Realisations of QIQT-H — runnable toy models

QIQT-H is best read as a boundary theory: a small postulate core (the $(\Phi,\lambda)$ ontology, quantum equilibrium, and finite regional capacity) plus a family of necessary conditions on the record structure of any quantum theory. A realisation is then a generative model — a constructive field theory, the Standard Model, a quantum-spacetime construction — whose regional data, encoded into the finite-capacity microstate space, satisfies those conditions. (See the theory and the formalization.)

The cleanest way to see what that means is to build the smallest realisations and check the boundary conditions exactly. Two short, fully runnable SymPy scripts do precisely that. Together they draw the exact line between what a finite realisation gives you and what the Einstein field equations actually require.

The minimal finite realisation

scripts/qiqth_minimal_realisation.py — one fermionic CAR mode (a qubit code $C=\mathbb{C}^2$) isometrically embedded, $V:\mathbb{C}^2\hookrightarrow\mathbb{C}^N$, into a finite microstate memory $\mathcal{H}_R=\mathbb{C}^N$. It is the smallest object that exercises every boundary condition — and, being one exact-CAR mode, it fits a finite sector perfectly. The script constructs the objects and verifies, symbolically and exactly, each condition (each tagged with its machine-checked Lean theorem):

• CAR fits exactly — $\{a,a^\dagger\}=1$ on $\mathbb{C}^2$ (a fermion is finite; a boson cannot be).
• The corner, and the tripwire — $P=VV^\dagger$ is an orthogonal projector, and the encoding is unital onto the corner: $\iota_V(1)=P\neq 1_{\mathcal{H}_R}$, never the ambient identity.
• Faithful read-back — $\operatorname{Tr}\big((V\rho V^\dagger)(VOV^\dagger)\big)=\operatorname{Tr}(\rho O)$.
• CAR in the corner — $\{\iota_V(a),\iota_V(a^\dagger)\}=P$.
• Born from typicality — the equiprobable measure over an equal-amplitude orthonormal fine-graining reproduces the Born weights $|c_k|^2$ (the toy returns $\{0:\tfrac12,\,1:\tfrac13,\,2:\tfrac16\}$).
• The area floor — $S_{\mathrm{vN}}(\rho)\le\log\dim C\le\log\dim\mathcal{H}_R=\text{area}$.
• Photon honesty — a truncated bosonic mode necessarily carries the explicit defect $[a,a^\dagger]=1-N\,|N{-}1\rangle\langle N{-}1|$ (exact CCR is impossible in finite dimension).

Three composition checks confirm the program stays consistent under combination:

• (a) Locality — disjoint regions’ observables commute, and the corner encoding preserves it: $[\iota_V(O_A),\iota_V(O_B)]=\iota_V([O_A,O_B])=0$ (because the encoding is a $\star$-homomorphism).
• (b) Tensor-network RT — a $3$-node weighted graph whose cut is the area primitive: purity $S(A)=S(A^c)$, subadditivity, the RT inequality $S_{\mathrm{vN}}(\rho_A)\le\mathrm{cut}(A)$, and min-cut violating the triangle inequality ($\lambda_{02}=5>4=\lambda_{01}+\lambda_{12}$) — so it is an area, never a distance.
• (c) The thermal half — a finite modular/Gibbs state with a genuine temperature (finite KMS), the exact entanglement first law $\delta S=\delta\langle K\rangle$, and — with the modular Hamiltonian $\propto$ area — the entropy–area variation $\delta S=\eta\,\delta A$.

That is as far as a finite realisation reaches: it carries the kinematic and the thermal boundary conditions — including $\delta S=\eta\,\delta A$ — but it yields no Einstein equations.

The minimal gravitational realisation — where GR appears

scripts/qiqth_gr_realisation.py — the continuum object that does reach GR: the free Klein–Gordon field on a pp-wave background, $ds^2=H(u,x,y)\,du^2+2\,du\,dv+dx^2+dy^2$ (matching the Lean capstone qiqt_gr_ppwave_showcase). The script verifies Jacobson’s “Einstein equation as an equation of state”, symbolically and end to end:

1. Geometry — Christoffel $\to$ Ricci $\to$ Einstein tensor: $R=0$, $G_{uu}=-\tfrac12\nabla^2_\perp H$, all other $G_{\mu\nu}=0$.
2. Conservation — the contracted Bianchi identity $\nabla^\mu G_{\mu\nu}=0$ (what fixes $\Lambda$ and lets a local Clausius relation become a field equation).
3. The null-cone lemma — a symmetric $S$ with $S_{\mu\nu}k^\mu k^\nu=0$ for all null $k$ must be $S=f\,g$ (verified on Minkowski with nine null directions). This is what turns the per-direction Clausius relation into a tensor equation.
4. The equation of state — $\delta Q=T\,\delta S$, with the QIQT-H inputs that are theorems ($\delta S=\eta\,\delta A$ from DifferentialAreaLaw, the $1/4$ from SakharovRatio) and the Unruh temperature discharged for the free field (Fock.OneParticleBW), closes to the explicit Einstein equation:

$-\tfrac12\,\nabla^2_\perp H \;=\; 8\pi G\,(\partial_u\phi)^2 .$

So GR is a property of the continuum realisation — the one that supplies the smooth metric and the Raychaudhuri area derivative $\delta A\leftrightarrow R_{kk}$ that no finite toy contains. It is a conditional derivation: the labelled residuals (the matter equation of motion, the capacity postulate (P4), the localization map, the reference-state identification, and the value of $G$) remain inputs.

What the two toys show together

• A finite realisation realises the kinematic + thermal boundary conditions — Born weights, the area floor, locality, the RT inequality, a finite temperature, and even $\delta S=\eta\,\delta A$.
• The Einstein field equations are not a boundary condition every realisation must satisfy. They are a property of the richer continuum realisation, which additionally supplies the smooth-background geometry (Raychaudhuri) and the Bisognano–Wichmann/Unruh temperature.

QIQT-H says what must hold of any region’s records; a realisation is what makes it hold — and a gravitational realisation is one that also carries the continuum structure from which the Einstein equations follow.

Both scripts are exact-symbolic (no floating point) and run to completion with every check passing. They live in scripts/ alongside the machine-checked Lean substrate.