

$$V_z^g\cong {Sym}^2\left(T_x^{*}X\right),$$

(157)

where g has Lorentzian signature and $\lambda >1/d$. Then

$${dim}_{\mathbb{R}}{V}_z^g={\displaystyle \frac{d(d+1)}{2}},sig{G}_g=\left({\displaystyle \frac{d(d-1)}{2}},d\right).$$

(158)

In $d=4$,

$${dim}_{\mathbb{R}}{V}_z^g=10,sig{G}_g=(6,4).$$

(159)

At $\lambda =1/d$, the trace direction is null.

Proof. 

Decompose

$$h=h_0+{\displaystyle \frac{{tr}_{g}h}{d}}g,{tr}_g{h}_0=0.$$

(160)

The two terms are $G_g$-orthogonal. In a Lorentzian orthonormal frame, the $d-1$ components with one temporal and one spatial index have negative norm, while all remaining directions have positive norm before the trace adjustment. The trace direction has norm

$$G_g(g,g)=d-\lambda d^2,$$

(161)

which becomes negative for $\lambda >1/d$. Hence

$$sig{G}_g=\left({\displaystyle \frac{d(d-1)}{2}},d\right),$$

(162)

giving $(6,4)$ for $d=4$. □

The metric-deformation geometry therefore supplies

$$Spin(V_z^g,G_{g_x})\cong Spin(6,4).$$

(163)

Positive definiteness of the particle density selects a fundamental symmetry $R_x$ satisfying

$$R_x^2=\mathbb{1},G_{g_x}(v,R_{x}v)>0\mathrm{for}\mathrm{all}v\ne 0.$$

(164)

This reduces the spin-frame symmetry to its maximal compact subgroup,

$$Spin\left(6\right)\times Spin\left(4\right)\cong SU\left(4\right)\times SU{\left(2\right)}_L\times SU{\left(2\right)}_R=G_{\mathrm{PS}}.$$

(165)

A compatible orthogonal complex structure $J_x$ satisfies

$$J_x^2=-\mathbb{1},G_{g_x}(J_{x}v,J_{x}w)=G_{g_x}(v,w),[J_x,R_x]=0.$$

(166)

The common stabilizer of $R_x$ and $J_x$ is

$$U\left(3\right)\times U\left(2\right).$$

(167)

Unimodularity then gives

$$S\left(U\left(3\right)\times U\left(2\right)\right)\cong {\displaystyle \frac{SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y}{{\mathbb{Z}}_6}}=G_{\mathrm{SM}}.$$

(168)

Its central generator is, up to normalization and sign,

$$Y=diag\left(-{\displaystyle \frac{1}{3}},-{\displaystyle \frac{1}{3}},-{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right).$$

(169)

Thus the particle-sector reduction is

$$\begin{array}{|c|}\hline Spin(6,4)\stackrel{R_x}{\to }{G}_{\mathrm{PS}}\stackrel{J_x}{\to }U\left(3\right)\times U\left(2\right)\stackrel{\mathrm{unimodularity}}{\to }{G}_{\mathrm{SM}}.\\ \hline\end{array}$$

(170)

The selected metric-deformation half-spin module consequently supplies the particle carrier of one Standard Model generation.

3.4. The Family Sector

By Definition 7, the right-Hermitian deformation space is

$$V_z^q=T_{q_x}{Herm}^{+}\left(W_{x,g}\right)\cong Herm\left(W_{x,g}\right),{dim}_{\mathbb{R}}{V}_z^q=4.$$

(171)

Its invariant form $K_{q_x}$ is already positive definite. The physical reduction is therefore supplied by unimodularity.

Define the space of determinant-one positive Hermitian forms by

$${Herm}_1^{+}\left(W_{x,g}\right):=\left\{q\in {Herm}^{+}\left(W_{x,g}\right):detq=1\right\}\cong {\displaystyle \frac{SL(2,\mathbb{C})}{SU\left(2\right)}}.$$

(172)

Since

$$\delta logdetq=tr\left(q^{-1}\delta q\right),$$

(173)

its tangent space is

$$V_z^{q,1}:=T_{q_x}{Herm}_1^{+}\left(W_{x,g}\right)=\left\{a\in Herm\left(W_{x,g}\right):tr\left(q_x^{-1}a\right)=0\right\}.$$

(174)

Thus

$$V_z^q=\mathbb{R}{q}_x\oplus V_z^{q,1},V_z^{q,1}\cong {Herm}_{0,q_x}\left(W_{x,g}\right),{dim}_{\mathbb{R}}{V}_z^{q,1}=3.$$

(175)

Theorem 2

(Three-family multiplicity). Let $E_{\mathrm{part}}$ be the complex one-generation particle carrier obtained in Section 3.3. Its extension by the unimodular right-Hermitian deformation space satisfies

$$\begin{array}{|c|}\hline E_{\mathrm{part}}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\cong E_{\mathrm{part}}{\otimes }_{\mathbb{C}}\left(\mathbb{C}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\right)\cong E_{\mathrm{part}}{\otimes }_{\mathbb{C}}{\mathbb{C}}^3\cong 3E_{\mathrm{part}}.\\ \hline\end{array}$$

(176)

The reduced carrier therefore contains three gauge-identical generations.

