Proposition 2. *For all $`\psi\in\Lambda`$,

\psi\in P \iff \psi\in\mathcal F(P) \iff C(P,\psi).
```*

</div>

Thus $`P`$ "explains" its members in the following limited but precise
sense:
``` math
\psi\in P \;\Rightarrow\; C(P,\psi).

For a law to “contain its own truthmaker” in the same formulation as the law itself, let $`L`$ be a fixed formal language and let $`\Lambda\subseteq\mathbb N`$ be a set of Gödel codes for $`L`$-sentences. Let $`p\in\Lambda`$ be the code of a sentence that expresses the meta-principle internally by quantifying over codes. Schematically, we may write

p \;\equiv\; \forall n\in\Lambda\;\Big(C(P,n)\ \Rightarrow\ \mathsf{Obtain}(n)\Big),

where $`\mathsf{Obtain}(n)`$ is read as “the law coded by $`n`$ obtains” (i.e. is among the selected/actualised laws).⁶

The “principle includes itself” statement is now the typed membership $`p\in P`$ (a sentence in a set of sentences), not $`P\in P`$ (a set a member of itself). Thus, the Foundation problem does not arise.

Our construction used $`\Psi=\mathcal P(\Lambda)`$, as a set of Gödel codes of candidate law-sentences, but the underlying pattern is more general. Let $`(T,\le)`$ be a partially ordered set of candidate theories (or “law-sets”). Intuitively, $`S\le T`$ means “$`T`$ has at least as much content/commitment/information as $`S`$”.

To model a self-subsuming principle, we need a way to say when an item (law, constraint, statement, datum) is admissible relative to a candidate theory.

D \;\subseteq\; T\times U,
\qquad
D(S,u)\ \text{read: "$u$ is admissible/endorsed given $S$"}.

When $`T`$ is a space of law-sets (e.g. $`T=\mathcal P(\Lambda)`$), one typically identifies $`U`$ with the same kind of objects (e.g. $`U=\Lambda`$) and regards $`S`$ as endorsing exactly those $`u`$ satisfying $`D(S,u)`$.⁷ This motivates the associated operator:

\mathcal F_D:T\to T,
\qquad
\mathcal F_D(S):=\{u\in U: D(S,u)\}.

\forall u\in U\;\;\big(u\in P \iff D(P,u)\big)

u\in P \iff D(P,u)
\quad\Longleftrightarrow\quad
u\in P \iff u\in \mathcal F_D(P),

Proposition 4 says any attempt to characterise a fundamental law-set $`P`$ by a membership rule that itself refers to $`P`$ is exactly a fixed point equation. Our $`C(P,\psi)`$ is one instance with $`D=C`$ and $`U=\Lambda`$. No monotonicity or lattice assumptions are required for Proposition 4 to hold.

By contrast, the existence of fixed points and the availability of a canonical choice, such as the least fixed point $`\mu \mathcal F`$, do require additional hypotheses. In the closure-style regime considered here, we assume $`\mathcal F`$ is monotone on a complete lattice, so that Tarski guarantees fixed points and yields distinguished extremal solutions. Non-monotone admissibility notions (for instance consistency or optimality constraints) may still admit fixed points but fall outside the scope of the Tarski canonicity argument. For this reason, to get the existence of fixed points systematically, one typically assumes a lattice-theoretic structure and monotonicity. At the beginning of this section, we have a complete lattice $`\Psi=\mathcal P(\Lambda)`$ and (by Assumption 1) a monotone operator $`\mathcal F`$. Hence, Tarski applies and yields least and greatest fixed points; in particular $`P=\mu\mathcal F`$ is well-defined.

Importantly, Tarski-style existence does not require uniqueness. In fact, the general conclusion is the opposite: Fixed points can form a rich lattice. If our aim is to analyse where such assumptions lead, this is a feature:

1. different fixed points correspond to different admissible global law-sets satisfying the same meta-criterion,⁸

2. the least and greatest fixed points give canonical extremal solutions,

3. studying the lattice of fixed points becomes a principled way to study the space of possible “fundamental” packages consistent with a given admissibility concept.

So the fixed point framework is naturally a constraint-analysis engine. The genuinely structural content is:

\text{(self-subsumption schema)} \quad\Rightarrow\quad \text{(fixed point equation)}.

Choosing which fixed point (least, greatest, or another) is an extra selection principle. Our choice of $`\mu \mathcal F`$ corresponds to a minimality stance.

The equation $`P=\mathcal F(P)`$ is still a circular characterisation. The fixed point method does not erase circularity; it replaces it with a non-vicious circularity⁹ resolved by a selection principle, that is, “choose the least fixed point $`P=\mu\mathcal F`$”.

Given $`\mathcal F`$, $`P=\mu\mathcal F`$ is canonical. But the choice of $`\mathcal F`$ (equivalently, of the admissibility criterion $`C`$) is a remaining datum. In another light, such a choice would not say much about the structural content of the theory, and would work as a choice of gauge.

