Quantum Gravity under the Biquaternion Framework — A Research Agenda

Introduction

This article is an agenda, not a result. Its question is narrower than "is gravity biquaternionic" and downstream of it. The companion agenda The Einstein Field Equations under the Biquaternion Framework — A Research Agenda asked whether the framework reproduces the Einstein field equations, and answered that the question is open and that even the linearised equation is transcribed rather than derived. This article asks the next question: given the gravitational kinematics the framework carries, where does the quantisation of the gravitational sector stand, and where does it stop?

Quantisation presupposes a classical theory — an action, a field, a conjugate momentum, a constraint structure. That presupposition is exactly what the framework's gravity series does not supply, and the honest answer is therefore that there is no quantisation of the gravitational sector here. The agenda's work is to say what kind of absence each part of the answer is, and what object or computation would close it. A statement that the sector "is not quantised" is nearly worthless on its own, because it is true of almost every unfinished programme; the value is in the separation that follows, and the separation is the result.

Two structural facts organise the article, and neither is a slogan.

The gravitational connection is two-sided. The covariant derivative that the local Lorentz action forces on the material sector is $$ D_\mu\tilde{Q}=\partial_\mu\tilde{Q}+\tilde\Gamma_\mu\tilde{Q}+\tilde{Q}\tilde\Gamma_\mu^{*} , $$ with $\tilde\Gamma_\mu$ in the six-dimensional Lie subspace. This is Curved Spacetime and the Biquaternion Framework's result, and the same article records that the natural abbreviation $D_\mu=\partial_\mu+[\tilde\Gamma_\mu,\cdot\,]$ is wrong precisely for the boosts. The gauge connection of the framework's other field theories is one-sided: $D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu$ in the abelian case with central coefficients, and $D_\mu=\partial_\mu+i\kappa\mathcal A_\mu$ acting by left multiplication in the non-abelian case, whose own article states that its connection "is one-sided" and "is not the two-sided gravitational connection." The gauge quantisation techniques that the Maxwell and Dirac articles transcribed were built for the one-sided object. That is why they do not transfer directly to the gravitational sector, and locating the obstruction precisely is one of the two jobs of this agenda.
The framework carries a finite-dimensional modular operator. In $\mathbb{B}\cong M_2(\mathbb{C})$ the modular theory of Tomita–Takesaki gives $\tilde K=-\log\tilde\rho\in\mathbb{M}_+$ for every faithful state, with $\mathrm{Tr}(e^{-\tilde K})=1$ and a flow generated by the commutator; at the wedge, Bisognano–Wichmann gives $\tilde K_W=2\pi G_1$ with $G_1=ie_1\in\mathbb{M}_+$ and a flow that is the boost of rapidity $2\pi s$. This is a route to a horizon temperature that never quantises a metric: the temperature is read off the modular structure of a state, not off a commutator of metric operators. It is genuinely different in kind from the route the rest of this article finds blocked, and the agenda has to state exactly what it gives and what it does not.

Three traps are named here because a reader will otherwise fall into them, and the article returns to each.

Trap (a): the perturbative graviton. This article does not present a perturbative graviton, a graviton propagator, or a loop expansion as existing in the framework. None does. There is no action, hence no quadratic kinetic operator, hence no propagator; the coupling constant is not fixed; and the natural single-biquaternion packaging annihilates exactly the two transverse-traceless amplitudes a graviton would carry. Every mention of a propagator or a loop below is a statement about what is absent and what would be needed.
Trap (b): the non-commutativity. $\mathbb{B}$ is a non-commutative algebra, but the non-commutativity is in the algebra of the field values, not in the coordinates. The material coordinates of $\mathbb{M}_-$ commute; there is no $[\hat x^\mu,\hat x^\nu]\neq0$. The framework is not a non-commutative geometry, and the promotion of its fields to operators is not a quantisation of spacetime. "Quantisation" here means, as in the Maxwell and Dirac articles, the promotion of a classical field to an operator-valued field on a Fock space — nothing more and nothing less.
Trap (c): the modular route and the Hawking temperature. The finite-dimensional modular operator does not yield a derivation of the Hawking temperature. It gives the modular Hamiltonian, the KMS property at unit modular temperature, and, at the wedge, the identification of the modular flow with the boost. It does not give the black-hole geometry, the surface gravity, or the identification of the wedge flow with a particular horizon's flow; those are imported. The section The Modular Route to a Horizon Temperature states the split in full.

The organisation is the three-way classification the agenda exists to make. The article first fixes what quantisation of the gravitational sector would require and what the framework supplies toward it. It then states the two-sided obstruction precisely and sets it against the one-sided gauge connection. The three central sections sort the sector into (i) established, (ii) obstacles with a known route, and (iii) obstacles with no route yet, each item carrying the object or computation that would settle it. A separate section states the modular route, and a ledger collects the classification in one table. The closing sections summarise and carry the inherited notation.

Conventions. The notation of the read-list articles is inherited without change, and nothing is renamed or rederived. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with quaternion basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, and scalar imaginary $i$ with $i^2=-1$ commuting with every $e_k$. The anti-Hermitian and Hermitian subspaces are $\mathbb{M}_-$ and $\mathbb{M}_+$, with $\mathbb{B}=\mathbb{M}_+\oplus\mathbb{M}_-$ and $i\mathbb{M}_\pm=\mathbb{M}_\mp$; $\mathbb{H}_{\mathbb{B}}$ is the real-quaternion subspace and $\mathbb{C}_{\mathbb{B}}=\mathrm{span}_\mathbb{R}\{e_0,ie_0\}$ the centre. The material basis is $\varepsilon_0=ie_0$, $\varepsilon_k=e_k$, with $\eta_{\mu\nu}=\langle\varepsilon_\mu,\varepsilon_\nu\rangle=\mathrm{diag}(-1,1,1,1)$ and bilinear form $\langle\tilde Q,\tilde P\rangle=\mathrm{Sc}(\tilde Q\tilde P^{\natural})$. The biquaternionic gradient is $\tilde\nabla=e_0\partial_{ict}+e_1\partial_x+e_2\partial_y+e_3\partial_z$, $\tilde\nabla^{\natural}$ is its quaternion conjugate, and $\Box=\tilde\nabla\tilde\nabla^{\natural}=\tilde\nabla^{\natural}\tilde\nabla$. The frame field is $\tilde E_\mu\in\mathbb{M}_-$ with $g_{\mu\nu}=\langle\tilde E_\mu,\tilde E_\nu\rangle$; the connection is $\tilde\Gamma_\mu$, valued in the six-dimensional traceless subspace $\mathrm{SL}(2,\mathbb{C})_{\mathbb{R}}=\mathrm{span}_\mathbb{R}\{e_k,ie_k\}$, with the rotations $J_k=e_k$ and the boosts $K_k=ie_k$. The trace formula is $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$, with $\mathrm{Tr}(e_0)=2$, inherited unchanged; the modular data are $S=J\Delta^{1/2}$, $\Delta=S^*S$, $\sigma_t(\tilde A)=\Delta^{it}\tilde A\Delta^{-it}=e^{-i\tilde Kt}\tilde A e^{i\tilde Kt}$ with $\tilde K=-\log\Delta$, and the wedge generator $G_1=ie_1\in\mathbb{M}_+$. Natural units $\hbar=c=k_B=1$ are used where a temperature or a coupling is displayed.