Proof. 

The space $V_z^{q,1}$ has real dimension three. Since $E_{\mathrm{part}}$ is complex, its tensor product over $\mathbb{R}$ with $V_z^{q,1}$ inherits complex multiplication through the first factor. Consequently,

$$\mathbb{C}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\cong {\mathbb{C}}^3.$$

(177)

The right-Hermitian factor is independent of the particle gauge action, so $G_{\mathrm{SM}}$ acts as

$${\rho }_{\mathrm{SM}}\otimes {\mathbb{1}}_3.$$

(178)

The three components therefore have identical Standard Model quantum numbers. □

The reduced right-Hermitian transport acts on this family multiplicity and will be denoted by $D_{\mathrm{fam}}$. General scalar transport on its complex rank-three realization admits

$$Y_u,Y_d,Y_e,Y_{\nu }\in {Mat}_3\left(\mathbb{C}\right),M_R\in {Mat}_3\left(\mathbb{C}\right),M_R=M_R^T.$$

(179)

Their relative diagonalizations give the CKM and PMNS matrices. Accordingly,

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\Sigma }^{\epsilon }(V_z^g,G_{g_x})& ⟶\mathrm{particle}\mathrm{content}\mathrm{of}\mathrm{one}\mathrm{generation},\hfill \\ \hfill V_z^q\stackrel{detq=1}{\to }{V}_z^{q,1}& ⟶\mathrm{three}-\mathrm{family}\mathrm{multiplicity},\hfill \\ \hfill \mathrm{scalar}\mathrm{transport}\mathrm{on}{\mathbb{C}}^3& ⟶\mathrm{masses}\mathrm{and}\mathrm{flavour}\mathrm{mixing}.\hfill \end{array}\\ \hline\end{array}$$

(180)

3.5. Assembly

The reductions of Section 3.3 and Section 3.4 give the physical fermion carrier

$${\mathcal{E}}_{\mathrm{f}}^{\left(3\right)}=\left[(S_L\otimes F_L)\oplus (S_R\otimes F_R)\right]{\otimes }_{\mathbb{C}}{\mathbb{C}}^3,$$

(181)

where

$$F_L=Q_L\oplus L_L,F_R=u_R\oplus d_R\oplus {\nu }_R\oplus e_R.$$

(182)

The Standard Model acts trivially on the family factor,

$${\rho }_{\mathrm{SM}}^{\left(3\right)}={\rho }_{\mathrm{SM}}\otimes {\mathbb{1}}_3.$$

(183)

Let ${\mathbb{D}}_x$, ${\mathbb{D}}_g$, and ${\mathbb{D}}_q$ denote, respectively, spacetime, metric-deformation, and right-Hermitian deformation transport. Let ${\Phi }_H$ be the odd scalar transport connecting the left- and right-handed particle sectors. The total generalized transport is

$$\begin{array}{|c|}\hline {\mathbb{D}}_{\mathrm{tot}}={\mathbb{D}}_x+{\mathbb{D}}_g+{\mathbb{D}}_q+{\Phi }_H,\\ \hline\end{array}$$

(184)

with corresponding first-order operator

$$D_{\mathrm{tot}}=D_x+D_g+D_q-{\Phi }_H.$$

(185)

Its curvature is

$$\begin{array}{cc}\hfill {\mathbb{D}}_{\mathrm{tot}}^2& ={\mathbb{D}}_x^2+{\mathbb{D}}_g^2+{\mathbb{D}}_q^2+[{\mathbb{D}}_x,{\mathbb{D}}_g]+[{\mathbb{D}}_x,{\mathbb{D}}_q]+[{\mathbb{D}}_g,{\mathbb{D}}_q]\hfill \\ \hfill & +[{\mathbb{D}}_x,{\Phi }_H]+[{\mathbb{D}}_g,{\Phi }_H]+[{\mathbb{D}}_q,{\Phi }_H]+{\Phi }_H^2.\hfill \end{array}$$

(186)

Pullback along the physical section $s=(g,q)$ identifies the reduced diagonal curvature sectors as

$$\begin{array}{cccccc}\hfill s^{*}{\mathbb{D}}_x^2& =R_g,\hfill & \hfill s^{*}{\mathbb{D}}_g^2& =F_A^{\mathrm{SM}},\hfill & \hfill s^{*}{\mathbb{D}}_q^2& =F_{\mathrm{fam}}.\hfill \end{array}$$

(187)

