Information Physics Institute

Observer Patch Holography specifies the hardware, specifies the protocol, runs both, and reads our universe off the output. An invitation to the Information Physics Institute.

TL;DR: Simulation theory has an argument and no machine. Observer Patch Holography (OPH) supplies the machine. The hardware is a finite network of twelve-port icosahedral patches. The software is one local move: find where two neighbouring patches disagree about their shared boundary and repair it. Nothing else is put in. No space, no clock, no wavefunction, no gauge group, no particle list. When the machine runs, three dimensions of space, one direction of time, quantum probability, the four laws of thermodynamics, a four-dimensional Lorentzian spacetime, Einstein’s equation, energy, mass, the Standard Model gauge group with one anomaly-free generation of matter, and the numerical values of the fine-structure constant and the cosmological constant come out as normal forms of the repair process. We think this is the right track because the output is the universe we live in. We also know where the construction has open bridges, and we would like the Institute’s help with them.

An argument without a machine

In 2003 Nick Bostrom published “Are You Living in a Computer Simulation?” in Philosophical Quarterly.[^1] The paper is a probability argument about posthuman civilisations and ancestor simulations. It contains no circuit diagram. Konrad Zuse had asked the engineering version in 1969 in Rechnender Raum, and Edward Fredkin, Stephen Wolfram and Gerard ’t Hooft each wrote down cellular automata that were supposed to underlie physics.[^2] None of those automata returns the Lorentz group, the Born rule, or SU(3)×SU(2)×U(1). The gap between “the universe might be a computation” and “here is the computation” stayed open for fifty years.

OPH closes that gap in the most literal way we could think of. We asked: what is the smallest machine whose output is this universe, in the sense that finite observers inside it would measure what we measure? Then we specified the hardware, wrote the protocol, proved what the protocol does, ran it on 65,536 patches, and compared the output to the world. This post walks through the machine and its outputs, and every section links to the popular book, the textbook site, the technical paper, and where one exists the machine-checked proof.

The hardware

One piece of reality is a patch. A patch has a local state, a set of records, and twelve boundary ports. The ports sit at the vertices of an icosahedron: twelve ports, thirty edges, twenty triangular faces, sixty proper symmetries forming the alternating group A₅. Patches are sewn together along shared ports into a federation whose outward-facing support is a sphere.

Twelve is forced. A closed triangulated surface on which every port touches five triangles has Euler characteristic 2 and therefore exactly twelve ports, and conversely a twelve-port carrier of this type must be a sphere. The sixty-element symmetry group is what the icosahedron gives you for free, and that group does an astonishing amount of work downstream: it fixes the number of spatial dimensions, the gauge Lie algebra, and the first angular rank at which propagation may become anisotropic.

This is the entire hardware specification. There is no background lattice, no ambient space in which the patches sit, and no global clock. Everything a patch knows about its neighbours arrives through twelve ports.

The protocol

The software is one move. Two neighbouring patches expose their overlap data on a shared seam. If the readings disagree, a local repair map rewrites the disagreeing patch. The map is the recovery channel of the collar between the patches: the state-preserving conditional expectation onto the algebra both sides already agree on, or its Petz approximation when the collar is only approximately Markov.[^3] Four physical demands (unital complete positivity, action only on the disputed algebra, preservation of conserved bookkeeping, respect for the state’s own modular flow) determine this map uniquely. Nobody gets to design the dynamics. The architecture emits it.

A repair is accepted only if it lowers disagreement on every seam it touches and lowers it strictly somewhere. Total disagreement across the network is scored by an inconsistency potential Φ, a weighted sum of seam mismatches, bounded below by zero. Every accepted repair strictly decreases Φ, and the network is finite, so repair terminates.

The theorem that makes the output objective is schedule independence. Under a transactional local-diamond condition (each commit revalidates what it read, conflicting commits have one canonical merge), Newman’s lemma from 1942 upgrades local confluence to global confluence: every maximal asynchronous repair schedule from the same initial state lands in the same consistent normal form.[^4] It does not matter who goes first. That unique endpoint is the public world. A gauge symmetry, in this language, is a relabelling of a patch’s internals that leaves its seam data untouched; physics lives in the quotient.

There is a certificate attached. Five clauses (endogenous update from recovery rules, observer-readable records on interfaces, a schedule-independent normal form, selected-branch elimination, implementation and clock closure) distinguish a machine that simulates from a system that merely sits in equilibrium.