What Quantisation of the Gravitational Sector Would Require

It is worth fixing the chain before measuring the framework against it, because "quantise gravity" names several different programmes with different inputs.

The canonical chain. A classical gravitational theory supplies an action $S[\tilde E_\mu,\tilde\Gamma_\mu]$; the Legendre transform supplies conjugate momenta; the momenta supply primary constraints; consistency of the constraints supplies secondary constraints; the first-class constraints supply a gauge freedom and a physical-state condition; the physical states supply a Hilbert space; and a quadratic expansion about a background supplies a propagator, a coupling, and a loop expansion.

The path-integral chain. A measure, an action, and a space of histories supply the sum over paths; the exponent's phase supplies the quantum weight; a gauge-invariant regulator supplies the non-perturbative definition. The framework's own path-integral article supplies the central unitary phase, locates the exponent $iS/\hbar$ in the material sector $\mathbb{M}_-$, and exhibits the Wick rotation — but it states plainly that it does not supply the measure, the action, or the space of paths.

The algebraic chain. A state and a net of local algebras supply the observables; the modular structure of the state supplies the flow and the thermal data. This is the chain the framework's horizon articles use, and it is the one chain the gravitational sector can enter without a metric quantisation.

Against these three chains, the framework's gravitational content is kinematical and only kinematical. Curved Spacetime and the Biquaternion Framework establishes that an arbitrary Lorentzian metric can be carried in a frame field $\tilde E_\mu\in\mathbb{M}_-$, that the algebra supplies the pointwise $SL(2,\mathbb{C})$, its vector representation, and its six-dimensional Lie algebra, and that a connection and its curvature can be carried as algebra-valued objects. The same article establishes, with equal force, that the algebra supplies no action, no field equation, and no representation of diffeomorphism invariance, so that nothing determines the frame, the connection, or the metric. The companion agenda on the field equations takes that absence as its subject and reaches the same conclusion; this agenda does not re-derive it.

The consequence for the question posed here is immediate and is the article's first finding:

The canonical chain for the gravitational sector has no first link. There is no classical gravitational dynamics to quantise, so canonical quantisation, the BRST/Faddeev–Popov route, and the perturbative expansion are not merely unfinished: they have no input. What the framework has, and what the rest of this agenda sorts, is (a) kinematic objects that a future quantisation could be built on, and (b) one route — the modular route — that produces thermal data without a classical metric theory at all.

This is not a quibble about ordering. In the Maxwell and Dirac cases the framework could transcribe a complete chain because a complete classical theory existed to transcribe; the articles' open items were about whether the algebra added anything, not about whether there was a theory. In the gravitational case the missing object is the classical theory itself, and every quantisation-specific obstacle below inherits that absence or is downstream of it. The items that are not so downstream — the two-sided obstruction, the bosonic Fock-space problem, and the modular route — are the quantisation-specific content of the agenda, and they are treated on their own.

One more discipline belongs here, because it is the failure mode of an agenda on this subject. The three-way classification below keeps apart (i) the kinematic computations the series has actually made, (ii) the obstacles for which the computation that would remove them can be named, and (iii) the obstacles for which no mechanism has been proposed. A reader who collapses (ii) into (i) will read "there is a known route" as "there is a result"; a reader who collapses (iii) into (ii) will read "no route yet" as "merely unfinished". The articles this agenda inherits are unusually careful about both distinctions, and the classification is kept as sharp as they keep it.

Why the Gauge Quantisation Does Not Transfer: One-Sided Against Two-Sided

The framework's two quantisation articles are both transcriptions of standard canonical quantisation, and both are honest that the algebra did not supply the quantisation. The Maxwell article's central finding is that the framework cannot fix the gauge naturally: the natural variable $\tilde F$ has no canonical partner, so quantisation must return to the potential $\tilde A$ and its redundancy $S=\mathrm{Sc}(\tilde\nabla^{\natural}\tilde A)$ with $S'=S-\Box\Gamma$, and no algebraic principle prefers one gauge to another. That obstruction is the closest existing precedent for the gravitational case, and it is worth stating precisely how much of it transfers and how much does not.

The gauge connection of the Maxwell and Yang–Mills articles is one-sided. In the abelian case the coefficients $A_\mu$ are central, so left and right multiplication agree and $$ D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu,\qquad D_\mu\tilde\Psi=\partial_\mu\tilde\Psi+\tfrac{iq}{\hbar}A_\mu\tilde\Psi, $$ which is left multiplication by the connection. In the non-abelian case the coefficients are algebra elements and left and right multiplication differ, but the covariant derivative is still one-sided on the fundamental field, $D_\mu=\partial_\mu+i\kappa\mathcal A_\mu$ with $D_\mu\Psi=\partial_\mu\Psi+i\kappa\mathcal A_\mu\Psi$; the Bianchi identity uses the adjoint action on an algebra-valued field, $D_\lambda \tilde{Q}=\partial_\lambda \tilde{Q}+i\kappa[\mathcal A_\lambda,\tilde{Q}]$, which is again a one-sided commutator. Gauge quantisation — the constraint algebra, the Faddeev–Popov determinant, the BRST complex, the Faddeev–Popov ghost with its one-sided covariant derivative — is formulated for exactly this one-sided object.

The gravitational connection is two-sided, and its article says why. The local Lorentz group acts on the material sector by $\tilde{Q}\mapsto\tilde\Lambda\tilde{Q}\tilde\Lambda^{*}$, so the infinitesimal action of a generator $G$ in the Lie subspace is $$ \delta_G\tilde{Q}=\tfrac12\big(G\tilde{Q}+\tilde{Q}G^{*}\big) , $$ which for a Hermitian boost generator $G=G^{*}$ is the anticommutator $\tfrac12\{G,\tilde{Q}\}$ and not the commutator. The same article records that the natural abbreviation $D_\mu=\partial_\mu+[\tilde\Gamma_\mu,\cdot\,]$ is wrong for exactly the boosts, which are the Lorentzian content of the theory. This was recomputed here: for $G_1=ie_1$ and a material element $\tilde{Q}=iq_0e_0+q_1e_1+q_2e_2+q_3e_3$, the two-sided generator reproduces the boost, $$ \frac{d}{d\psi}\Big|_0\tilde\Lambda\tilde{Q}\tilde\Lambda^{*}=\tfrac12\{G_1,\tilde{Q}\},\qquad [G_1,\tilde{Q}]=\big(0,\,0,\,-2i q_3,\,2i q_2\big), $$ so the commutator vanishes on the boost plane $q_2=q_3=0$, and off it it is a different generator — it is not the boost. A quantisation built on the commutator form would therefore quantise the wrong generator on the one case the Lorentzian theory needs.