The product-compatible physical reduction sets the mixed deformation curvatures to their reference values. The family curvature may similarly be declared unsurprising by taking

$$F_{\mathrm{fam},0}=F_{\mathrm{fam}}.$$

(188)

The scalar transport has the usual left–right form

$${\Phi }_H=\left(\begin{array}{cc}0& {\Phi }_{LR}\left(H\right)\\ {\Phi }_{LR}{\left(H\right)}^{†}& 0\end{array}\right),H\sim {(\mathbf{1},\mathbf{2})}_{1/2},\tilde{H}=i{\sigma }_2{H}^{*},$$

(189)

where its family endomorphisms are

$$Y_u,Y_d,Y_e,Y_{\nu }\in {Mat}_3\left(\mathbb{C}\right),M_R=M_R^T\in {Mat}_3\left(\mathbb{C}\right).$$

(190)

Choose the reference

$${\mathbb{F}}_{0,\mathrm{tot}}={\displaystyle \frac{1}{2{\ell }^2}}g\overline{)\wedge }g+F_{\mathrm{fam},0}+C_H.$$

(191)

The pulled-back universal transition jets are

$$\begin{array}{cc}\hfill s^{*}{\mathfrak{J}}_{ϵ}\Psi & =s^{*}\left(\Psi +iϵD_{\mathrm{tot}}\Psi ,\sqrt{ϵ}{\mathbb{D}}_{\mathrm{tot}}^2\right),\hfill \end{array}$$

(192)

$$\begin{array}{cc}\hfill s^{*}{\mathfrak{J}}_r\Psi & =s^{*}\left(\Psi ,\sqrt{ϵ}{\mathbb{F}}_{0,\mathrm{tot}}\right).\hfill \end{array}$$

(193)

Their accumulated surprisal is

$${\mathcal{I}}_{\mathrm{tot}}=-i{S}_{\mathrm{tot}},$$

(194)

where

$$\begin{array}{|c|}\hline S_{\mathrm{tot}}={\int }_X{\mathrm{d}}^{4}x\sqrt{-g}{s}^{*}\left[Re〈\Psi |D_{\mathrm{tot}}\Psi 〉+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}_{\mathrm{tot}}^2-{\mathbb{F}}_{0,\mathrm{tot}}∥}^2\right].\\ \hline\end{array}$$

(195)

The operator sectors contribute as follows:

$$\begin{array}{cccc}\hfill D_x& ⟼\mathrm{Dirac}\mathrm{kinetic}\mathrm{terms},\hfill & \hfill {\mathbb{D}}_x^2& ⟼C^2,R^2,E_4,M_{\mathrm{P}}^{2}R,\Lambda ,\hfill \end{array}$$

(196)

$$\begin{array}{cccc}\hfill D_g& ⟼\mathrm{particle}\mathrm{transport},\hfill & \hfill {\mathbb{D}}_g^2& ⟼G_{\mu \nu }^A{G}^{A\mu \nu },W_{\mu \nu }^I{W}^{I\mu \nu },B_{\mu \nu }{B}^{\mu \nu },\hfill \end{array}$$

(197)

$$\begin{array}{cccc}\hfill D_q& ⟼\mathrm{family}\mathrm{transport}\mathrm{on}{\mathbb{C}}^3,\hfill & \hfill {\Phi }_H& ⟼\mathrm{Higgs},\mathrm{Yukawa},\mathrm{and}\mathrm{Majorana}\mathrm{transport}.\hfill \end{array}$$

(198)

Moreover,

$$\begin{array}{cc}\hfill [{\mathbb{D}}_x+{\mathbb{D}}_g,{\Phi }_H]& ⟼-{\left(D_{\mu }^{\mathrm{SM}}H\right)}^{†}{D}^{\mu ,\mathrm{SM}}H,\hfill \end{array}$$

(199)

$$\begin{array}{cc}\hfill {\mathbb{D}}_q,{\Phi }_H]& ⟼\mathrm{flavour}\mathrm{mixing},\hfill \end{array}$$

(200)

$$\begin{array}{cc}\hfill {\Phi }_H^2-C_H& ⟼-m_H^2{H}^{†}H-{\lambda }_H{\left(H^{†}H\right)}^2,\hfill \end{array}$$

(201)

$$\begin{array}{cc}\hfill {〈R_g,{\Phi }_H^2〉}_{\mathrm{mix}}& ⟼-\xi R{H}^{†}H.\hfill \end{array}$$

(202)

Consequently, accumulated surprisal of the pulled-back and reduced transition jets gives the three-generation Standard Model coupled to quadratic gravity, collected explicitly in Section 3.6.

