The Ladder of Gaps Physics A: Stability Ladder and Statistical Mechanics Phase Transitions– An Axiomatic Derivation from Information Conservation to Energy Barriers
This paper constructs a four-tier axiomatic system of the stability ladder toreveal the underlying logic of phase transitions in statistical mechanics. Startingfrom information conservation (L0) and short-range interactions (L1), we rigorously prove, using DLR measure theory and large deviation principles in the thermodynamic limit, that the coexistence of macroscopically distinguishable states insystems of spatial dimension not less than two necessarily requires a strictly positiveenergy barrier in the free energy landscape (L2). In one-dimensional systems, information protection is instead realized via topological boundary conditions. Takingthe two-dimensional Ising model as a paradigm, we reconstruct the ontological basisof the existence of a phase transition in Onsager’s exact solution from the axioms,and extend the framework to general lattice systems and quantum many-bodymodels. The derivation does not rely on mean-field approximations or specific calculations of the renormalization group, thereby providing a necessary explanationfor the universality of phase transitions based on the logic of information conservation. In critical dynamics and quantum phase transitions, the barrier conceptis generalized to a dynamical barrier or quantum gap, demonstrating the broaderapplicability of the stability ladder.
This paper constructs a four-tier axiomatic system of the stability ladder toreveal the underlying logic of phase transitions in statistical mechanics. Startingfrom information conservation (L0) and short-range interactions (L1), we rigorously prove, using DLR measure theory and large deviation principles in the thermodynamic limit, that the coexistence of macroscopically distinguishable states insystems of spatial dimension not less than two necessarily requires a strictly positiveenergy barrier in the free energy landscape (L2). In one-dimensional systems, information protection is instead realized via topological boundary conditions. Takingthe two-dimensional Ising model as a paradigm, we reconstruct the ontological basisof the existence of a phase transition in Onsager’s exact solution from the axioms,and extend the framework to general lattice systems and quantum many-bodymodels. The derivation does not rely on mean-field approximations or specific calculations of the renormalization group, thereby providing a necessary explanationfor the universality of phase transitions based on the logic of information conservation. In critical dynamics and quantum phase transitions, the barrier conceptis generalized to a dynamical barrier or quantum gap, demonstrating the broaderapplicability of the stability ladder.
This paper places the condensed-matter quantum many-body system withinthe framework of a four-tier stability ladder axiomatic system, and systematicallydemonstrates the existence of a finite gap in the bulk excitation spectrum of topological order and the pairing gap of charged superconductor quasiparticles. Thecore logical chain shows that under the condition that the degenerate ground-statespace carries nonlocal information encoding, if the system satisfies causal closurewith local interactions, then in the thermodynamic limit the excitation spectrummust possess a strictly positive energy threshold; otherwise, the axiom of information conservation would be violated by local perturbations. Unlike known stability theorems (gap implies stability), this paper proves the converse proposition information encoding structure implies the existence of a gap. By rigorouslydistinguishing between the broad and narrow definitions of topological long-rangeentanglement, we place gapless metallic phases, quantum critical points, non-Fermiliquids, etc., outside the legitimate boundary of the axiomatic framework, explicitlystating that information conservation in these systems is achieved by alternativemechanisms such as topological invariants of the Fermi surface. The exact solutionof the Kitaev honeycomb model and BCS mean-field theory are used to verify thelogical self-consistency of the derivation, and the elimination of Goldstone modes bythe Anderson-Higgs mechanism is precisely pinpointed as the watershed betweencharged superconductivity and neutral superfluidity. The appendices systematically organise the Majorana fermion representation, the area law of topological entanglement entropy, Gor’kov Green’s functions, and quasi-adiabatic continuationtechniques, providing complete mathematical support for the axiomatic derivation.
This paper applies the four-tier stability ladder axiomatic system to classicaland quantum information coding theory, demonstrating that under the joint constraints of the information conservation design goal and the physical constraint oflocal noise, a coding system must necessarily exhibit the structural feature of anonzero error correction radius. The core argument uses a proof by contradictionto rigorously show: if the minimum distance of the code is less than 3, then localerrors can propagate without attenuation in the code space, causing mixing of logical states and directly violating the information conservation axiom. At the formalfoundation level, a precise mathematical definition of coding systems is established,and a bidirectional equivalence theorem between the information conservation axiom and the Knill–Laflamme condition is proved, with a complete classification ofboundary cases. Using the quantum surface code as an example, the ontologicalfoundation of the quantum error correction threshold is reconstructed from thestability axioms, and the framework is extended to general stabilizer codes andclassical linear codes. The entire derivation does not rely on any specific code construction, decoding algorithm, or explicit calculation of channel capacity, providingan axiomatic explanatory structure for the universality of the error correction radius in coding theory.