Three consequences for quantisation follow, and they are the article's structural findings.

The ghost and BRST structure would have to be rebuilt two-sidedly. The Faddeev–Popov construction for a local Lorentz gauge field treats the ghost as a Lie-algebra-valued field transforming in the adjoint, with the one-sided covariant derivative above. On the material sector the local Lorentz transformation is two-sided, so the ghost's own transformation law and its covariant derivative are not the Yang–Mills ones; the framework has not constructed the two-sided analogue, and no article in the series attempts it.
The gauge-fixing problem is inherited and enlarged. Maxwell's obstruction was that the algebra gives no reason to prefer $S=0$ to any other gauge. Gravity has two gauge freedoms to fix — the six-dimensional local Lorentz (rotor) freedom and the four-dimensional linearised diffeomorphism freedom — and the algebra supplies a rotor action for the first but nothing at all for the second. The Maxwell precedent therefore transfers to the local Lorentz half and stops at the diffeomorphism half.
The constraint algebra is not the Yang–Mills one. The gravitational Hamiltonian and momentum constraints are the generators of the four diffeomorphisms, not of the six local Lorentz transformations, and the framework has no representation of $\mathrm{Diff}(M)$ to write them with. The local Lorentz constraints, whose generators the algebra does contain, are the ones for which a Maxwell-style analysis is even formulable.

The distinction is worth stating in one line, because it is the reason this agenda is not a rerun of the Maxwell article. Maxwell is blocked by a redundancy the algebra does not remove; gravity is blocked by that redundancy plus a symmetry, diffeomorphism invariance, that the algebra does not represent, and by a connection that is two-sided where the quantisation techniques are one-sided. The first is an obstacle of the kind the Maxwell article identified; the second and third have no counterpart there.

Established: What the Quantisation Programme Can Build On

Everything in this section is a kinematic or an algebraic fact recomputed in the series; none of it is a quantisation of the gravitational sector. The section is deliberately short, because the established content relevant to this agenda is short. Where an item is inherited rather than recomputed here, the parent is named.

The kinematical fibre of tetrad gravity. Curved Spacetime and the Biquaternion Framework establishes, and the field-equation agenda re-derives as its kind-(ii) content, the following. Any Lorentzian metric can be carried by a frame field $\tilde E_\mu\in\mathbb{M}_-$ through $g_{\mu\nu}=\langle\tilde E_\mu,\tilde E_\nu\rangle$; the local gauge group is a pointwise $SL(2,\mathbb{C})$ acting by $\tilde{Q}\mapsto\tilde\Lambda\tilde{Q}\tilde\Lambda^{*}$; the sixteen frame functions modulo the six rotor parameters give the ten components of the metric; the Lie algebra is the six-dimensional traceless subspace $\mathrm{span}_\mathbb{R}\{e_k,ie_k\}$; and a connection one-form $\tilde\Gamma_\mu$ and its curvature two-form can be carried as algebra-valued objects. This is the fibre in which any local quantisation would live.

The two-sided covariant derivative. $D_\mu\tilde{Q}=\partial_\mu\tilde{Q}+\tilde\Gamma_\mu\tilde{Q}+\tilde{Q}\tilde\Gamma_\mu^{*}$, with the infinitesimal generator $\tfrac12(G\tilde{Q}+\tilde{Q}G^{*})$ and the anticommutator form $\tfrac12\{G,\cdot\,\}$ for Hermitian $G$; the commutator fails for boosts. A rotor field's logarithmic derivative $\tilde\Lambda^{\natural}\partial_\mu\tilde\Lambda$ is a pure-gauge Lie-algebra connection with vanishing curvature. Both are established in Curved Spacetime and recomputed above; both are kinematics.

The one-sided gauge connection, as the contrast. $D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu$ in the abelian case (central coefficients, left multiplication) and $D_\mu=\partial_\mu+i\kappa\mathcal A_\mu$ in the non-abelian case (left multiplication on the fundamental, adjoint commutator on algebra-valued fields), from the covariant-derivative and non-abelian articles. The contrast is not a detail of notation; it is the reason the gauge quantisation does not transfer.

The finite-dimensional modular operator. In $\mathbb{B}\cong M_2(\mathbb{C})$, for every faithful state $\tilde\rho\in\mathbb{M}_+$ the Tomita–Takesaki construction gives $S=J\Delta^{1/2}$, $\Delta=S^*S>0$, and $$ \tilde K=-\log\tilde\rho\in\mathbb{M}_+,\qquad \sigma_t(\tilde A)=\Delta^{it}\tilde A\Delta^{-it}=e^{-i\tilde Kt}\tilde A e^{i\tilde Kt},\qquad \frac{d}{dt}\Big|_0\sigma_t(\tilde A)=-i[\tilde K,\tilde A], $$ with $\mathrm{Tr}(e^{-\tilde K})=1$ and, by the inherited trace formula, $\omega(\tilde A)=2\,\mathrm{Sc}(e^{-\tilde K}\tilde A)$. This is The Modular Hamiltonian in Biquaternionic Form's content, and it was re-verified here on a faithful state and on the thermal state: $\tilde K$ is Hermitian, $\mathrm{Tr}(e^{-\tilde K})=1$, $Z=\mathrm{Tr}(e^{-\beta G_1})=2\cosh\beta$, and for $\tilde\rho\propto e^{-\beta G_1}$ the modular Hamiltonian $\tilde K=\beta G_1+(\log Z)e_0$ commutes with $G_1$. The finite-dimensional flow is inner, generated by the commutator.

The wedge modular Hamiltonian. The Modular Hamiltonian in Biquaternionic Form and The Bisognano–Wichmann Theorem under the Biquaternion Framework give, at the right wedge, $G_1=ie_1\in\mathbb{M}_+$ with $G_1^2=e_0$, $\tilde K_W=2\pi G_1$, equivalently $\Delta_W=e^{-2\pi G_1}$, and a modular flow that is the boost by rapidity $2\pi s$. The flow's action on $\mathbb{M}_-$ is the two-sided rotor conjugation, not the commutator $[\tilde K_W,\cdot\,]$. Bisognano–Wichmann is imported, not derived; the generator identification is exact and the flow identification is not available inside the finite-dimensional algebra.

The nongravitational quantisations, as transcriptions. The Maxwell article transcribes the Legendre transform, the first-class constraints $\pi^0\approx0$ and $\mathrm{div}\,\boldsymbol\pi\approx0$, the Gupta–Bleuler construction, and the two transverse polarisations, and finds that the framework cannot fix the gauge naturally. The Dirac article transcribes the second-class constraint $\pi-i\psi^\dagger\approx0$, the equal-time anticommutation relations, the Fock space, and Pauli exclusion, and finds that the $\mathbb{Z}/2$ grading had to be imposed. Neither article derives its quantisation from $\mathbb{B}$.