3.6. The Action

Evaluating the pulled-back and reduced sectors assembled in Section 3.5 gives

$${\mathcal{I}}_{\mathrm{tot}}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{1}{ϵ}}\mathcal{K}\left(s^{*}{\mathfrak{J}}_{ϵ}\Psi ,s^{*}{\mathfrak{J}}_r\Psi \right)=-i{S}_{\mathrm{SM}+\mathrm{QG}}^{\left(3\right)}.$$

(203)

Here $s=(g,q):X\to {\mathcal{C}}_X$ is the physical section. Positivity, compatible complex structure, and unimodularity reduce the metric-deformation half-spin module to the one-generation Standard Model particle carrier and the right-Hermitian deformation module to the complex rank-three family carrier. The resulting action is

$$\begin{array}{cc}\hfill S_{\mathrm{SM}+\mathrm{QG}}^{\left(3\right)}& ={\int }_X{\mathrm{d}}^{4}x\sqrt{-g}[a{C}_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma }+b{R}^2+c{E}_4+{\displaystyle \frac{M_{\mathrm{P}}^2}{2}}R-\Lambda \hfill \\ \hfill & -{\displaystyle \frac{1}{4g_3^2}}{G}_{\mu \nu }^A{G}^{A\mu \nu }-{\displaystyle \frac{1}{4g_2^2}}{W}_{\mu \nu }^I{W}^{I\mu \nu }-{\displaystyle \frac{1}{4g_1^2}}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ \hfill & +{\displaystyle \sum _{n=1}^3}\sum _f{\overline{\psi }}_{f,n}i{\gamma }^{\mu }{D}_{\mu }^{\mathrm{SM}}{\psi }_{f,n}-{\left(D_{\mu }^{\mathrm{SM}}H\right)}^{†}{D}^{\mu ,\mathrm{SM}}H-\xi R{H}^{†}H\hfill \\ \hfill & -m_H^2{H}^{†}H-{\lambda }_H{\left(H^{†}H\right)}^2\hfill \\ \hfill & -\left({\overline{Q}}_L{Y}_u\tilde{H}{u}_R+{\overline{Q}}_L{Y}_{d}H{d}_R+{\overline{L}}_L{Y}_{e}H{e}_R+{\overline{L}}_L{Y}_{\nu }\tilde{H}{\nu }_R+\mathrm{h}.\mathrm{c}.\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left(\overline{{\nu }_R^c}{M}_R{\nu }_R+\mathrm{h}.\mathrm{c}.\right)],\hfill \end{array}$$

(204)

where

$$\tilde{H}=i{\sigma }_2{H}^{*}.$$

(205)

The fermion fields are vectors in the family space

$${\mathcal{F}}_{\mathrm{fam}}=\mathbb{C}{\otimes }_{\mathbb{R}}{V}^{q,1}\cong {\mathbb{C}}^3.$$

(206)

Consequently,

$$Y_u,Y_d,Y_e,Y_{\nu },M_R\in {Mat}_3\left(\mathbb{C}\right),M_R=M_R^T.$$

(207)

The Yukawa matrices need not be simultaneously diagonalizable; their relative diagonalizations give the CKM and PMNS matrices.

Because ${\nu }_R$ is a Standard Model singlet, the Majorana block $M_R$ is gauge invariant. The choice $M_R=0$ gives Dirac neutrinos, whereas $M_R\ne 0$ permits a type-I seesaw regime. Its value remains part of the scalar transport or reference data.

The action (204) is local and contains operators of mass dimension at most four. Its curvature-square gravitational sector gives the ultraviolet propagator behaviour required for power-counting renormalizability. Perturbative quantization proceeds through the standard background-field and BRST construction for quadratic gravity (Salvio 2018,Stelle 1977). One expands

$$g_{\mu \nu }={\overline{g}}_{\mu \nu }+\kappa h_{\mu \nu },A_{\mu }={\overline{A}}_{\mu }+a_{\mu },$$

(208)

and supplements the action by the corresponding gauge-fixing and ghost terms,

$$S_{\mathrm{gf}}=S_{\mathrm{SM}+\mathrm{QG}}^{\left(3\right)}+S_{\mathrm{g}.\mathrm{f}.}^{\mathrm{diff}}+S_{\mathrm{g}.\mathrm{f}.}^{\mathrm{SM}}+S_{\mathrm{gh}}^{\mathrm{diff}}+S_{\mathrm{gh}}^{\mathrm{SM}}.$$

(209)

After four-derivative gravitational gauge fixing, the spin-two propagator has the schematic form

$${\Delta }^{\left(2\right)}\left(k\right)\propto {\displaystyle \frac{P^{\left(2\right)}}{k^2\left(1-k^2/m_2^2\right)}}=P^{\left(2\right)}\left[{\displaystyle \frac{1}{k^2}}-{\displaystyle \frac{1}{k^2-m_2^2}}\right],$$

(210)

with the pole prescription understood. Hence

$${\Delta }^{\left(2\right)}\left(k\right)\sim {\displaystyle \frac{1}{k^4}},\left|k^2\right|⟶\infty .$$

(211)

The opposite residue of the massive spin-two pole is represented by an opposite-norm sector. The resulting kinematical carrier is therefore of Hilbert–Krein form,

$$K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}},{\eta }_{universe}={\mathbb{1}}_{\mathrm{SM}}^{\left(3\right)}\otimes {\eta }_{\mathrm{QG}}.$$

(212)

Its intensity-positive measurement interpretation is developed in Section 4.