Physical Instantiation

The fixed point normal form is a meta-constraint template. To apply it to physics, fix (i) a candidate domain of laws $`\Sigma`$ and (ii) a lattice of theory-packages $`\mathcal T`$ (such as $`\mathcal P(\Sigma)`$, or as in the Examples section below, lattice of linear subspaces when $`\Sigma`$ is a vector space) together with an admissibility operator $`\mathcal F:\mathcal T\to\mathcal T`$ encoding the chosen constraints. A package counts as admissible precisely when it is closed:

P=\mathcal F(P).

In physics, certain requirements function less like contingent dynamical laws and more like constitutive constraints. Examples include unitarity, symmetry principles, locality/causality constraints, renormalisability, etc. A natural reading of our framework is:

A package counts as a physical theory only if it is closed under the implications of these admissibility constraints.

Thus, $`\mathcal F(S)`$ returns the admissible completion determined by the constraints extracted from $`S`$.¹⁰ As an intuitive example, let $`\mathcal T`$ be a space of partial “theory data” ordered by “information content”. Define $`\mathcal F(S)`$ to be the admissible completion operator of $`S`$ under a certain set of admissibility constraints, which ought to be chosen by the physicist according to the subject in question. Then, fixed points are solutions that give canonical admissible packages; that is, $`S`$ is admissible iff $`\mathcal F(S)=S`$.

In this reading, our framework organises the landscape such that the fixed points are the admissible theories, and extremal solutions ($`\mu \mathcal F`$, $`\nu \mathcal F`$) give canonical minimal structures.

Therefore, one can state a “metaprinciple” as follows:

Meta-principle: Any proposal in which “the laws of nature” are characterised as exactly those statements satisfying an admissibility criterion that itself depends on the totality of laws has, in general, fixed point form $`P=\mathcal F(P)`$, with a canonical minimal choice, $`\mu \mathcal F`$.

A General Logical Architecture

While the logical schema outlined guarantees fixed points for any monotone operator, we must identify a physically motivated operator that satisfies this property. As noted previously, simple consistency checks are often non-monotone and metatheoretically complex. In this section, we propose that the interplay between Dynamical Laws and Symmetry Groups provides the robust, monotonic structure required to ground self-subsuming principles.

Let us define two abstract domains of discourse for a physical theory:

Next, we posit a binary relation $`R\subseteq \Sigma\times \Gamma`$, where $`\phi\,R\,G`$ means that the law $`\phi`$ respects the symmetry group $`G`$.¹² This relation induces two canonical mappings:

\operatorname{Inv}(G)\;:=\;\{\phi\in \Sigma:\ \phi\,R\,G\},

\operatorname{Sym}(S)\;:=\;\bigvee \{\,G\in\Gamma:\ S\subseteq \operatorname{Inv}(G)\,\},

With these definitions, one has the Galois correspondence

S\subseteq \operatorname{Inv}(G)\quad\Longleftrightarrow\quad G\subseteq \operatorname{Sym}(S),

so $`\operatorname{Inv}`$ and $`\operatorname{Sym}`$ form an antitone Galois connection between packages and symmetry groups. In particular, $`\operatorname{Inv}`$ is antitone and $`\operatorname{Sym}`$ is antitone, hence their composition is monotone. The logic follows directly from the reversal of the inclusion order twice:

\begin{equation}
    S_1 \subseteq S_2 \implies \operatorname{Sym}(S_2) \subseteq \operatorname{Sym}(S_1) \implies \operatorname{Inv}(\operatorname{Sym}(S_1)) \subseteq \operatorname{Inv}(\operatorname{Sym}(S_2)).
\end{equation}

We then obtain a canonical invariance-based completion operator

\begin{equation}
\label{eq:invariance-closure}
\mathcal F(S)\;:=\;\operatorname{Inv}(\operatorname{Sym}(S)).
\end{equation}

This operator is monotone on $`\mathcal T`$, so by Tarski’s theorem it admits fixed points. Fixed points $`P=\mathcal F(P)`$ are precisely packages that are symmetry-closed relative to the chosen candidate domain and admissible symmetry family.

More general admissibility operators may be built by composing $`\operatorname{Sym}`$ and $`\operatorname{Inv}`$ with additional monotone meta-maps, such as the localisation map on $`\Gamma`$, encoding a gauge principle in the QED example below. Such operators remain monotone but need not satisfy $`S\subseteq \mathcal F(S)`$, so a seed may select an admissible completion without being contained in it.

Cases with More Structural Constraints

It is important to emphasise that the specific Law–Symmetry duality need not be the only structural constraint on a physical theory. In practice, mature theories must satisfy multiple independent requirements, such as symmetry (gauge, Lorentz, diffeomorphism, etc.), causality, and unitarity, simultaneously. A strength of the fixed point schema is that such requirements can be composed into a single admissibility operator.