The capacity of the algebra for modes. Fock Space and Creation/Annihilation Operators in Biquaternionic Form establishes, by the dimension count $\dim_\mathbb{C}\mathbb{B}=4=\dim_\mathbb{C}M_2(\mathbb{C})$, that $\mathbb{B}$ carries exactly one fermionic mode and that no second one fits; and, by $\mathrm{Tr}([\tilde a,\tilde a^\dagger])=0$, that no bosonic mode exists in $\mathbb{B}$ at all — a canonical commutator would have to be a central scalar and the trace of a commutator vanishes. The $(-1)^F$ grading of the single mode is the element $ie_3\in\mathbb{M}_+$. This last item matters directly for the gravitational sector, which is bosonic.

What is not established, stated as a first-class negative. There is no action and no field equation for the frame or the connection; consequently there is no conjugate momentum, no constraint algebra, no gauge-fixing principle, no physical-state condition, and no Hilbert or Fock space for the gravitational sector. There is no perturbative graviton, no propagator, and no loop expansion, and no coupling constant. There is no microstate count and no horizon Hilbert space. None of these is "unwritten"; each is absent, and the three-way classification below is the statement of what kind of absence each is.

Obstacles with a Known Route

A "known route" means here that the object does not exist, but the computation that would produce it is identifiable and its inputs are named. The distinction from the established list is that these items require a construction the series has not made; the distinction from the next section is that one can say what the construction is.

The canonical analysis of a linearised frame action. Known: the framework has the Maxwell template in full — a Legendre transform with a singular metric, a primary constraint, a secondary constraint, a first-class algebra, and a physical-state condition — and it has the linearised frame kinematics on which the analogue would act. Open: whether a candidate linearised frame action yields the same kind of first-class structure, and what its constraint algebra is. Would settle it: the object is the conjugate momentum $\tilde\pi_\mu$ to the frame perturbation $\delta\tilde E_\mu$ and the constraint algebra it generates; the computation is the Dirac–Bergmann analysis of a candidate action, checked against the classical count and against the requirement that the physical-state condition remove the unphysical directions without removing the two transverse-traceless ones. This item is blocked on the action, which belongs to the field-equation agenda and is not re-derived here; it is upstream, not independent.

A gauge-fixing functional for the local Lorentz freedom. Known: the gauge freedom of the frame is $16=10+6$, with a six-dimensional local Lorentz kernel; the six generators $J_k=e_k$, $K_k=ie_k$ lie in the algebra, and their action on the material sector is the two-sided rotor action. Open: whether any condition built from the algebra's own operations — the trace, the biquaternion norm, the bilinear form — selects one representative per orbit. Would settle it: the object is a section of the local Lorentz orbit, an algebraic condition on $\delta\tilde E_\mu$ that meets each orbit once; the computation is to exhibit the condition, verify that it removes exactly six of the sixteen frame components, and verify that the ten-component symmetric perturbation survives with its own residual freedom intact. The Maxwell finding makes a positive answer non-obvious: the analogous problem for the central $U(1)$ had no algebra-native solution, and here the group is larger and acts two-sidedly. This is the half of the gauge-fixing problem the algebra could in principle address.

A bosonic Fock space for the frame perturbation. Known: the Maxwell article constructed the bosonic Fock space as the symmetric algebra of the polarised solution space, and established that the framework contributes the module and the notation but no native ladder algebra; the Fock article proves the corresponding negative for $\mathbb{B}$ itself, that no bosonic mode exists inside the algebra. Open: whether the symmetric algebra of the frame perturbation's solution space can be built with a physical-state condition that leaves the two transverse-traceless polarisations. Would settle it: the object is the mode space of the linearised frame equation and the physical-state condition on its symmetric Fock space; the computation is to construct the mode algebra for a candidate dispersion, impose the constraints, and check that the physical subspace is positive-definite and two-dimensional. It is blocked twice over: on the action, which would supply the dispersion, and on the carrier, since the natural single-biquaternion packaging annihilates the transverse-traceless amplitudes the physical subspace is supposed to contain.

The modular-operator route to a horizon temperature. Known: the framework has the finite-dimensional modular Hamiltonian, the KMS property at unit modular temperature, and the wedge identification $\tilde K_W=2\pi G_1$ with the modular flow the boost of rapidity $2\pi s$; the Unruh and Hawking articles use exactly this to read a temperature off a geometric flow. Open: how far the identification extends from the wedge to a general horizon, and how the finite-dimensional result is to be embedded in a field algebra. Would settle it: the object is the modular Hamiltonian of the field state restricted to the horizon region; the computation is to show that its modular flow is the horizon Killing flow and that the KMS inverse temperature is $2\pi$ in the rapidity normalization, equivalently $2\pi c/\kappa$ in the acceleration normalization. This route is genuinely different from a metric quantisation — it never promotes $g_{\mu\nu}$ to an operator — and it is treated on its own in the next section, where its limits are stated.

Obstacles with No Route Yet

A "no route yet" means the object does not exist and no mechanism producing it has been proposed. The items here are not more difficult in degree than the previous section's; they are different in kind, and the difference is that one cannot say what to compute.

A perturbative expansion: a graviton, a propagator, a loop expansion. No route. A perturbative expansion requires a quadratic action about a background, a coupling constant, and a field to expand — here, a carrier of the two transverse-traceless polarisations. The framework has none of the three. The field-equation agenda records that the coupling $8\pi G/c^4$ is imported and not fixed by the algebra, that no action exists, and that no bilinear carrier of the symmetric traceless rank-two content is known; the gravitational-waves article records that the natural single-biquaternion packaging collapses to the trace and annihilates exactly the two propagating amplitudes. The object that would settle it is a bilinear map from the frame perturbation into the symmetric traceless rank-two representation together with a quadratic action normalised to a coupling constant; with neither in hand, there is no route. A loop expansion would require in addition a gauge-invariant regulator, which is the next item. No loop expansion is presented as existing here, and none should be inferred from the framework's renormalization-group article, which treats scalar quartic and non-abelian couplings and not the gravitational sector.

A representation of diffeomorphism invariance. No route. The gravitational Hamiltonian and momentum constraints are the generators of $\mathrm{Diff}(M)$, and the framework has no representation of $\mathrm{Diff}(M)$; the algebra carries the form of the linearised transformation $\delta\tilde E_\mu\mapsto\delta\tilde E_\mu-\partial_\mu\tilde\Xi$ but not its origin, as the field-equation agenda states. Without it the gravitational constraint algebra cannot be written, and the quantum analogue of Gauss's law — the condition that selects physical states — is not even formulable. Would settle it: a construction, in the algebra's operations, of a representation of $\mathrm{Diff}(M)$ on frame fields — equivalently a biquaternion version of the gauged-translation or Poincaré structure in which the soldering form closes the algebra — or a proof that none exists. The standard tetrad statement that the local translations are a symmetry of the soldering and not of the action is the setting to test; no attempt is recorded in the series.