4. Intensity Measurements

One difficulty remains. The quadratic-gravity action is perturbatively renormalizable and admits asymptotically free trajectories (Avramidi and Barvinsky 1985,Fradkin and Tseytlin 1982,Salvio 2018,Stelle 1977), but its standard covariant quantization contains a massive spin-two pole with opposite residue. This pole is not an incidental excitation that can simply be removed: schematically,

$${\displaystyle \frac{1}{k^2(1+k^2/m_2^2)}}={\displaystyle \frac{1}{k^2}}-{\displaystyle \frac{1}{k^2+m_2^2}},$$

(213)

so the same pole that produces the indefinite sector is tied to the $k^{-4}$ ultraviolet decay underlying perturbative renormalizability.

Retaining the complete spectrum therefore places the massive spin-two mode in a negative-metric sector, conventionally called the Stelle ghost. Since the Standard Model sector is Hilbert-positive, the resulting kinematical carrier is

$$\begin{array}{|c|}\hline K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}},{\eta }_{universe}={\mathbb{1}}_{\mathrm{SM}}^{\left(3\right)}\otimes {\eta }_{\mathrm{QG}},\\ \hline\end{array}$$

(214)

where ${\eta }_{\mathrm{QG}}$ acts as $+\mathbb{1}$ on the positive sectors and as $-\mathbb{1}$ on the massive spin-two sector.

The usual objection conflates measurement with a globally normalized probability measure, and as such incorrectly claims that measurements are not possible on a Krein space. The literature is a prescription of attempts to remove the ghost without damaging the renormalizable character of the theory. However, a simpler realization resolves the issue—the physical interpretation is merely that Hilbert spaces describe quantum systems with external observers, and Krein spaces describe quantum systems with internal observers.

A probability measure satisfies three Kolmogorov requirements: nonnegativity, countable additivity on disjoint channels, and normalization of an exhaustive family to unity. However, detector records prior to normalization are intensities, which require only the first two:

$$I_A\left(\omega \right)\ge 0,I_{{\sum }_m{A}_m}\left(\omega \right)=\sum _m{I}_{A_m}\left(\omega \right).$$

(215)

Whenever a registered family $\mathcal{D}=\left\{A_m\right\}$ has finite nonzero total intensity, it defines ordinary probabilities by

$$p(m\mid \mathcal{D},\omega )={\displaystyle \frac{I_{A_m}\left(\omega \right)}{{\sum }_n{I}_{A_n}\left(\omega \right)}}.$$

(216)

These satisfy all three Kolmogorov requirements within the registered experiment. This is sufficient for quantum measurement on the Krein carrier.

The order isomorphism

$$A⟼E:=\eta A$$

(217)

identifies intensity-positive Krein observables with ordinary positive Hilbert-space effects, so measurement families, state updates, and instruments transfer directly between the two descriptions. The indefinite form remains a conserved signed charge, while every detector record is a nonnegative intensity and every registered experiment with finite nonzero total admits the usual probabilistic interpretation.

The objection that Krein spaces cannot be measured, hence cannot be physical, is dissolved in the details. In a Krein space, the observer retains access to intensity measurements via detector counts, but cannot normalize over all possible outcomes of the quantum system. Consequently, the observer can never prove it measured the totality of the system, and as such we say it is internal to it. Krein spaces are to internal observers what Hilbert spaces are to external observers. The kinematical carrier

$$K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}},{\eta }_{universe}={\mathbb{1}}_{\mathrm{SM}}^{\left(3\right)}\otimes {\eta }_{\mathrm{QG}},$$

(218)

places the observer externally to the particle sector (i.e., ${\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}$ is a Hilbert space) but internally to the universe (i.e., $K_{\mathrm{QG}}$ and $K_{universe}$ are Krein spaces).

We now construct this intensity-positive measurement structure without altering the action, propagator, or spectrum.

The intensity cone.