In the abstract discussion, the package lattice $`\mathcal T`$ need not be a powerset $`\mathcal P(\Sigma)`$. In the Examples section, for instance, $`\Sigma`$ is a vector space of candidate Lagrangian terms and $`\mathcal T`$ is taken to be the complete lattice of linear subspaces of $`\Sigma`$. In such cases, an operator defined naively by

S\longmapsto \{\psi\in\Sigma:\ C(S,\psi)\}

need not land in $`\mathcal T`$. To define admissibility operators uniformly on an arbitrary package lattice $`\mathcal T`$, we introduce a package closure map that projects subsets of $`\Sigma`$ to the least package containing them.

\langle X\rangle_{\mathcal T}\;:=\;\bigwedge\{\,P\in\mathcal T:\ X\subseteq P\,\}\in\mathcal T,
\qquad X\subseteq \Sigma.

If $`\mathcal T=\mathcal P(\Sigma)`$ then $`\langle X\rangle_{\mathcal T}=X`$. If $`\Sigma`$ is a vector space and $`\mathcal T`$ is the lattice of linear subspaces, then $`\langle X\rangle_{\mathcal T}=\mathrm{span}(X)`$.

Each physical principle may be encoded as a monotone operator

\mathcal F_i:\mathcal T\to\mathcal T,

for instance via a predicate $`C_i(S,\psi)`$ by setting

\mathcal F_i(S)\;:=\;\Big\langle \{\psi\in\Sigma:\ C_i(S,\psi)\}\Big\rangle_{\mathcal T}.

Given a family of monotone operators $`\{\mathcal F_i\}_i`$, define their combined operator by the meet in $`\mathcal T`$:

\mathcal F_{\mathrm{tot}}(S)\;:=\;\bigwedge_i \mathcal F_i(S).

In the examples, $`\mathcal T`$ is closed under intersections, so this meet is simply $`\bigcap_i \mathcal F_i(S)`$.

Thus, on a complete lattice of theory-packages $`\mathcal T`$, Tarski’s theorem applies to $`\mathcal F_{\mathrm{tot}}`$. In particular, $`\mathcal F_{\mathrm{tot}}`$ admits fixed points, and its least fixed point $`\mu\mathcal F_{\mathrm{tot}}`$ gives a canonical minimal package stable under all encoded principles.

This shows that the fixed point view is not tied to any single physical hypothesis, but provides a general logical architecture: Once a domain of candidates $`\Sigma`$ and admissibility constraints are specified, “the theory” is a structure in the lattice of theory-packages. Additional principles, known or yet to be discovered, correspond to refining $`\mathcal F_{\mathrm{tot}}`$, which updates the resulting stable packages accordingly. Of course, choosing or discovering such criteria is itself a philosophical problem to be discussed, which will not be attempted here.

Fixed points of the operator $`P=\mathcal F(P)`$ may be called closed theory-packages relative to $`\Sigma`$. The laws in $`P`$ determine a symmetry content $`\operatorname{Sym}(P)`$, and $`\operatorname{Inv}(\operatorname{Sym}(P))`$ returns the set of laws compatible with that symmetry content within the chosen candidate domain. Thus, $`P`$ contains no terms that break its admitted symmetries, and it is complete with respect to the chosen convention. Tarski’s theorem guarantees a least fixed point, which is the minimal closed package relative to $`\mathcal F`$ (and to $`\Sigma`$).

In summary, the Galois connection provides a mathematically well-behaved admissibility operator built from invariance, avoiding the metatheoretic difficulties of treating raw “consistency checking” as the primary admissibility filter.

Examples

In the physics examples below, we show that once one fixes a natural candidate domain and encodes standard physical constraints (symmetry, locality, and a truncation criterion) as a monotone completion operator, familiar theories such as QED and GR can be presented as fixed points of that operator. For more information on the subjects, the reader is kindly directed to .

I. Quantum Electrodynamics

Operator domain.

Fix four-dimensional Minkowski spacetime.¹³ Let $`\Sigma_{\le 4}`$ denote the vector space of local polynomial Lagrangian densities built from a Dirac spinor $`\psi(x)`$ and a vector potential $`A_\mu(x)`$, and finitely many derivatives, subject to:

1. a power-counting bound of dimension $`\le 4`$ ;

2. an equivalence relation $`\sim`$ identifying densities that differ by a total derivative and by standard “inessential” redundancies caused by integration by parts.

We write $`[\mathcal O]\in \Sigma_{\le 4}`$ for the equivalence class of a density $`\mathcal O`$. This implements the idea that physics is insensitive to Lagrangian terms differing by boundary terms.

Theory packages.

A theory package $`S`$ will be represented as a linear subspace $`S\subseteq \Sigma_{\le 4}`$, spanned by a chosen generating set of operator classes. In particular, if $`[\mathcal O_1],[\mathcal O_2]\in S`$, then $`a[\mathcal O_1]+b[\mathcal O_2]\in S`$ for all scalars $`a,b`$, where the linear closure is treated as background structure instead of a part of the admissibility operator.¹⁴

Symmetry space.