A non-perturbative quantisation: measure, action, and a gauge-invariant regulator. No route. The framework's path-integral article supplies the central unitary phase, the location of the exponent $iS/\hbar$ in the material sector, and the Wick rotation, and states plainly that it does not supply the measure, the action, or the space of paths. For the gravitational sector the deficit is larger, because even the action is absent and the configuration space of frame fields has no measure or regulator that preserves the local rotor invariance. (The QCD agenda records the same non-perturbative gap for the gauge sector, and names the same three prerequisites — a $\mathbb{B}$-valued action, a measure, and a gauge-invariant regulator.) Would settle it: a $\mathbb{B}$-valued action for the frame and connection, a measure on the space of frame fields, and a regulator invariant under the local rotor action.

A microstate count, and a horizon Hilbert space. No route. Black Hole Thermodynamics in Biquaternionic Form states that $S=A/4G$ is a thermodynamic and geometric statement in this framework, obtained from the first law and the Euclidean action, and that the framework supplies no microstate count and no horizon Hilbert space. An entropy functional on the states of the informational sector has been proposed, $\tilde S(\tilde\rho)=-2\,\mathrm{Sc}(\tilde\rho\log\tilde\rho)$; the informational article records it as requiring verification, and it is not the black-hole entropy in any case, being an entropy of a two-state system rather than a count of horizon states. Would settle it: a state count whose logarithm reproduces $A/4G$; no candidate construction exists, and the article is explicit that none is claimed.

A type III field algebra with an outer modular flow. No route yet. The finite-dimensional modular flow is inner, on a type I$_2$ factor that is semifinite and admits a trace; the wedge algebra of a quantum field theory is type III, admits no trace and no density matrix, and its modular flow is outer. The two statements "$\tilde K\in\mathbb{M}_+$" and "$\tilde K$ generates the modular flow" hold in both settings, but by different mechanisms — an inner commutator in finite dimension, a geometric outer flow at the wedge — and the gap between them is the type gap. Realising the wedge requires an infinite-dimensional algebra built on $\mathbb{B}$-modules; the modular and Bisognano–Wichmann articles name this, and no construction is attempted anywhere in the series. Would settle it: the object is an infinite-dimensional von Neumann algebra on $\mathbb{B}$-modules whose modular flow is outer and on which the field vacua of the horizon articles can be represented.

Empirical content for the gravitational sector. No route. Nothing in the framework's gravity series predicts a deviation from standard general relativity; a quantisation could in principle produce one, but there is no quantisation to do so. This item is owned by The Empirical Status of the Biquaternion Framework and is not developed here; the gravitational part of it is the action and its coupling, both of which belong to the field-equation agenda.

The Modular Route to a Horizon Temperature: What It Gives and What It Does Not

The perturbative route is blocked at its first link. There is a second route in the framework that reaches thermal data without quantising a metric, and its status is different enough to be stated on its own. It is the framework's finite-dimensional modular operator, and it is the one place in this agenda where the algebraic chain of the introduction is entered rather than the canonical or path-integral chains.

What it gives. For every faithful state $\tilde\rho\in\mathbb{M}_+$ the modular construction supplies a Hermitian $\tilde K=-\log\tilde\rho$, a modular flow $\sigma_t$, and the KMS property at unit modular temperature; the inherited trace formula makes the Gibbs form a scalar extraction, $\omega(\tilde A)=2\,\mathrm{Sc}(e^{-\tilde K}\tilde A)$. At the right wedge, Bisognano–Wichmann identifies the modular flow with the boost: $\tilde K_W=2\pi G_1$ with $G_1=ie_1\in\mathbb{M}_+$, and the flow advances the rapidity by $2\pi$ per unit modular parameter. The Unruh and Hawking articles read a temperature off exactly this normalization — the KMS inverse temperature is $2\pi$ when the flow parameter is the rapidity, and $2\pi c/\kappa$ when the horizon's surface gravity fixes the normalization. The temperature is fixed by the state's modular structure and a geometric flow, not by a commutator of metric operators.

Why it is genuinely a different route. In a metric quantisation the temperature is a property of a quantised field on a background: one promotes $g_{\mu\nu}$, or its perturbation, to an operator, builds a Fock space, and extracts a thermal spectrum from mode mixing. Here none of that occurs. No mode of the metric is quantised, no propagator is required, and the factor $2\pi$ is the Bisognano–Wichmann normalization of a geometric flow. The route is algebraic rather than perturbative, and it is the only route in this agenda that reaches a horizon temperature at all.

What it does not give. It is not a derivation of the Hawking temperature, and three things are imported rather than produced.

The geometry. The metric, the horizon, and the surface gravity $\kappa$ are imported; the framework's own local-scale route produces no black-hole exterior, and the frame route is selected by nothing. Without $\kappa$ there is no temperature to read off.
The flow identification. Bisognano–Wichmann is a theorem about the vacuum and a wedge; its extension to a bifurcate Killing horizon is the content of the Hawking effect, and it is imported. The framework checks the generator $G_1$ and the $2\pi$; it does not derive the identification of the modular flow with a particular horizon's flow.
The state. The modular flow is defined for a given faithful state, and the framework supplies a family of states but no dynamics that selects one. The distinguished tracial state gives $\tilde K=(\log 2)e_0$ and a trivial flow; the boost-thermal state $e^{-\beta G_1}/Z$ must be put in by hand.

A fourth limit is structural rather than interpretive. The finite-dimensional algebra is a type I$_2$ factor: its modular flow is inner, generated by the commutator, and its states are density matrices on a finite-dimensional space. The wedge algebra of a quantum field theory is type III; its modular flow is outer, it admits no trace and no density matrix, and it cannot be represented inside $\mathbb{B}$. So the generator identification is exact in finite dimension while the flow identification is not available there — the framework verifies $2\pi G_1$ and cannot host the outer flow it generates. This is the type gap the modular and Bisognano–Wichmann articles record, and it is inherited here rather than repaired.

The summary of the route is therefore asymmetric, and the asymmetry is the point. The modular route reaches a horizon temperature — genuinely, and by a logic that never quantises a metric — and it does not derive the Hawking temperature, because the geometry, the horizon-flow identification, the state, and the field-algebra setting are all inputs. What it establishes is that the framework can house the KMS statement once the geometry and the state are given; what it does not establish is the geometry, the state, or the type III setting.

The Path-Integral Programmes: Regge Calculus, Causal Dynamical Triangulations, Spinfoam

The ledger's non-perturbative item — a measure, an action, and a gauge-invariant regulator — is not an abstract want. There are three concrete standard programmes that attempt exactly it, and it is worth stating what each is and how the framework's ingredients map onto it, because the mapping is definite and the shortfall is exactly the item's.