Write K for $K_{universe}$ and $\eta$ for ${\eta }_{universe}$, so that

$${\eta }^{†}=\eta ,{\eta }^2=\mathbb{1},[\varphi ,\psi ]=(\varphi ,\eta \psi ),$$

(219)

with $(·,·)$ the positive pairing defining the topology. The Krein adjoint is

$$A^{♯}=\eta A^{†}\eta ,$$

(220)

and the bounded Krein-self-adjoint observables form

$${\mathfrak{A}}^{\mathrm{sa}}=\{A:A^{♯}=A\}.$$

(221)

Definition 10

(Intensity-positive observable). An observable $A\in {\mathfrak{A}}^{\mathrm{sa}}$ is intensity-positiv when

$$[\psi ,A\psi ]\ge 0$$

(222)

for every $\psi \in K$. The resulting cone is

$$\mathcal{C}:=\left\{A\in {\mathfrak{A}}^{\mathrm{sa}}:[\psi ,A\psi ]\ge 0\mathrm{for}\mathrm{all}\psi \right\}.$$

(223)

A preparatio is a nonzero continuous real-linear functional ω on ${\mathfrak{A}}^{\mathrm{sa}}$ satisfying $\omega \left(A\right)\ge 0$ on $\mathcal{C}$, and the intensity of a channel $A\in \mathcal{C}$ is

$$I_A\left(\omega \right):=\omega \left(A\right)\ge 0.$$

(224)

Vectors define preparations by

$${\omega }_{\psi }\left(A\right):=[\psi ,A\psi ].$$

(225)

No normalization is imposed.

Lemma 2

(Structure). $A^{♯}=A$ if and only if $\eta A$ is Hilbert-self-adjoint, and

$$[\psi ,A\psi ]=(\psi ,\eta A\psi ).$$

(226)

Hence

$$A⟼E:=\eta A$$

(227)

is a linear bijection from ${\mathfrak{A}}^{\mathrm{sa}}$ onto the Hilbert-self-adjoint operators, carrying $\mathcal{C}$ onto the ordinary positive cone. Detector channels are therefore $A=\eta E$ with $E\ge 0$.

Proof. 

The condition $A^{♯}=A$ is

$$A^{†}\eta =\eta A,$$

(228)

equivalently ${\left(\eta A\right)}^{†}=\eta A$. The displayed identity is immediate, and ${\eta }^2=\mathbb{1}$ inverts the map. If both A and $-A$ belong to $\mathcal{C}$, then $\eta A=0$, hence $A=0$. □

Proposition 2

(Generation and tomography). η is an order unit for $\mathcal{C}$: for every $B\in {\mathfrak{A}}^{\mathrm{sa}}$,

$$-M\eta \le B\le M\eta ,M=\parallel \eta B\parallel .$$

(229)

Consequently,

$${\mathfrak{A}}^{\mathrm{sa}}=\mathcal{C}-\mathcal{C},$$

(230)

and preparations agreeing on $\mathcal{C}$ agree everywhere.

Proof. 

$M\eta \mp B\in \mathcal{C}$ if and only if

$$M\mathbb{1}\mp \eta B\ge 0,$$

(231)

which holds for $M=\parallel \eta B\parallel$. Writing

$$B=B_{+}-B_{-},B_{\pm }={\displaystyle \frac{1}{2}}(M\eta \pm B)\in \mathcal{C},$$

(232)

shows that the cone generates the full observable space. □

A signed observable value is therefore a difference of two nonnegative records, not a negative detector event.

The dichotomy.

In a Hilbert space one observable is simultaneously conserved and positive. An indefinite pairing separates these roles.

Theorem 3

(Conservation and positivity are exclusive). Let $H^{♯}=H$ generate the Krein-unitary evolution

$$U\left(t\right)=e^{-iHt}.$$

(233)

Then:

(i) 

$\mathbb{1}$ is conserved, but $\mathbb{1}\in \mathcal{C}$ only if $\eta =\mathbb{1}$;

(ii) 

$\eta \in \mathcal{C}$, but its reading is conserved if and only if $[\eta ,H]=0$;

(iii) 

no countable family $\left\{A_m\right\}\subset \mathcal{C}$ satisfies ${\sum }_m{A}_m=\mathbb{1}$, whereas, with

$$P_{\pm }={\displaystyle \frac{1}{2}}(\mathbb{1}\pm \eta ),A_{\pm }=\eta P_{\pm },$$

(234)

one has

$$A_{+}+A_{-}=\eta ,A_{+}-A_{-}=\mathbb{1};$$

(235)

(iv) 

if the Krein-unitary group acts irreducibly, the only element of $\mathcal{C}$ invariant under all of it is 0.

Proof. 

For (i), intensity positivity of $\mathbb{1}$ would require $\eta \ge 0$, which fails unless the carrier is definite. For (ii), $H^{♯}=H$ implies

$$H^{†}=\eta H\eta .$$

(236)

Hence the Hilbert norm is conserved for every state if and only if $H^{†}=H$, equivalently $[\eta ,H]=0$. For (iii), if ${\sum }_m{A}_m=\mathbb{1}$, then

$$\sum _m\eta A_m=\eta$$

(237)

would be an increasing limit of positive operators and hence positive, contradicting indefiniteness. The identities for $A_{\pm }$ follow from ${\eta }^2=\mathbb{1}$. For (iv), invariants lie in the commutant, which by irreducibility consists of scalar multiples of $\mathbb{1}$; no nonzero such multiple lies in $\mathcal{C}$ when $\eta$ is indefinite. □

Corollary 1

(Krein charge and energy). The identity has the conserved signed reading

$$Q\left(\omega \right):=\omega \left(\mathbb{1}\right)=I_{+}\left(\omega \right)-I_{-}\left(\omega \right),Q\left({\omega }_{\psi }\right)=[\psi ,\psi ].$$

(238)

When $[\eta ,H]=0$ and $H\ge 0$, the observable $\eta H$ belongs to $\mathcal{C}$ and has nonnegative conserved reading

$$I_{\eta H}\left({\omega }_{\psi }\right)=(\psi ,H\psi ).$$

(239)

Energy intensity and signed energy charge are therefore distinct.