Running it

The public simulator at simulation.floatingpragma.io runs 65,536 patches through the protocol and shows thirty-three panels in six acts: consensus, space, time, quantum, matter, spacetime. Its own first sentence is the point of this post: “There is no space in the substrate, no clock, and no wavefunction. There is a finite set of patches, a disagreement register on every shared seam, and one local move that lowers disagreement.” The flagship paper’s diagnostic tower goes one level further, a level-six geodesic icosahedral rung with 81,920 patch rows and 122,880 adjacencies, every receipt pinned and reproducible.[^5]

Each panel is labelled exact, measured, or declared, so you can see which pictures are theorems, which are computed at runtime, and which are drawn to illustrate a claim. The source is public.[^6]

What comes out

Everything below is an output. None of it is in the hardware specification or the protocol.

Three dimensions of space

Linearise repair around a consistent state and you get one operator on twelve port readings. Icosahedral symmetry forces that operator to respect the decomposition of the twelve-port module into irreducible blocks, P₁₂ = 1 ⊕ 3 ⊕ 3′ ⊕ 5, so Schur’s lemma reduces a twelve-by-twelve matrix to four numbers. The response splits into a slow band of rank three, a middle band of rank five, and a fast band of rank three. Only the slow band retains content across many repair passes, so only the slow band can carry position. The Gram matrix of the twelve port vectors satisfies G² = 4G with trace 12, hence has rank exactly three. Long-lived repair records live in a three-dimensional space. No dimension was assumed anywhere upstream. The integer frames built from port labels are dense in a three-dimensional Euclidean carrier, which is where the continuum comes from.

Time, and why it runs one way

Time has two faces in the machine. The reversible one is the modular flow parameter of a cap, which will turn into boosts. The irreversible one is record time: the sequence of accepted repairs. Each repair strictly lowers Φ, which is bounded below, so record time has a direction by construction. A repair is idempotent and discards the specific disagreement it settled, keeping only the fact that agreement was reached. That information loss is why no rule in the protocol runs the sequence backward. The felt present is the current normal form; the past is committed repairs; the future is unresolved disagreement.

Quantum mechanics

A measurement is a record write. The records an observer can access form the central, commutative subalgebra of the observer’s algebra, and the finite event algebra of those records is where the quantum identities live. The theorem is stronger than Gleason’s: every nonnegative normalised valuation that is additive on coexisting effects is the trace against a unique density operator, in every finite dimension, with no continuity axiom. Pr(E) = Tr(ρP_E) is the only weight function compatible with observer agreement, because a context-dependent weight would let one observer change another’s report without touching the shared record. The Lüders update ρ ↦ P_E ρ P_E / Tr(ρP_E) is the state after the record is written. Correlations obey |S_CHSH| ≤ 2√2, and the bound is attained exactly on an explicit entangled witness on the slot-split interface. An isometry that copies two distinct sharp states from a common blank forces their overlap to be zero or one, which is no-cloning.

The measurement problem dissolves rather than resolves: nothing collapses, a record gets written and the rest of the network repairs toward consistency with it.

Thermodynamics

The third axiom of OPH says the realised state carries no structure beyond what agreement forces. Apply it to states and you get the Gibbs family by information projection; two patches in repair equilibrium share a modular temperature, which is the zeroth law. Apply it to transitions and the repair step becomes weighted conditional resampling onto the fibre of the repaired public datum, a channel that contracts relative entropy. The second law is therefore a data-processing theorem, ΔS ≥ Δ⟨K⟩ on reference-preserving channels, with Landauer’s erasure bound as a corollary. The first law is the exact bookkeeping split ΔU = δQ + δW plus a cross term that vanishes on ordered strokes. The third law follows from finite capacity: a rank-deficient zero-temperature state cannot be reached by faithful repair in finitely many steps, and the residual entropy is k_B log g₀. Crooks and integral fluctuation identities hold exactly, and a Green–Kubo matrix with an exact cutoff remainder gives linear transport. Unruh and Hawking temperatures are the same modular temperature read on different collars.

A four-dimensional Lorentzian spacetime

Every accepted repair is a commit that certifies which earlier records it read. The authenticated read-from relation, transitively closed, is a locally finite strict partial order: a causal set in exactly the sense of Bombelli, Lee, Meyer and Sorkin (1987), except that here the order has a physical source instead of being postulated.[^7] Two commits that never read each other are unrelated, whatever order a scheduler ran them in.