Regge calculus. Replace the smooth manifold by a simplicial complex and put the curvature on its hinges: in a piecewise-flat triangulation the curvature of a hinge (a triangle in four dimensions) is the deficit angle by which the surrounding simplices fail to close, and the gravitational action is the sum over hinges of the hinge volume times the deficit angle — the Regge action, which tends to the Einstein–Hilbert action as the triangulation is refined. The path integral is then a sum over edge lengths (and, in the dynamical version, over triangulations) weighted by $e^{iS_{\mathrm{Regge}}/\hbar}$. The framework's nearest relative is the lattice regulator of Lattice Gauge Theory and the Biquaternion Path Integral: the same discretisation strategy, applied to the gauge field rather than to the metric. What the framework has is the linearised kinematical fibre — the frame $\delta\tilde E_\mu$, the metric as its bilinear form, the connection, the curvature — and the lattice pattern; what it lacks is a discrete $\mathbb{B}$-valued action for the frame and a map from the frame to the simplicial data, which is the missing action of the field-equation agenda in one more guise.

Causal dynamical triangulations. Keep the triangulation but fix a foliation into spacelike slices and allow only simplices that respect the causal order, so that every configuration is a Lorentzian, causally well-behaved history. The causality restriction is not cosmetic: it is what makes the sum over triangulations converge, the unrestricted Euclidean sum being dominated by degenerate configurations. The framework's nearest relative is its causal structure — the null cones of the $ict$ metric, the causal order, the causally complete regions of The Functional Integral in Biquaternionic Form — which is exactly the structure CDT imposes by hand on the triangulation. But CDT's foliation and causality are properties of the triangulation and of the global time function, not of the algebra, and the framework has neither a triangulation nor a time function to foliate. It supplies the cone metric that says what "causal" means; it does not supply the sum.

Spinfoam models. Sum over two-complexes — the spacetime histories of spin networks — whose boundary data are representation-theoretic: a spin network assigns to each edge of a graph an irreducible representation of the Lorentz group (or of its $SU(2)$ subgroup) and to each vertex an intertwiner, and the spinfoam amplitude is built from the representation theory and the constraints. The framework's nearest relative is its rotor group: the local $SL(2,\mathbb{C})$ of Curved Spacetime and the Biquaternion Framework, its representation theory in the Lorentz-group articles, the modules over $\mathbb{B}$, and the two-sided rotor action whose anticommutator generates the boosts. The representation-theoretic boundary data of a spinfoam are precisely the kind of labels the framework's Lorentz-group articles construct, and the local Lorentz structure of a spinfoam vertex is the two-sided structure the framework's frame carries. What is absent is everything else: the two-complex, the vertex amplitude, the simplicity and closure constraints that reduce the representation theory to geometry, and the measure.

The causal character of the boundary, and the spinor states. Inside the spinfoam programme the boundary data are graded by the causal character of a triangle and of the tetrahedron that carries it, and the grading fixes both the group and the representation. A triangle whose holonomy is written $h_{\bar a b}$ lies in $SU(2)$ when the triangle and its tetrahedron are space-like, and in $SU(1,1)$ in the two remaining cases — a space-like triangle of a time-like tetrahedron, and a time-like triangle of a time-like tetrahedron; the labels are the spin $j\in\tfrac12\mathbb{N}$ of a unitary $SU(2)$ representation in the first case, the parameters $j\in-\tfrac12\mathbb{N}$ of the discrete series of $SU(1,1)$ in the second, and the parameters $j\in\mathbb{R}_+$ of the continuous series in the third. The distinction is not a matter of labels: $SU(2)$ is compact and $SU(1,1)$ is not, so the time-like boundary carries the non-compact case, exactly as a boost does in the rotor group of this framework.

A recent external paper reads these data through the same algebra this corpus uses. J. D. Simão observes that the equations of the semiclassical EPRL-type amplitude admit a common biquaternionic structure — the complexified quaternions, with the two commuting conjugations and the matrix realisation of the companion articles — and uses it to model the three boundary spaces on spinors rather than on vectors:

the sphere $S^2$, on one Weyl spinor, by $v^i=\langle z|\sigma^i|z\rangle$ with $\langle z|z\rangle=1$;
the two-sheeted time-like hyperboloid $H^\pm\subset\mathbb{R}^{1,2}$, on one Weyl spinor, by the indefinite pairing $v^i=\tfrac12\langle z|\sigma_3\varsigma^i|z\rangle$ with $\varsigma=(\sigma_3,i\sigma_2,-i\sigma_1)$, normalised so that $\eta_{ij}v^iv^j\ge0$;
the one-sheeted hyperboloid $H^{sl}$, on a pair of Majorana spinors, by $v^i=\langle z_2|\sigma_3\varsigma^i|z_1\rangle$, a real structure commuting with $SU(1,1)$ taking the place of the complex conjugation of the first two cases.

The bilinears are then shown to close, under a symplectic form on the spinor space, into the Poisson algebra of $\mathfrak{so}(1,2)\cong\mathfrak{su}(1,1)$, displayed in the paper as $\{v^i,v^j\}=-\varepsilon^{ijk}\eta_{kl}v^l$; the closure — up to the conventional overall sign of the symplectic form — was recomputed here on generic elements, and it closes. The main result of the paper is that the Majorana spinor space, subject to an area-matching constraint, is symplectomorphic to the cotangent bundle $T^*SU(1,1)$, generalising the twisted-geometry identification of $T^*SU(2)$ with two copies of $\mathbb{C}^2$ from the space-like case. It is worth recording why the Majorana pair and not a single Weyl spinor enters at $H^{sl}$: the paper argues that the unified Weyl-type treatment of the two causal types forces a positive area spectrum where a time-like area must be negative, and that the pair is what repairs the sign.

The relevance to this agenda is precise and limited. The framework's rotor group is $SL(2,\mathbb{C})$, and $SU(1,1)$ is its subgroup fixing a space-like direction — the double cover of the $SO(1,2)$ subgroup, and the same group that appears in this corpus as the dynamical group $Sp(2,\mathbb{R})$ of the oscillator's squeezings. Simão's construction therefore supplies, for the spinfoam boundary, a published instance of the identification that this agenda can only speculate about: a phase space of the framework's own unitary group realised on spinors of the framework's own module, with a constraint — the area matching — playing the role that makes the identification work. What it does not supply is anything the ledger lists as missing: the two-complex, the vertex amplitude, the simplicity and closure constraints, the measure, and the action. It is an external result about the standard theory, cited here as the closest external realisation the series has found of the ledger's non-perturbative item, and not as a route through it.