This paper systematically applies the stability ladder axiomatic system to faulttolerant quantum computation. Starting from two foundational axioms—the conservation of logical quantum information and the locality of quantum operations—itre-derives the necessary existence of a fault-tolerance threshold. The core argumentshows that under the constructive completeness condition that a quantum computing system must support arbitrarily deep reliable computation, hierarchical encoding is the minimal sufficient structure that coordinates the axioms. This structureinherently possesses a superlinear compression property of the error propagationmapnearthe origin, which via a fixed-point analysis necessarily yields a strictly positive threshold gap. This conclusion repositions the fault-tolerance threshold froma numerical discovery of specific code constructions to an analytical consequence ofthe axiomatic system, providing a unified framework for quantum fault-tolerancetheory rooted in information ontology. Using the concatenation of surface codesand the threshold theorem as an example, this paper establishes a logical divisionof labor between axiomatic derivation and constructive analysis: the former answers ”why the threshold must exist,” while the latter answers ”how the thresholdis concretely realized.”
Recent rigorous spectral analysis of stochastic transfer operators on non-associativealgebras shows that when the associator norm of the algebra is positive, the spectralgap of the system is locked in the thermodynamic limit by a strictly positive lowerbound depending only on the algebra dimension and an intrinsic constant. Thispaper points out that directly applying this abstract algebraic theorem to physicalpredictions for the Kitaev honeycomb model suffers from three fundamental gaps:the lack of a strict correspondence between the transfer operator spectral gap andthe Hamiltonian energy gap, the impossibility of directly generalizing one-bond estimates on a one-dimensional chain to the thermodynamic limit of two-dimensionalhoneycomb lattice points, and the dimensional gap between the low-dimensionalHilbert space of physical spin-1/2 degrees of freedom and the high-dimensional algebraic structure of octonions. To fill these logical gaps, this paper sequentiallyestablishes a quantum Cheeger-type correspondence between the imaginary-timeevolution transfer operator and the Hamiltonian energy gap, proposes a syntheticdimension scheme encoding four honeycomb lattice spin degrees of freedom into anoctonion degree of freedom, eliminates the lattice point dependence in the thermodynamic limit through a multi-bond cooperative variational principle and LiebRobinson quasi-locality, and rigorously derives the linear scaling relation betweenthe effective associator norm and the perturbation strength using quasi-degenerateperturbation theory. Based on these bridges, this paper derives three quantitative physical predictions without free fitting parameters for the Kitaev honeycombmodel with three-body perturbations: a rigid plateau of the magnon excitationgap, an exponential suppression of the low-temperature spin thermal conductivity,and the maintenance of the area law for entanglement dynamics. This paper further designs a complete experimental test scheme using a superconducting qubitarray, covering independent measurement of the effective associator norm, gapspectroscopy, thermal transport reconstruction, and a randomized measurementprotocol for entanglement entropy, and pre-specifies falsification conditions including statistical thresholds and operationalized trigger criteria, ensuring the theoryis falsifiable. The three-element bridging methodology of ”algebraic theoremphysical model– experimental test” constructed in this paper provides a reusabletemplate for converting rigidity theorems of other abstract algebraic structures intotestable quantum simulation predictions.
We study the informational underpinnings of thermodynamics and statistical mechanics, using an abstract framework, general probabilistic theories, capable of describing arbitrary physical theories. This allows one to abstract the informational content of a theory from the concrete details of its formalism. In this framework, we extend the treatment of microcanonical thermodynamics, namely the thermodynamics of systems with a well-defined energy, beyond the known cases of classical and quantum theory, formulating two necessary requirements for a well-defined thermodynamics. We adopt the recent approach of resource theories, where one studies the transitions between states that can be accomplished with a restricted set of physical operations. We formulate three different resource theories, differing in the choice of the restricted set of physical operations. To bridge the gap between the objective dynamics of particles and the subjective world of probabilities, one of the core issues in the foundations of statistical mechanics, we propose four information-theoretic axioms. They are satisfied by quantum theory and more exotic alternatives, including a suitable extension of classical theory where classical systems interact with each other creating entangled states. The axioms identify a class of theories where every mixed state can be modelled as the reduced state of a pure entangled state. In these theories it is possible to introduce well-behaved notions of majorisation, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. The three resource theories define the same notion of resource if and only if, on top of the four axioms, the dynamics of the underlying theory satisfy a condition called "unrestricted reversibility". Under this condition we derive a duality between microcanonical thermodynamics and pure bipartite entanglement.
This paper reinterprets the thermodynamic limit of statistical mechanics as aprojective dynamics process driven by the accumulation of informational potentialdifferences. By rigorously constructing the condensation operator as a sequenceof orthogonal projections on finite-dimensional Hilbert spaces, it is proved thatwhen the cumulative informational potential difference crosses the ln2 threshold,the projection sequence converges in the strong operator topology on the inductive limit Hilbert space to a limit projection, i.e., the system undergoes projectiveclosure. The commutativity between the regularised limit and the thermodynamiclimit is shown to be equivalent to the joint uniform convergence of the family ofprojection operators with respect to the regularisation parameter and the hierarchical index. Within this framework, the locking of Lee-Yang zeros on the imaginaryaxis of the complex magnetic field is shown not to be a consequence of finite-sizeextrapolation or reflection positivity, but rather a necessary consequence, in thethermodynamic limit, of the gap opening of the effective Hamiltonian when its realpart is non-zero and of the extremal condition of the dissipative cohomology. Thisconclusion relies on four premises: the axiom of information conservation, the axiom of computability, the ferromagnetic condition, and the Hermitian structure ofthe connection.