Frequencies.

For a registered family $\mathcal{D}=\left\{A_m\right\}\subset \mathcal{C}$ of mutually exclusive channels with finite nonzero total, define

$$p(m\mid \mathcal{D},\omega )={\displaystyle \frac{I_{A_m}\left(\omega \right)}{{\sum }_n{I}_{A_n}\left(\omega \right)}}.$$

(240)

Then

$$p(m\mid \mathcal{D},\omega )\ge 0,\sum _{m}p(m\mid \mathcal{D},\omega )=1.$$

(241)

Probabilities are normalized ratios of registered intensities. On the Standard Model factor of (214), where ${\eta }_{\mathrm{SM}}=\mathbb{1}$, an exhaustive positive family is available and the ordinary Born rule is recovered.

5. Conclusion

This work proposed least surprisal as a first-principles procedure for constructing physical actions. Given displacement and reference jets, transition surprisal extracts the action through

$$\mathcal{I}=\underset{ϵ\to 0}{lim}{\displaystyle \frac{1}{ϵ}}\mathcal{K}\left({\mathfrak{J}}_{ϵ}\psi ,{\mathfrak{J}}_r\psi \right)=-iS,$$

(242)

where

$$S={\int }_X{\mathrm{d}}^{d}x\sqrt{\left|g\right|}\left[Re{\langle \psi \mid D\psi \rangle }_x+{\displaystyle \frac{s_F}{2}}{∥{\mathbb{D}}^2-{\mathbb{F}}_0∥}_x^2\right].$$

(243)

Thus $e^{-\mathcal{I}}=e^{iS},$ and stationarity of accumulated surprisal gives the equations of motion. Applied to established carriers and references, the same construction recovers the Dirac, Yang–Mills, instanton, Standard Model, and Einstein–Stelle operator sectors.

The transition jets also select their arena. Dimensionless compatibility of their matter and curvature slots requires $d=4$, while positivity of the Dirac density selects Lorentzian signature. For an amplitude parametrized only by spacetime,

$$\psi \left(x\right)\in {Cl}^{+}(T_{x}X,g_x),$$

(244)

accumulated surprisal gives Dirac matter coupled to quadratic gravity.

Our main result extends this construction to the deformation geometries exposed by the commuting left and right actions

$$\psi \left(x\right)⟼S\psi \left(x\right)U,S\in Spin(3,1),U\in U{\left(2\right)}_R.$$

(245)

These actions preserve a Lorentzian form $g_x$ on $T_{x}X$ and a positive Hermitian form $q_x$ on the complex rank-two right module. Their configuration spaces have tangent deformation spaces

$$\begin{array}{cc}\hfill V_z^g& =T_{g_x}{Met}_{1,3,x}^{\mathrm{spin}}\left(X\right)\cong {Sym}^2\left(T_x^{*}X\right),\hfill \end{array}$$

(246)

$$\begin{array}{cc}\hfill V_z^q& =T_{q_x}{Herm}^{+}\left(W_{x,g}\right)\cong Herm\left(W_{x,g}\right).\hfill \end{array}$$

(247)

The first carries the DeWitt form $G_{g_x}$, while the second carries the positive invariant form $K_{q_x}$.

These deformation geometries furnish the deformation carrier

$$\begin{array}{|c|}\hline {\left(\mathcal{E}\right)}_z={Cl}^{+}(T_{x}X,g_x){\otimes }_{\mathbb{R}}{\Sigma }^{\epsilon }(V_z^g,G_{g_x}){\otimes }_{\mathbb{R}}{V}_z^q,\epsilon \in \{+,-\}.\\ \hline\end{array}$$

(248)

The corresponding deformation amplitude

$$\Psi =\Psi (x,g_x,q_x)$$

(249)

carries generalized transport

$${\mathbb{D}}_{\mathrm{tot}}={\mathbb{D}}_x+{\mathbb{D}}_g+{\mathbb{D}}_q+\Phi .$$

(250)

Accumulated surprisal of its transition jets determines the universal deformation action associated with the original amplitude $\psi$.