The common denominator, and the verdict. All three programmes require the same three things the ledger names: a $\mathbb{B}$-valued action for the frame — discrete in Regge and CDT, encoded in a vertex amplitude in spinfoam — a measure and regulator invariant under the local rotor action, and a representation of diffeomorphism invariance or an accepted replacement for it. The path-integral programmes are therefore not a separate route to quantum gravity; they are the explicit form of the ledger's non-perturbative item, and the framework's ingredients map onto them in a definite way, the rotor group onto the spinfoam boundary data, the causal cones onto CDT's causality, and the lattice regulator pattern onto Regge's discretisation. None of the three is constructed here, none is claimed, and the mapping is stated so that the shortfall is the named item and not a vaguer absence.

The Ledger

The table collects the agenda and sorts each item into the three-way classification. "Established" means recomputed in the series and inherited here, or recomputed for this article; "known route" means the object does not exist but the computation that would produce it is identifiable; "no route yet" means no mechanism has been proposed.

Item	Status	Object or computation that would settle it
Kinematical fibre of tetrad gravity: frame $\tilde E_\mu\in\mathbb{M}_-$, metric $g_{\mu\nu}=\langle\tilde E_\mu,\tilde E_\nu\rangle$, local $SL(2,\mathbb{C})$, Lie algebra, connection $\tilde\Gamma_\mu$, curvature	Established	—
Two-sided covariant derivative $D_\mu\tilde{Q}=\partial_\mu\tilde{Q}+\tilde\Gamma_\mu\tilde{Q}+\tilde{Q}\tilde\Gamma_\mu^{*}$; anticommutator generator of boosts; commutator fails on the boost plane	Established (recomputed)	—
One-sided gauge connection: abelian central $D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu$ and non-abelian $D_\mu=\partial_\mu+i\kappa\mathcal A_\mu$	Established	—
Finite-dimensional modular operator: $S$, $\Delta=S^*S>0$, $\tilde K=-\log\tilde\rho\in\mathbb{M}_+$, $\mathrm{Tr}(e^{-\tilde K})=1$, trace formula	Established (recomputed)	—
Wedge modular Hamiltonian $\tilde K_W=2\pi G_1$, $G_1=ie_1$, modular flow = boost of rapidity $2\pi s$	Established (Bisognano–Wichmann imported)	—
Maxwell and Dirac canonical quantisations as transcriptions; gauge-fixing obstruction	Established (transcribed)	—
Capacity of $\mathbb{B}$: exactly one fermionic mode, no bosonic mode	Established	—
Action, field equation, graviton, propagator, loop expansion, horizon Hilbert space	Absent (first-class negative)	see below
Canonical analysis of a linearised frame action	Known route, blocked on the action	Conjugate momentum $\tilde\pi_\mu$ and constraint algebra (Dirac–Bergmann)
Gauge-fixing functional for the local Lorentz freedom	Known route, non-obvious	A section of the six-dimensional rotor orbit; verify it removes $6$ of $16$ and preserves $10$
Bosonic Fock space for the frame perturbation	Known route, blocked on action and carrier	Mode space plus physical-state condition; positive-definite two-dimensional physical subspace
Modular flow to a horizon temperature	Known route; scope stated above	Modular Hamiltonian of the horizon state; show its flow is the horizon Killing flow and $\beta=2\pi c/\kappa$
Perturbative expansion: graviton, propagator, loops	No route yet	Bilinear carrier into the symmetric traceless rank-two representation plus a quadratic action with a coupling
Representation of diffeomorphism invariance; gravitational constraint algebra	No route yet	A representation of $\mathrm{Diff}(M)$ on frame fields built from the algebra, or a no-go
Non-perturbative quantisation: measure, action, regulator	No route yet	A $\mathbb{B}$-valued action, a measure on frame fields, a rotor-invariant regulator
Path-integral quantum gravity: Regge calculus, causal dynamical triangulations, spinfoam	No route yet	The same three prerequisites in their programme-specific form: a discrete $\mathbb{B}$-valued frame action, a sum over triangulations or two-complexes, and a rotor-invariant measure and regulator
Microstate count and horizon Hilbert space	No route yet	A state count whose logarithm reproduces $A/4G$
Type III field algebra with an outer modular flow	No route yet	An infinite-dimensional von Neumann algebra on $\mathbb{B}$-modules with an outer modular flow
Empirical content for the gravitational sector	No route yet	A prediction distinguishing the framework from standard physics

What Would Settle It, in Order

The items admit an ordering by what is reachable without first solving the field-equation agenda, and the order is this agenda's judgement of it. Each step's status is recorded so that a partial advance is recognisable for what it is.

An action for the frame and connection. The prerequisite for the entire canonical and path-integral program, and explicitly owned by The Einstein Field Equations under the Biquaternion Framework — A Research Agenda. It is listed first because everything in the canonical chain waits on it, but it is not a quantisation result and the field-equation agenda does not duplicate it here.
A gauge-fixing functional for the local Lorentz freedom. An algebra-only computation, independent of the action: exhibit a condition on $\delta\tilde E_\mu$ that meets each rotor orbit once. It is the smallest computation in this agenda that could succeed while the action is still absent, and a negative result would be a real result about the algebra.
The modular flow identification in the field algebra. The route that does not need the action: construct or rule out the infinite-dimensional algebra on $\mathbb{B}$-modules whose modular flow is the horizon flow, or establish how far the finite-dimensional identification extends. A positive answer would turn the framework's one working thermal route into a framework result about the flow rather than only about the generator.
The canonical analysis of a linearised frame action. Requires step 1: the Legendre transform, the constraint algebra, and the physical-state condition, checked against the classical count.
A bilinear carrier for the two transverse-traceless polarisations. Independent of the action: the equivariant map into the symmetric traceless rank-two representation, which a propagator and a physical-state condition both require. Without it there is no field to quantise.
The bosonic Fock space and physical-state condition. Requires steps 1 and 5: the mode algebra, the subsidiary condition, and the positive-definite two-dimensional physical subspace.
A representation of diffeomorphism invariance. The object behind the gravitational constraints; construct it or prove it cannot exist. Steps 4 and 6 are incomplete without it.
The non-perturbative construction and the microstate count. The measure, the regulator, and the state count, in whichever order they become reachable. The concrete forms of the measure-and-regulator problem are the path-integral programmes of the preceding section — Regge calculus, causal dynamical triangulations, and spinfoam — and each needs the same three inputs.

Two disciplines apply to every item, and both are the standing disciplines of the corpus rather than special to this agenda. First, no item in the "known route" list may be reported as established until the named computation is exhibited; the gap between identifying a route and walking it is exactly the gap this classification exists to keep visible. Second, every candidate result must be recomputed on a case chosen independently of the one that suggested it — a generic frame rather than the symmetric ansatz, a second state rather than the one that prompted the formula — and no perturbative graviton, propagator, or loop may be reported as existing in the framework, because none does.