Let $`\Gamma`$ be a poset of symmetry groups acting on the fields, ordered by inclusion. Since we fix Minkowski spacetime and restrict attention to Lorentz-scalar densities, spacetime symmetries are treated as background constraints and omitted from $`\Gamma`$ (although it is trivial to incorporate them, as well as the discrete symmetries).¹⁵ Since $`\psi`$ carries no additional flavour or internal index, the relevant internal symmetry transformations are constant, non-spacetime field redefinitions $`\psi\mapsto U\psi`$ that preserve the free Dirac density $`\bar\psi(i\gamma^\mu\partial_\mu-m)\psi`$. Preservation of the mass term requires $`\bar\psi\psi\mapsto \bar\psi\psi`$, and preservation of the kinetic term requires $`\bar\psi\gamma^\mu\partial_\mu\psi\mapsto \bar\psi\gamma^\mu\partial_\mu\psi`$. These two requirements imply that $`U`$ commutes with the Dirac bilinear structure, hence $`U`$ can only act as multiplication by a constant phase. Imposing unitarity restricts this to $`U=e^{i\theta}`$, so the continuous internal symmetry group is $`U(1)_{\mathrm{glob}}`$. The free theory has a global symmetry, $`U(1)_{\mathrm{glob}}`$, whereas with the addition of the gauge principle as one of the criteria, we also have a local one, $`U(1)_{\mathrm{loc}}`$. Therefore,

\Gamma \;:=\;\{\,\mathbf{1},\,U(1)_{\mathrm{glob}},\,U(1)_{\mathrm{loc}}\,\},
\qquad
\mathbf{1}\subset U(1)_{\mathrm{glob}} \subset U(1)_{\mathrm{loc}}.

Hence $`\Gamma`$ has a greatest element given by

\top_\Gamma \;=\; U(1)_{\mathrm{loc}}.

Invariant map.

For $`G\in\Gamma`$, define

\operatorname{Inv}(G)\;:=\;\{\, [\mathcal O]\in \Sigma_{\le 4} : [\mathcal O]\ \text{is invariant under}\ G \,\}.

Here, invariance means that $`[\mathcal O]`$ is invariant if for one (equivalently any) representative $`\mathcal{O}`$, the transformed density differs from $`\mathcal O`$ by $`\sim`$.

Symmetry map.

For a package $`S\subseteq \Sigma_{\le 4}`$, define

\operatorname{Sym}(S)\;:=\;\max\{\,G\in\Gamma:\ S\subseteq \operatorname{Inv}(G)\,\}.

Since $`U(1)_{\mathrm{glob}}\subset U(1)_{\mathrm{loc}}`$, this maximum exists.¹⁶

Galois correspondence.

With the above definitions one has the standard antitone correspondence:

S\subseteq \operatorname{Inv}(G)\quad\Longleftrightarrow\quad G\subseteq \operatorname{Sym}(S),

so $`\operatorname{Inv}`$ and $`\operatorname{Sym}`$ form a Galois connection between operator packages and symmetry groups.

The step from global to local symmetry is not obtained automatically from $`\operatorname{Sym}(S)`$; it represents an additional physical principle, the gauge principle, encoded here as a map on symmetry groups.

Localisation operator.

Introduce an order-preserving map, $`\operatorname{Loc}:\Gamma\to\Gamma`$, as

\operatorname{Loc}(\mathbf{1})=\mathbf{1},\qquad
\operatorname{Loc}\big(U(1)_{\mathrm{glob}}\big)=U(1)_{\mathrm{loc}},\qquad
\operatorname{Loc}\big(U(1)_{\mathrm{loc}}\big)=U(1)_{\mathrm{loc}}.

It should be noted that, in this example, $`\operatorname{Loc}`$ fixes the top element,

\operatorname{Loc}(\top_\Gamma)=\top_\Gamma,

reflecting the idea that once the symmetry is already local, localisation does nothing further. An element $`\alpha(x) \in U(1)_\mathrm{loc}`$ acts by

\begin{equation}
\label{eq:u1loc}
\psi(x)\mapsto e^{i\alpha(x)}\psi(x),
\;
\bar\psi(x)\mapsto \bar\psi(x)e^{-i\alpha(x)},
\;
A_\mu(x)\mapsto A_\mu(x)-\frac{1}{e}\partial_\mu\alpha(x),
\end{equation}

where the coupling $`e`$ is fixed so that $`D_\mu=\partial_\mu+ieA_\mu`$ transforms covariantly.