A physical section

$$s=(g,q):X⟶{\mathcal{C}}_X$$

(251)

pulls this universal theory back to spacetime. Positive polarization, compatible complex structure, and unimodularity reduce the metric-deformation spin symmetry according to

$$Spin(6,4)\stackrel{R_x}{\to }{G}_{\mathrm{PS}}\stackrel{J_x}{\to }U\left(3\right)\times U\left(2\right)\stackrel{\mathrm{unimodularity}}{\to }{G}_{\mathrm{SM}}.$$

(252)

The selected metric-deformation half-spin module thereby supplies the particle content of one Standard Model generation.

The right-Hermitian deformation geometry is already positive, while the complete matter carrier inherits complex linearity from its particle factor. Right-sector unimodularity imposes

$$det{q}_x=1,tr\left(q_x^{-1}\delta q_x\right)=0,$$

(253)

and hence removes the trace deformation. The surviving family deformation space is

$$V_z^{q,1}\cong {Herm}_{0,q_x}\left(W_{x,g}\right),{dim}_{\mathbb{R}}{V}_z^{q,1}=3.$$

(254)

Because the particle carrier $E_{\mathrm{part}}$ is complex and the right-Hermitian factor commutes with the Standard Model action,

$$\begin{array}{|c|}\hline E_{\mathrm{part}}{\otimes }_{\mathbb{R}}{V}_z^{q,1}\cong E_{\mathrm{part}}{\otimes }_{\mathbb{C}}{\mathbb{C}}^3\cong 3E_{\mathrm{part}}.\\ \hline\end{array}$$

(255)

The right-Hermitian deformation sector therefore supplies three gauge-identical generations. General scalar transport on this multiplicity factor supplies the complex Yukawa and Majorana matrices, whose relative diagonalizations give the CKM and PMNS matrices.

The reduced transport1 separates the resulting physical sectors: $D_x$ supplies the Dirac kinetic terms, while ${\mathbb{D}}_x^2$ supplies the gravitational terms; $D_g$ supplies particle transport, while ${\mathbb{D}}_g^2$ supplies the Yang–Mills terms; $D_q$ supplies family transport; and $\Phi$, its covariant derivatives, and its mixed curvatures supply the Higgs, Yukawa, Majorana, and nonminimal scalar couplings. Accumulated surprisal of the pulled-back and reduced jets therefore gives the three-generation Standard Model coupled to quadratic gravity, including

$$C_{\mu \nu \rho \sigma }{C}^{\mu \nu \rho \sigma },R^2,E_4,M_{\mathrm{P}}^{2}R,\Lambda .$$

(256)

The resulting four-dimensional action is local and power-counting renormalizable.

Quadratic gravity retains an opposite-residue massive spin-two sector, represented kinematically by a Krein space. The resulting carrier is

$$K_{universe}={\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}\widehat{\otimes }{K}_{\mathrm{QG}},{\eta }_{universe}={\mathbb{1}}_{\mathrm{SM}}^{\left(3\right)}\otimes {\eta }_{\mathrm{QG}}.$$

(257)

It places the observer externally to the particle physics sector (i.e., ${\mathcal{H}}_{\mathrm{SM}}^{\left(3\right)}$ is a Hilbert space) and internally to the universe (i.e., $K_{\mathrm{QG}}$ and by extension $K_{universe}$ are Krein spaces). The intensity-positive measurement theory developed here assigns nonnegative detector readings to this Hilbert–Krein carrier while retaining its conserved signed structure.

The construction consequently closes as

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \mathrm{least}\mathrm{surprisal}& ⟶\mathrm{Lorentzian}3+1\mathrm{spacetime}\hfill \\ \hfill & ⟶\mathrm{Dirac}--\mathrm{Hestenes}\mathrm{amplitude}\hfill \\ \hfill & ⟶\mathrm{left}\mathrm{and}\mathrm{right}\mathrm{deformation}\mathrm{geometries}\hfill \\ \hfill & ⟶\mathrm{canonical}\mathrm{deformation}\mathrm{carrier}\mathrm{and}\mathrm{action}\hfill \\ \hfill & \stackrel{\mathrm{pullback},R_x}{\to }\mathrm{four}-\mathrm{generation}\mathrm{Pati}-\mathrm{Salam}+\mathrm{quadratic}\mathrm{gravity}.\hfill \\ \hfill & \stackrel{J_x,\mathrm{unimodularity}}{\to }\mathrm{three}-\mathrm{generation}\mathrm{Standard}\mathrm{Model}+\mathrm{quadratic}\mathrm{gravity}.\hfill \end{array}\\ \hline\end{array}$$

(258)

The arena supplies the Dirac–Hestenes carrier, the carrier exposes its native deformation geometries, and accumulated surprisal of their transition jets supplies the action governing spacetime and its contents. Least surprisal thereby provides a first-principles route to a candidate matter–gravity parent theory whose physically reduced realization is the three-generation Standard Model coupled to quadratic gravity.