Summary

The quantisation of the gravitational sector does not exist in this framework, and the result of this agenda is the separation of that absence into what is established, what has a known route, and what has no route yet. Quantisation presupposes a classical gravitational theory, and Curved Spacetime and the Biquaternion Framework establishes that the framework carries the kinematical fibre of tetrad gravity — frame, metric, local $SL(2,\mathbb{C})$, Lie algebra, connection, curvature — and no action, no field equation, and no representation of diffeomorphism invariance. The canonical chain therefore has no first link: there is no conjugate momentum, no constraint algebra, no physical-state condition, and no propagator to build. No perturbative graviton, no graviton propagator, and no loop expansion exists here, and none is presented; the coupling is imported and the natural single-biquaternion packaging annihilates exactly the two transverse-traceless amplitudes a graviton would carry.

Two structural facts organise the separation. The first is that the gravitational connection is two-sided, $D_\mu\tilde{Q}=\partial_\mu\tilde{Q}+\tilde\Gamma_\mu\tilde{Q}+\tilde{Q}\tilde\Gamma_\mu^{*}$, with the infinitesimal generator $\tfrac12(G\tilde{Q}+\tilde{Q}G^{*})$ and the anticommutator form for boosts; the natural commutator abbreviation fails exactly for the boosts, and this was recomputed. The gauge connection of the framework's other field theories is one-sided — central left multiplication in the abelian case, left multiplication on the fundamental and an adjoint commutator on algebra-valued fields in the non-abelian case — so the constraint algebra, the Faddeev–Popov ghost, and the BRST complex built for that object do not transfer. Maxwell's obstruction, that the framework cannot fix the gauge naturally, is inherited for the local Lorentz freedom and stops at the diffeomorphism freedom, which the algebra does not represent at all. The second fact is that the framework has a finite-dimensional modular operator, and it is a genuinely different route to a horizon temperature: the temperature is read off the modular flow, not off a commutator of metric operators, and no metric is quantised. It gives the modular Hamiltonian $\tilde K=-\log\tilde\rho\in\mathbb{M}_+$, the KMS property at unit modular temperature, the trace formula $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$, and the wedge identification $\tilde K_W=2\pi G_1$ with the modular flow the boost of rapidity $2\pi s$. It does not derive the Hawking temperature: the geometry, the surface gravity, the identification of the modular flow with the horizon flow, and the state are all imported, and the finite-dimensional type I$_2$ algebra cannot host the outer modular flow of the type III wedge.

The ledger's three columns are the deliverable. Established: the kinematical fibre, the two-sided covariant derivative, the one-sided gauge connection as the contrast, the finite-dimensional modular operator and wedge Hamiltonian, the nongravitational quantisations as transcriptions, and the capacity of $\mathbb{B}$ for one fermionic mode and no bosonic mode. Known route, blocked on inputs: the canonical analysis of a linearised frame action, a gauge-fixing functional for the local Lorentz freedom, a bosonic Fock space for the frame perturbation, and the modular-flow identification at a horizon. No route yet: a perturbative expansion, a representation of diffeomorphism invariance and the gravitational constraint algebra, a non-perturbative measure and gauge-invariant regulator — whose concrete forms are the path-integral programmes of Regge calculus, causal dynamical triangulations, and spinfoam, each needing the same three inputs — a microstate count and horizon Hilbert space, a type III algebra with an outer modular flow, and empirical content. Each item carries the object or computation that would settle it, and the honest boundary — not "gravity is not quantised" but which absence is which kind — is the result.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_k^2=-e_0$
$i$	Scalar imaginary, $i^2=-1$
$\mathbb{M}_+,\mathbb{M}_-$	Informational (Hermitian) and material (anti-Hermitian) subspaces
$\mathbb{H}_{\mathbb{B}},\mathbb{C}_{\mathbb{B}}$	Real-quaternion subspace; centre $\mathrm{span}_\mathbb{R}\{e_0,ie_0\}$
$\varepsilon_0=ie_0,\ \varepsilon_k=e_k$	Material basis; $\eta_{\mu\nu}=\langle\varepsilon_\mu,\varepsilon_\nu\rangle=\mathrm{diag}(-1,1,1,1)$
$\langle\tilde Q,\tilde P\rangle=\mathrm{Sc}(\tilde Q\tilde P^{\natural})$	Bilinear form on $\mathbb{M}_-$
$\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$	Trace formula; $\mathrm{Tr}(e_0)=2$
$\tilde\nabla=e_0\partial_{ict}+e_k\partial_k$, $\Box=\tilde\nabla\tilde\nabla^{\natural}$	Biquaternionic gradient and d'Alembertian
$\tilde E_\mu\in\mathbb{M}_-$	Frame field; $g_{\mu\nu}=\langle\tilde E_\mu,\tilde E_\nu\rangle$
$J_k=e_k$, $K_k=ie_k$	Rotation and boost generators; $\mathrm{SL}(2,\mathbb{C})_{\mathbb{R}}=\mathrm{span}_\mathbb{R}\{e_k,ie_k\}$
$\tilde\Gamma_\mu$	Connection one-form, valued in the Lie subspace
$D_\mu\tilde{Q}=\partial_\mu\tilde{Q}+\tilde\Gamma_\mu\tilde{Q}+\tilde{Q}\tilde\Gamma_\mu^{*}$	Two-sided covariant derivative on $\mathbb{M}_-$
$\tfrac12(G\tilde{Q}+\tilde{Q}G^{*})$	Two-sided infinitesimal generator; $\tfrac12\{G,\tilde{Q}\}$ for Hermitian $G$
$D_\mu=\partial_\mu+\tfrac{iq}{\hbar}A_\mu$, $D_\mu=\partial_\mu+i\kappa\mathcal A_\mu$	One-sided gauge covariant derivatives (abelian, non-abelian)
$\delta\tilde E_\mu$, $h_{\mu\nu}$	Frame perturbation and metric perturbation; $16=10+6$
$S=J\Delta^{1/2}$, $\Delta=S^*S$	Tomita operator and modular operator
$\tilde K=-\log\Delta=-\log\tilde\rho\in\mathbb{M}_+$	Modular Hamiltonian (finite-dimensional model)
$\sigma_t(\tilde A)=\Delta^{it}\tilde A\Delta^{-it}=e^{-i\tilde Kt}\tilde A e^{i\tilde Kt}$	Modular flow; $-i[\tilde K,\tilde A]$ in finite dimension
$\tilde K=\beta\tilde H+(\log Z)e_0$	Gibbs-state modular Hamiltonian
$G_1=ie_1\in\mathbb{M}_+$	Boost generator; $G_1^2=e_0$
$\tilde K_W=2\pi G_1$	Wedge modular Hamiltonian (Bisognano–Wichmann)
$\kappa$, $T=\hbar\kappa/(2\pi c k_B)$	Surface gravity and Hawking temperature (imported geometry)
$A$, $S=A/4G$	Horizon area and Bekenstein–Hawking entropy