Height in that order (the longest authenticated parent chain) supplies a time coordinate. The rank-three slow band supplies space. Adjoin a real axis for height to the three-dimensional carrier and you have W = ℝ ⊕ V_src, four-dimensional because dimension adds on a direct sum, with quadratic form Q = t² − g(x, x) of signature (1, 3) because a squared height enters positive and a squared distance enters negative. Every unit direction û in the carrier gives a null vector (1, û). The source sphere, the celestial sphere an astronomer sees, and the boundary of the light cone are the same set of directions. A closed convex cone invariant under the icosahedral rotations and under boosts along one axis must be a Lorentz cone; the light cone is selected without a light postulate. Commit counts in a ledger interval converge to Alexandrov volume, the ordering fraction converges to 1/10, which read backwards is dimension four, and Malament plus Hawking–King–McCarthy then say order plus number is a Lorentzian metric, uniquely.[^8] General covariance is a theorem about relabelling commits.

Lorentz invariance and special relativity

Every cap on an observer’s sphere carries a modular Hamiltonian whose flow satisfies the KMS condition. Bisognano and Wichmann showed in 1975 that the modular flow of a maximum-entropy cap state is the conformal dilation of the cap toward its own centre.[^9] The orientation-preserving conformal group of the two-sphere is PSL(2, ℂ) ≅ SO⁺(3, 1). So a modular flow, read through that isomorphism, is a Lorentz boost, and the space of future unit timelike directions is the hyperboloid H³ = SO⁺(3, 1)/SO(3), three-dimensional. Lorentz symmetry admits exactly one invariant quadratic form, the Minkowski interval; time dilation and length contraction are theorems about it. The speed limit has a discrete arithmetic underneath: a crossing step is timelike exactly when 4 < h², a block of k rests plus one crossing is timelike exactly when 2 < (k + 1)h, which forces at least two rests per crossing, and the resulting speed 2/((k + 1)h) has supremum one, approached and never attained. The ratio of the two thresholds is 2φ² = 3 + √5. The golden ratio shows up because the icosahedron does.

General relativity and the cosmological constant

Ted Jacobson showed in 1995 that Einstein’s equation follows from δQ = T dS on local Rindler horizons if you assume the entropy–area law and the Unruh temperature.[^10] OPH derives both inputs. Split a small ball into interior, collar and exterior. The generalised entropy is the interior von Neumann entropy plus a quarter of the collar area in units of Newton’s constant. At consensus equilibrium that entropy is stationary under admissible variations. The entanglement first law converts entropy variation into modular energy, the null-stress bridge converts modular energy into the null component of a stress tensor, the small-ball integral yields the 00-component, and overlapping observers in every timelike direction extend it to the full tensor. Null tomography gives G − κT = φ g pointwise; the contracted Bianchi identity plus conservation force φ constant. The result is

R_ab − ½ R g_ab + Λ g_ab = 8πG T_ab,

with 8πG coming out of the small-ball integral itself. The equivalence principle is an output: every observer builds a chart from local modular flow in the identical way. In the weak-field limit the spherically symmetric flux at constant shell charge falls off as r⁻², with the exponent equal to carrier dimension minus one, the dimension being the rank-three theorem rather than an assumption.

Local physics is blind to Λ because null probes cannot see a term proportional to the metric. The horizon fixes it. Reading the public record capacity N of the whole observer system as de Sitter horizon entropy in nats gives N = 3π/(Λℓ_P²). The capacity closure returns N ≈ 3.30 × 10¹²², and the resulting Λ ≈ 1.09 × 10⁻⁵² m⁻² sits within 0.4% of the value inferred from supernovae, CMB and BAO. A large but finite horizon is what a nearly flat accelerating universe looks like from inside. The same identity gives the dark-energy equation of state w(a) = −1 + ⅓ d ln N / d ln a: fixed capacity means (w₀, w_a) = (−1, 0) exactly, with no continuous freedom, and any w < −1 would mean the universe is losing memory. DESI can kill this.