The localisation map is not itself one of the admissibility predicates $`C_i`$, nor one of the induced package-operators $`\mathcal F_i`$, because it does not act on theory packages $`S`$ or on candidate laws $`\psi`$. Rather, $`\operatorname{Loc}`$ is part of the background structure used to define an admissibility principle associated with local gauge symmetry. Concretely, one may introduce a predicate

C_{\mathrm{gauge}}(S,[\mathcal O])\;:=\;\Big([\mathcal O]\in \operatorname{Inv}\big(\operatorname{Loc}(\operatorname{Sym}(S))\big)\Big),

which reads “relative to the background package $`S`$, the candidate operator $`[\mathcal O]`$ is admissible because it is invariant under the localized symmetry extracted from $`S`$.” The corresponding induced operator is then

\mathcal F_{\mathrm{gauge}}(S)\;:=\;\{[\mathcal O]\in \Sigma_{\le 4}:\ C_{\mathrm{gauge}}(S,[\mathcal O])\},

so $`\operatorname{Loc}`$ enters the schema via the definitional content of $`C_{\mathrm{gauge}}`$ and hence of $`\mathcal F_{\mathrm{gauge}}`$. In the present example, we fold this into the composite operator $`\mathcal F_{\mathrm{QED}}(S)=\operatorname{Inv}(\operatorname{Loc}(\operatorname{Sym}(S)))`$.

Total admissibility operator.

Define the operator on packages

\begin{equation}
\label{eq:Fqed}
\mathcal F_{\mathrm{QED}}(S)\;:=\;\operatorname{Inv}\!\big(\operatorname{Loc}(\operatorname{Sym}(S))\big)\ \subseteq\ \Sigma_{\le 4}.
\end{equation}

By construction, $`\mathcal F_{\mathrm{QED}}`$ is monotone with respect to $`\subseteq`$. Intuitively, this operator (i) diagnoses the symmetries of $`S`$, (ii) enforces their localisation, and (iii) outputs the theory of renormalisable operators compatible with this constraint.

The computation from a free matter seed

Let the seed package be the span of the free Dirac kinetic term:

P_0\;:=\;\mathrm{span}\Big\{\big[\bar\psi\, i\gamma^\mu\partial_\mu\psi\big]\Big\}\ \subseteq\ \Sigma_{\le 4}.

Under [eq:u1loc] one has

\partial_\mu(e^{i\alpha(x)}\psi)
=
e^{i\alpha(x)}\big(\partial_\mu\psi+i(\partial_\mu\alpha)\psi\big),

so the free term $`\bar\psi i\gamma^\mu\partial_\mu\psi`$ acquires an extra contribution proportional to $`\partial_\mu\alpha`$. Hence it is invariant for $`\alpha=\text{const}`$ but not for general $`\alpha(x)`$, and therefore

\operatorname{Sym}(P_0)\;=\;U(1)_{\mathrm{glob}}.

To incorporate gauge principle, apply $`\operatorname{Loc}`$:

\operatorname{Loc}(\operatorname{Sym}(P_0))\;=\;\operatorname{Loc}\big(U(1)_{\mathrm{glob}}\big)\;=\;U(1)_{\mathrm{loc}}.

Now, to compute $`\operatorname{Inv}(U(1)_{\mathrm{loc}})\cap \Sigma_{\le 4}`$, we list the renormalisable invariant operator classes:

1. Fermion mass. The class $`\big[\bar\psi\psi\big]`$ is invariant since $`e^{-i\alpha}e^{i\alpha}=1`$:

   MATH

   Collapse Copy

   \bar\psi\psi \mapsto \bar\psi e^{-i\alpha} e^{i\alpha}\psi=\bar\psi\psi.

   Click to expand and view more

2. Photon mass A Proca term $`\big[A_\mu A^\mu\big]`$ is not invariant under $`A_\mu\mapsto A_\mu-\frac{1}{e}\partial_\mu\alpha`$, so $`\big[A_\mu A^\mu\big]\notin \operatorname{Inv}(U(1)_{\mathrm{loc}})`$. Therefore, the photon mass is excluded.

3. Minimal coupling. Define the covariant derivative

   MATH

   Collapse Copy

   D_\mu := \partial_\mu + ieA_\mu.

   Click to expand and view more

   Using [eq:u1loc],

   MATH

   Collapse Copy

   D_\mu(e^{i\alpha}\psi)
   =
   (\partial_\mu+ieA_\mu)\,e^{i\alpha}\psi
   =
   e^{i\alpha}\big(\partial_\mu+ie(A_\mu-\tfrac{1}{e}\partial_\mu\alpha)\big)\psi
   =
   e^{i\alpha}D_\mu\psi,

   Click to expand and view more

   so $`D_\mu\psi`$ transforms like $`\psi`$. Hence the class $`\big[\bar\psi\, i\gamma^\mu D_\mu\psi\big]`$ is invariant. Expanding it, we have the familiar interaction term,

   MATH

   Collapse Copy

   \bar\psi\, i\gamma^\mu D_\mu\psi
   =
   \bar\psi\, i\gamma^\mu \partial_\mu\psi
   \;-\;
   e\,\bar\psi\gamma^\mu A_\mu\psi.

   Click to expand and view more

4. Gauge-field kinetic term. The field strength

   MATH

   Collapse Copy

   F_{\mu\nu}:=\partial_\mu A_\nu-\partial_\nu A_\mu

   Click to expand and view more

   is invariant because the shift piece cancels by commutativity of partial derivatives as

   MATH

   Collapse Copy

   \partial_\mu(A_\nu-\tfrac{1}{e}\partial_\nu\alpha)-\partial_\nu(A_\mu-\tfrac{1}{e}\partial_\mu\alpha)
   =
   \partial_\mu A_\nu-\partial_\nu A_\mu.