Energy, mass, momentum

The machine has one quantitative resource, the inconsistency potential, and energy is the rate at which a region’s records are processed by repair per unit of its own modular clock: E = (ℏ/τ) H_mod. Positivity follows because Φ cannot go below zero, which is the spectrum condition of quantum field theory. Conservation follows because the repair rule is the same at every patch and every tick. Additivity follows because independent regions have independent states. Mass is the standing cost of holding a self-sustaining record pattern in place while its records go nowhere. On the Lorentz module this gives E² = (pc)² + (mc²)², with E = mc² exactly at rest and the massless branch on the null cone. A massless pattern has zero standing cost, so its whole ledger entry is displacement, one seam per tick, which is why light moves at c. The dynamical half of the mass–energy identity is the slope with which internal energy enters inertia: frame covariance of the momentum family and, independently, a midpoint refinement of the worldline both select slope one, so the inertial coefficient of a composite is m₁ + m₂ + E₁ + E₂ plus the binding defect. That is the shape a mass defect takes. Planck’s E = hf falls out because the modular clock counts in units of action.

Matter and the forces

The complete reversible response of a patch is a twelve-dimensional Lie algebra of anti-Hermitian operators, one direction per port. The second axiom says that every proper symmetry of the carrier is implemented from inside, by the response group itself. That single condition forces the Lie type. The A₅ action fixes only the uniform port line, so the centre is one-dimensional; the semisimple remainder has dimension eleven, and eleven decomposes over compact simple algebras in exactly one way, 8 + 3. So

𝔤 ≅ 𝔲(1) ⊕ 𝔰𝔲(2) ⊕ 𝔰𝔲(3),

with the su(2) ideal carrying the 3 and the su(3) ideal carrying 3′ ⊕ 5 of the icosahedral module. Three colours are the eight-dimensional ideal. The Tannaka reconstruction of transportable sectors reaches the same Lie type from the opposite direction. The global form is a quotient: on the matter representation the common kernel of SU(3)×SU(2)×U(1) is ℤ₆, so the realised group is S(U(3)×U(2)), and the differential of that action is why hypercharge is quantised. An exhaustive scan of the 1,024 sub-menus of the exterior algebra leaves exactly two rank-fifteen chiral anomaly-free masks, exchanged by charge conjugation: one Standard Model generation, Q ⊕ uᶜ ⊕ eᶜ ⊕ dᶜ ⊕ L, with the textbook hypercharges and even Witten parity. The family band of the icosahedral module triples it. The product adjoint contains no X or Y generator, so there is no minimal grand-unified exchange channel: the proton is stable and there are no monopoles.

Particles are refinement-stable transport patterns riding the port response. The photon is the propagating mode of the unique central direction, massless with two polarisations. Gluons are modes on non-central bands charged under the field they carry, which is why colour confines. W and Z are non-central modes that displace the electroweak vacuum vector and pay for it with a mass. Matter sits on the conjugate rank-fifteen chiral projector pair, and orthogonality of the projectors is the Pauli principle. The Higgs is the radial oscillation of the reference the other four types are read against, hence spin zero. A sixth type would need a place in the decomposition that the module does not provide. The Yang–Mills mass gap is the repair gap: the finite spectral gap between the fast and slow bands transfers to the continuum Hamiltonian on the compact-gauge branch.

The numbers

The machine has two closure equations and no fitted continuous parameter. The pixel closure P = φ + √π / A_T(P) says that a screen cell’s area in area quanta equals its electromagnetic coupling, lifted from the golden-ratio entropy equilibrium by exactly one observation step. The capacity closure ties N to Λ as above. Both are contractions with certified unique fixed points; seed the pixel iteration with a deliberately bad guess and it reaches twelve digits in ten steps. Everything else is transported from P and N.

Quantity	Machine output	Measured
Inverse fine-structure constant α⁻¹(0)	137.03566	137.035999
Cosmological constant Λ	1.093 × 10⁻⁵² m⁻²	1.089 × 10⁻⁵² m⁻²
Planck-to-weak hierarchy v/E⋆	2.02 × 10⁻¹⁷, from P alone	matches, with Higgs naturality defect exactly 0
W and Z masses	80.33 GeV, 91.12 GeV	80.37 GeV, 91.19 GeV
Higgs and top masses	125.20 GeV, 172.35 GeV	125.09 GeV, 172.69 GeV
Koide invariant	exactly 2/3 from icosahedral face circulants	2/3 to five digits
Tau mass	1776.969 MeV, window 70 eV wide	1776.93 ± 0.09 MeV

The α row is the one we want argued over. The residual, 3.4 × 10⁻⁴ absolute, is ten thousand times the experimental uncertainty. The pixel equation contains no measured number, but the physical transport from the cell to the Thomson limit, including the hadronic vacuum polarisation, is the lane that has to close before the fixed point counts as a prediction rather than a diagnostic. The hierarchy problem, by contrast, has no residual to argue about: the weak scale is emitted by the source branch as P⋆^{-1/2} exp[−2π/(4α_U(P⋆))], and the Higgs mass is a normal-form readout with zero naturality defect, so there are no partner particles to look for.

Problems the machine answers

Problem	What the machine says	Where
Measurement problem	Measurement is a record write; Born weights are the unique agreement-compatible valuation; nothing collapses	learn ch. 10–11 · book ch. 19
Arrow of time	Repair strictly lowers a bounded potential and discards what it settled	learn ch. 9 · book ch. 16
Why 3 + 1 dimensions	Rank-three slow band plus ledger height; signature from squared height versus squared distance	learn ch. 3, 15 · book ch. 15, 7
Why Lorentz invariance	Conformal group of the observer’s sphere is SO⁺(3,1); boosts are modular flows	learn ch. 7 · book ch. 17
Why gravity is geometry	Generalised-entropy stationarity on collars, with both Jacobson inputs derived	learn ch. 16 · book ch. 24
Cosmological constant problem	Λ is a global capacity readout, invisible to local physics; N ≈ 3.3 × 10¹²²	learn ch. 17 · book ch. 28
Dark matter and dark energy	Non-luminous modular charge on collars; deep-regime law v⁴ = G M_b a₀ with flat rotation curves; w(a) from capacity	dark sector · book ch. 32
Electroweak hierarchy	Weak scale emitted by the pixel branch; Higgs naturality defect zero	learn ch. 25 · book ch. 29
Why this gauge group	Twelve ports plus internal implementation of A₅ force u(1) ⊕ su(2) ⊕ su(3)	learn ch. 21–22 · book ch. 25
Charge quantisation, proton stability, no monopoles	ℤ₆ quotient fixes the charge lattice; product adjoint has no X/Y bosons	learn ch. 23, 27 · book ch. 26
Yang–Mills mass gap	The gap is the repair gap between fast and slow bands	paper · book ch. 25
Value of α	Interval-certified pixel fixed point, 2.5 × 10⁻⁶ from CODATA; endpoint transport open	learn machine ch. 13 · book ch. 28
Why these laws and not others	The description that builds the machine and the description read from inside it are one system; the universe is its own fixed point	learn ch. 28 · book ch. 35, 36

The continuum, with things moving in it

Here is the whole picture assembled, because this is the point of the article.

Start with patches. There is no space and no time, and there are no fields. Each patch has twelve ports and a set of records, and each shared seam has a disagreement register. Repair runs. Disagreement falls. The network settles into the unique normal form that every schedule reaches.

Refine. Sew more patches into the sphere, at level after level of the geodesic icosahedron. The integer frames built from port labels sit inside a three-dimensional Euclidean carrier and become dense in it. That carrier is space. It is Euclidean because the twelve port vectors have the Gram matrix they have, and it is three-dimensional because only the rank-three slow band survives repair. Stack the ledger heights on top of it and the four-dimensional carrier with signature (1, 3) appears, the light cone selected by icosahedral rotations and boost invariance, the count of commits between two events converging to the Alexandrov volume of the interval between them. Malament’s theorem then hands you a Lorentzian metric, unique up to the conformal factor the counting supplies. In the refinement limit the finite ledger looks, from inside, like a smooth spacetime. General covariance is the statement that the ledger does not care what you call the commits.

Put something in it. A particle is a pattern in the port response that survives repair at every level of refinement. It has a definite spike in the spectral density, which is its mass, and it propagates with positive energy. Its motion is record drift across seams: the pattern’s records are re-established one seam over at the next tick, and the cost of that drift is momentum. A massless pattern drifts one seam per tick and can do nothing else, so it travels at c. A massive pattern pays a standing cost to hold itself together, which is its rest energy, and the arithmetic of rests and crossings keeps its speed below one. Two such patterns near each other change each other’s repair bias, and that bias, read by an observer, is a force: geometric repair is gravity, compact internal-transport repair is gauge force. Where records pile up, generalised entropy stationarity bends the chart, and the bent chart is curvature. An electron orbiting a proton, a photon crossing the room, a galaxy holding on to its stars, a planet keeping its orbit, are the same thing at different scales: refinement-stable patterns drifting across seams in a net of patches that keeps repairing itself into agreement.

Nothing in that paragraph was assumed. Space is what the slow band of repair looks like at large refinement. Time is the height of the commit ledger. Matter is what survives repair. Motion is where the records go next. The universe is the normal form of its own consistency.