   Click to expand and view more

   Therefore $`\big[-\tfrac14 F_{\mu\nu}F^{\mu\nu}\big]\in \operatorname{Inv}(U(1)_{\mathrm{loc}})`$.

5. No further renormalisable invariants up to equivalence. Dimension-$`5`$ operators are excluded by the $`\le 4`$ truncation. A term $`F_{\mu\nu}\tilde F^{\mu\nu}`$ is a total derivative in the abelian case and is removed by the equivalence relation. Gauge-fixing terms are not in $`\operatorname{Inv}(U(1)_{\mathrm{loc}})`$; they enter only at the quantisation stage.

The package.

The image of the seed under $`\mathcal F_{\mathrm{QED}}`$ is

\mathcal F_{\mathrm{QED}}(P_0)=\operatorname{Inv}(U(1)_{\mathrm{loc}})\subseteq \Sigma_{\le 4},

whose minimal generating set may be taken as

\begin{equation}
\label{eq:Pqed}
P_{\mathrm{QED}}
=
\mathrm{span}\Big\{
\big[\bar\psi\, i\gamma^\mu D_\mu\psi\big],\;
\big[\bar\psi\psi\big],\;
\big[-\tfrac14 F_{\mu\nu}F^{\mu\nu}\big]
\Big\}.
\end{equation}

In Lagrangian form, this is exactly QED:

\mathcal L_{\mathrm{QED}}
=
\bar\psi(i\gamma^\mu D_\mu-m)\psi
-\frac14 F_{\mu\nu}F^{\mu\nu}.

Fixed point property.

The package $`P_{\mathrm{QED}}`$ is stable under the admissibility operator:

\mathcal F_{\mathrm{QED}}(P_{\mathrm{QED}})=P_{\mathrm{QED}},

since $`\operatorname{Sym}(P_{\mathrm{QED}})=U(1)_{\mathrm{loc}}`$, and $`P_{\mathrm{QED}}`$ contains precisely the renormalisable invariants under that group within $`\Sigma_{\le 4}`$.

P_{\mathrm{QED}}=\operatorname{Inv}(U(1)_{\mathrm{loc}})\subseteq \Sigma_{\le 4}

\mathcal F_{\mathrm{QED}}(\bot_\Sigma)
=
\operatorname{Inv}(\operatorname{Loc}(\operatorname{Sym}(\bot_\Sigma)))
=
\operatorname{Inv}(U(1)_{\mathrm{loc}})
=
P_{\mathrm{QED}}.

\mathcal F_{\mathrm{QED}}(P_{\mathrm{QED}})
=
\operatorname{Inv}(\operatorname{Loc}(\operatorname{Sym}(P_{\mathrm{QED}})))
=
\operatorname{Inv}(U(1)_{\mathrm{loc}})
=
P_{\mathrm{QED}},

The “matter seed” $`P_0=\mathrm{span}\{[\bar\psi i\gamma^\mu\partial_\mu\psi]\}`$ is not itself a gauge-invariant operator class, so it need not satisfy $`P_0\subseteq P_{\mathrm{QED}}`$. What the construction shows is rather that applying the completion operator $`\mathcal F_{\mathrm{QED}}`$ to $`P_0`$ produces the least fixed point in one step:

\mathcal F_{\mathrm{QED}}(P_0)=\operatorname{Inv}(U(1)_{\mathrm{loc}})=P_{\mathrm{QED}}=\mu \mathcal F_{\mathrm{QED}}.

Thus the example exhibits QED as the canonical least fixed point selected by the monotone operator $`\mathcal F_{\mathrm{QED}}`$ on the package lattice.

Thus, it is shown that QED, in this form, can be presented as a least fixed point schema under the composed metalanguage assumptions of symmetry, locality in the sense of gauge principle, and closedness under local invariants within the renormalisable operator domain.

II. General Relativity

The QED example showed how a local internal symmetry selects interaction terms in a renormalisable operator domain under the chosen admissibility operator. General Relativity provides an analogous illustration. Once one fixes “metric geometry” as the kinematical arena and imposes diffeomorphism invariance together with locality and a derivative-order restriction, the Einstein–Hilbert action can be presented as a minimal stable package.

Geometric background.

Let $`M`$ be a smooth, connected, oriented $`4`$-manifold. The dynamical field is a Lorentzian metric $`g_{\mu\nu}`$ on $`M`$. Diffeomorphisms $`\phi\in \operatorname{Diff}(M)`$ act on $`g`$ by pullback:

g \longmapsto \phi^\ast g.

Candidate domain.

Let $`\Sigma_{\le 2}`$ be the vector space of local scalar densities built from $`g_{\mu\nu}`$ and finitely many derivatives, restricted to at most two derivatives of the metric,¹⁷ modulo the standard equivalence relation $`\sim`$ identifying densities that differ by a total derivative,

\mathcal L \sim \mathcal L + \nabla_\mu V^\mu.

We write $`[\mathcal L]\in \Sigma_{\le 2}`$ for an equivalence class. The corresponding action functional is $`S[g]=\int_M d^4x\, [\mathcal L]`$ under appropriate assumptions so that boundary terms can be ignored.

Theory packages.

A theory package is represented by a linear subspace $`S\subseteq \Sigma_{\le 2}`$ spanned by a chosen set of density-classes. Linearity is again treated as background structure, so closure under linear combinations does not need to be enforced by the admissibility operator.

Symmetry space.

For the GR example, the maximal group acting on the metric field is the group of diffeomorphisms of $`M`$, $`\operatorname{Diff}(M)`$. Therefore, let $`\Gamma`$ be

\begin{equation}
    \Gamma := \{\mathbf{1}, \operatorname{Diff}(M) \}.
\end{equation}

So the “largest” symmetry allowed by the chosen framework is diffeomorphism invariance.

Invariant map.

Define

\operatorname{Inv}(G)\;:=\;\{\, [\mathcal L]\in\Sigma_{\le 2}\ :\ [\mathcal L]\ \text{is invariant under}\ G\,\},

where invariance means $`\mathcal L(g)`$ and $`\mathcal L(\phi^\ast g)`$ differ by a total derivative, and hence lie in the same equivalence class for all $`\phi\in G`$.

Symmetry map.

For a package $`S\subseteq\Sigma_{\le 2}`$ define

\operatorname{Sym}(S)\;:=\;\max\{\,G\in\Gamma:\ S\subseteq \operatorname{Inv}(G)\,\}.

Since $`\Gamma=\{\mathbf 1,\operatorname{Diff}(M)\}`$ is a finite chain, this maximum exists.

With these definitions one has the standard antitone correspondence

S\subseteq \operatorname{Inv}(G)\quad\Longleftrightarrow\quad G\subseteq \operatorname{Sym}(S),

so $`\operatorname{Inv}`$ and $`\operatorname{Sym}`$ form a Galois connection between operator packages and symmetry groups.

Admissibility operator.

Define the GR admissibility operator as

\begin{equation}
\label{eq:Fgr}
\mathcal F_{\mathrm{GR}}(S)\;:=\;\operatorname{Inv}\!\big(\operatorname{Sym}(S)\big)\ \subseteq\ \Sigma_{\le 2}.
\end{equation}

This operator is monotone with respect to $`\subseteq`$ by general properties of the Galois correspondence.

Seed.

The “seed” is the kinematical commitment to a metric field, together with no further dynamical density-terms:

P_0 := \{0\}\ \subseteq\ \Sigma_{\le 2}.

Because $`P_0`$ contains no nontrivial densities, every diffeomorphism vacuously preserves all elements of $`P_0`$, hence

\operatorname{Sym}(P_0)=\operatorname{Diff}(M).

Therefore

\mathcal F_{\mathrm{GR}}(P_0)=\operatorname{Inv}(\operatorname{Diff}(M)),

so computing the image reduces to classifying the diffeomorphism-invariant density-classes in $`\Sigma_{\le 2}`$.

A key geometric fact is that diffeomorphism-invariant local scalar densities built from the metric are constructed from curvature tensors and their covariant derivatives, rather than from raw coordinate derivatives $`\partial g`$. In particular, curvature starts at second derivative order, so there are no genuinely invariant first-derivative scalar densities built only from $`g`$ and $`\partial g`$.

Within the truncation $`\Sigma_{\le 2}`$ this yields:

1. Order 0. The only scalar density available is the volume form:

   MATH

   Collapse Copy

   [\sqrt{-g}]\ \in\ \Sigma_{\le 2}.

   Click to expand and view more

   Thus the associated action is the cosmological term

   MATH

   Collapse Copy

   S_\Lambda[g] := \int_M d^4x\, \sqrt{-g}.

   Click to expand and view more

2. Order 1. Formally, $`\partial g`$ is not a tensor, so no coordinate-invariant scalar density can be formed from $`g`$ and $`\partial g`$ alone. Intuitively, one may always choose locally inertial coordinates at a point, so that $`\partial_\lambda g_{\mu\nu}=0`$ there.

3. Order 2. At second derivative order the curvature tensor appears. The only non-trivial scalar density, up to normalisation and equivalence of terms with extra total derivatives, is generated by the Ricci scalar $`R`$:

   MATH

   Collapse Copy

   [\sqrt{-g}\,R]\ \in\ \Sigma_{\le 2},
   \qquad
   S_{\mathrm{EH}}[g]:=\int_M d^4x\, \sqrt{-g}\, R.

   Click to expand and view more

   Any other curvature scalar quadratic in curvature (e.g. $`R^2`$, $`R_{\mu\nu}R^{\mu\nu}`$) lies beyond the $`\le 2`$-derivative truncation because it contains four derivatives of the metric in total.