Where we need you

We think we are on the right track. The reason is simple: we built a machine from three axioms with no physics in them, ran it, and the output is the universe we live in, structure by structure and constant by constant. That is either the first working implementation of simulation theory or a very elaborate coincidence, and we would rather find out which with the help of people who think differently than we do.

The construction has five open bridges, and they are the same five in every paper. The abstract three-dimensional carrier has to be identified with physical position and given a physical length. The internal repair tick and the modular ledger have to be calibrated against a laboratory clock and a laboratory energy. The finite port response has to be identified with a physical gauge current, together with a matter action and a continuum operator limit. The spacetime, modular, stress, entropy, and vacuum premises have to be realised by one common refinement tower instead of several independently declared ones. And the action has to be produced by the source rather than declared. Closing any one of these turns a row of the results table from a reconstruction implication into a physical theorem. The α endpoint transport and the family multiplicity are the two quantitative lanes we would most like a fresh pair of eyes on.

Everything is public. The flagship paper states every claim with its type (finite theorem, reconstruction implication, diagnostic, frozen prediction). The repository carries more than ten thousand machine-checked Lean theorems with no admitted proofs, exact rational arithmetic in place of floating point, pinned simulation receipts, a falsification program with mature mathematical and physical falsifiers, a frozen-prediction ladder whose kill bands are registered and timestamped before the comparison data is examined, and a collection of common objections with our answers.[^11] If you think an axiom smuggles in what it claims to output, if you can build a countermodel to the local diamond, if you know how to do the hadronic transport to the Thomson limit properly, if you see a cleaner route from the family band to three generations, or if you simply think the whole thing is wrong in an interesting way, open an issue or write to us.[^12] Thinking different is good. That is what the Institute is for.

If you are interested, please visit the Github repository, and join our R&D Telegram group.

License and patent policy

This essay links back to the canonical OPH license and open-use anti-patent covenant:

References

[^1]: Nick Bostrom, “Are You Living in a Computer Simulation?”, Philosophical Quarterly 53 (211), 2003.

[^2]: Konrad Zuse, Rechnender Raum, Vieweg, 1969 (English translation “Calculating Space”, MIT Project MAC, 1970). Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002. Gerard ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, Springer, 2016.

[^3]: Dénes Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 1986. Omar Fawzi and Renato Renner, “Quantum conditional mutual information and approximate Markov chains”, Communications in Mathematical Physics 340, 2015.

[^4]: M. H. A. Newman, “On theories with a combinatorial definition of ‘equivalence’”, Annals of Mathematics 43, 1942. B. Mueller et al., “Reality as a Consensus Protocol”, Observer Patch Holography paper stack.

[^5]: B. Mueller et al., “Finite Observer Consensus as a Reconstruction Principle: Normal Forms, the Standard Model Lie Type, and a Route to the Einstein Field Equation”, flagship paper of the Observer Patch Holography stack, release r2039.

[^6]: OPH Mini-Universe and the oph-physics-sim repository; protected-consensus evidence bundle at icosa_82k_protected_consensus_20260827_r1.

[^7]: Luca Bombelli, Joohan Lee, David Meyer and Rafael D. Sorkin, “Space-time as a causal set”, Physical Review Letters 59, 1987.

[^8]: David B. Malament, “The class of continuous timelike curves determines the topology of spacetime”, Journal of Mathematical Physics 18, 1977. S. W. Hawking, A. R. King and P. J. McCarthy, “A new topology for curved space–time which incorporates the causal, differential, and conformal structures”, Journal of Mathematical Physics 17, 1976.

[^9]: Joseph J. Bisognano and Eyvind H. Wichmann, “Duality condition for a Hermitian scalar field”, Journal of Mathematical Physics 16, 1975.

[^10]: Ted Jacobson, “Thermodynamics of Spacetime: The Einstein Equation of State”, Physical Review Letters 75, 1995. Ted Jacobson, “Entanglement Equilibrium and the Einstein Equation”, Physical Review Letters 116, 2016.

[^11]: OPH Falsification Program; Frozen-Prediction Ladder; Common Objections; Lean proof index; reproduction guide.

[^12]: Closure issues on GitHub; OPH hub at floatingpragma.io/oph; textbooks at learn.floatingpragma.io; the book at oph-book.floatingpragma.io; corresponding author bernhard@floatingpragma.ai.