Therefore, within $`\Sigma_{\le 2}`$,

\operatorname{Inv}(\operatorname{Diff}(M))
=
\mathrm{span}\Big\{\, [\sqrt{-g}],\ [\sqrt{-g}\,R]\,\Big\}.

Total package.

Define the GR package

\begin{equation}
\label{eq:Pgr}
P_{\mathrm{GR}}
:=
\mathrm{span}\Big\{\, [\sqrt{-g}],\ [\sqrt{-g}\,R]\,\Big\}
\ \subseteq\ \Sigma_{\le 2}.
\end{equation}

Equivalently, at the level of actions one may parameterise elements of $`P_{\mathrm{GR}}`$ by coefficients, as

S[g]
=
\int_M d^4x\, \sqrt{-g}\,\big(\alpha R - 2\beta\big),

with $`\alpha,\beta`$ constants to be identified empirically with $`1/(16\pi G)`$ and $`\Lambda/(8\pi G)`$ after normalisation.

Fixed point property.

By construction, $`P_{\mathrm{GR}}`$ is stable:

\mathcal F_{\mathrm{GR}}(P_{\mathrm{GR}})=P_{\mathrm{GR}},

because $`\operatorname{Sym}(P_{\mathrm{GR}})=\operatorname{Diff}(M)`$ and $`\operatorname{Inv}(\operatorname{Diff}(M))\cap\Sigma_{\le 2}=P_{\mathrm{GR}}`$.

Least fixed pointhood.

Since $`\mathcal F_{\mathrm{GR}}`$ is monotone on a complete lattice of packages (by the Galois correspondence), Tarski’s theorem guarantees a least fixed point. In this example we compute directly that $`\mathcal F_{\mathrm{GR}}(P_0)=P_{\mathrm{GR}}`$, and $`P_{\mathrm{GR}}`$ is a fixed point, hence it is the least fixed point above $`P_0`$.

Hence, relative to the metric kinematics, diffeomorphism invariance, locality, and the two-derivative truncation, the Einstein–Hilbert plus cosmological term is the minimal stable residue. Therefore, GR is captured as the minimal stable package within the chosen domain, $`\Sigma_{\le 2}`$.

Conclusion

We began with a simple thought that the laws of nature might be characterised as exactly those candidates that satisfy an admissibility condition whose evaluation itself depends on the totality of laws. Taken extensionally, this idea either becomes ill-typed (laws versus law-sets) or collapses into self-membership once levels are forced to coincide. The first part of the paper made this failure explicit: An extensional reconstruction cannot sustain the intended explanatory reading without either type collapse or trivial self-endorsement.

The repair is to keep types distinct and treat “theory packages” as elements of a structured space of packages. In that setting, self-subsumption is not a paradoxical set-theoretic construction but a fixed point equation

P=\mathcal F(P).

Under monotonicity on a complete lattice, Tarski’s theorem guarantees fixed points and supplies a canonical minimal choice, the least fixed point $`P=\mu\mathcal F`$. This does not remove circularity; it replaces it by a controlled closure condition, where $`P`$ is determined as the minimal package stable under an explicitly specified admissibility operator.

To connect the abstract schema to physics, we proposed reading admissibility as completion under constitutive constraints. The key structural lesson is that closure is the invariant content. A package counts as an admissible theory precisely when it is stable under the completion map. This aligns naturally with an invariance-style conception of objectivity. What is “the theory” is what survives (is closed under) the admissibility transformations one treats as constitutive. In the examples, we instantiated $`\mathcal F`$ via an invariance–symmetry correspondence, and showed how QED and GR arise as least fixed points once one fixes a candidate domain and encodes symmetry, locality in the gauge sense, and a truncation criterion.

Several limitations are built into the present formulation. First, the framework does not tell us which candidate domain $`\Sigma`$ or which admissibility operator $`\mathcal F`$ is correct; those encode substantive physical and philosophical choices. Second, the examples rely on simplifying truncations (renormalisable operators in QED; a two-derivative expansion in GR), so they illustrate the fixed point architecture rather than capture the full content of the theories. Third, monotonicity is a strong hypothesis. Many attractive criteria (consistency checks, optimality constraints, some notions of “minimality”) are not monotone, and fall outside the direct scope of the Tarski canonicity argument.

These constraints also suggest natural directions for future work:

1. Enrich the package lattices $`\mathcal T`$ to incorporate quotients by physical equivalence (gauge, duality, field redefinitions) as part of the package structure.

2. Study non-monotone admissibility notions and alternative fixed point principles, clarifying when and how canonicity survives without monotonicity.

3. Extend the invariance-based construction to more realistic settings (effective field theory bases, anomaly constraints, renormalisation group closure, and bootstrap-style constraint packages), where the set of admissible completions may form a nontrivial lattice of fixed points.

The main takeaway is structural: Whenever one characterises a “fundamental” package by a rule of membership that itself depends on that package, the resulting formulation is, in general, a fixed point problem. Making this explicit separates two questions that are often conflated. The logical question of how to represent self-subsumption without contradiction, and the substantive question of which admissibility transformations should count as constitutive of physical law.