Theorems
The framework's derivations prove 661 named results (theorems, propositions, and corollaries) across 72 derivations. Each traces back to the three axioms (and any declared structural postulates), with proof steps available on each derivation page.
661
Named results
72
Derivations
14
Domain groups
9
Lean 4 verified
Axioms
Prop 1.4 C5 implies C4 Prop 2.2 Non-negativity Prop 2.3 Independence characterization Prop 2.4 Symmetry Prop 2.5 Chain rule Prop 4.3 Cauchy slices exist Prop 5.1 Ontological closure Prop 5.2 Subadditivity forces relational structure Cor 5.3 Necessity of strict subadditivity Thm 6.1 Existence of models
Thm 0.2 State space is a finite-dimensional compact smooth manifold Thm 0.0 Smooth structure Thm 0.1 Invariant Riemannian metric Prop 2.3 Drift bound Prop 2.5 Exact closure gives persistence Thm 3.1 Persistence requires exact closure Prop 3.2 Minimal period Prop 4.2 Loop is a smooth embedding Prop 4.4 Orbit decomposition Thm 5.1 Loop closure ↔ Noether pair Prop 5.2 Relationship between Axioms 2 and 3 Prop 7.2 Positive minimum Prop 8.1 Natural frequency Prop 8.2 Action-energy relation Cor 8.3 Planck-Einstein relation
Foundation
Thm 1.1 Forced aperiodicity Thm 2.1 Substitution matrix Prop 3.1 Constitutive universality decomposition Prop 4.1 Scale-independent packing coefficient Prop 1.1 Periodicity collapses C5 and individuation Prop 2.0 Observer networks are Delone sets Prop 2.1 Disorder breaks constitutive universality and the no-boundary condition Thm 1.1 Forced aperiodicity Prop 4.1 Bootstrap as substitution Prop 4.2 Two tile types Prop 5.1 Primitivity from bootstrap closure Thm 5.2 Pisot eigenvalue required Thm 2.1 Metallic mean parameter space Prop 3.1 CUP decomposition Cor 3.2 Solomyak guarantees half of CUP Prop 4.1 Scale-independent packing coefficient
Prop 1.1 What the axioms require of coherence Prop 2.1 What the axioms require of observers Prop 3.1 Neither layer alone is complete Prop 3.2 Co-formation, not emergence Prop 3.3 The joint constraint is selective Prop 4.2 The crystallization fraction as compatibility condition Prop 4.3 The packing coefficient as the bridge Prop 5.1 Reformulation of open problems
Minimal Observer
Thm 2.1 Equal rest frequency Cor 2.2 Equal mass Thm 3.1 Charge conjugation Cor 3.2 Multiple charges Prop 4.1 Dual self/non-self structure Cor 4.2 Dissolution operators Thm 5.1 Pair persistence Prop 5.2 Re-creation after annihilation Prop 5.3 Virtual pairs Prop 6.2 Properties of $C$ Prop 7.1 Self-conjugate / neutral observers
Prop 1.2 Coherence content requires non-trivial boundary Thm 2.1 Single observer has zero coherence Thm 3.1 Multiplicity Cor 3.2 Observer in the complement Prop 4.1 Mutual definition Prop 4.2 Pair creation is necessary Prop 5.1 Coherence budget Cor 5.2 Relational coherence is non-trivial Prop 6.2 Composite observer satisfies the observer definition Thm 6.3 No new dual required Prop 6.4 Relational invariants are self-conjugate Cor 6.5 No infinite regress Prop 7.1 C5 is vacuous on pairs Thm 7.2 Pairs are insufficient Cor 7.3 The observer network Cor 7.4 Simultaneous condensation Cor 7.5 Pre-geometric $t_0$
Interactions
Prop 1.2 Each level needs new algebraic structure Lean 4 Thm 2.2 Cayley-Dickson is the unique norm-preserving doubling Thm 2.3 Bootstrap forces Cayley-Dickson doubling Prop 3.1 Level 0 is $\mathbb{R}$ Prop 4.1 Pairwise interaction forces $\mathbb{C}$ Prop 5.1 Triple interaction forces $\mathbb{H}$ Prop 6.1 Quadruple interaction forces $\mathbb{O}$ Thm 7.1 Sedenion obstruction Lean 4 Cor 7.2 Three forces and no more Prop 8.1 Algebraic properties and physical consequences
Prop 4.2 Phase is the unique transferable quantity in Type I Thm 5.1 Exhaustive classification Prop 6.1 Type I transfers only phase Prop 7.1 Passage is self-reverse Prop 7.3 Decay coherence accounting Prop 7.5 Decoherence coherence accounting Prop 7.7 Dissolution coherence accounting Prop 8.1 Concentration vs. distribution
Thermodynamics
Prop 2.1 Strict positivity Thm 3.1 Existence of minimum cycle cost Prop 3.3 Quantization of action Prop 4.1 Action-energy-time relation Cor 4.2 Planck-Einstein relation Thm 5.1 Stationary action from coherence resonance Cor 6.2 Position-momentum uncertainty Cor 6.3 Energy-time uncertainty Prop 6.4 Structural interpretation
Dimensions
Spacetime
Prop 1.2 G as a ratio Prop 2.1 Fisher normalization of $T_{\mu\nu}$ Cor 2.2 Einstein equations with explicit $\hbar$ Prop 3.1 Unruh temperature from loop closure Prop 3.2 Clausius relation from coherence conservation Thm 3.3 Einstein equations from thermodynamics Cor 3.4 G from entropy density Prop 4.1 Gravitational self-consistency bound Prop 4.2 Saturation defines the Planck scale Prop 5.1 Dimensional independence Prop 6.2 Dimensional obstacle Prop 7.1 Structural consistency Prop 8.1 Area-scaling S1 would become a theorem Prop 9.1 Clifford normalization is conventional Prop 9.2 Fisher $\hbar$ does not propagate to the gravity sector Cor 9.3 Spinor/tetrad route does not determine $G$ Prop 10.1 Simultaneous condensation is forced Prop 10.2 Subadditivity reduces the net entropy Prop 10.3 Transcendental optimization Prop 10.4 Circularity diagnosis Prop 11.1 Pre-geometric condensation Prop 11.2 $t_0$ is not a Type III interaction Thm 11.4 Fixed-point characterization of $\ell_{\min}$ Prop 11.5 Variational characterization Prop 12.1 Periodicity trivializes C5 Prop 12.2 Disorder violates constitutive universality Cor 12.3 Aperiodic order is forced Prop 12.4 Substitution matrix constraints Prop 12.5 Packing coefficient from inflation factor Thm 12.6 Multi-scale self-consistency
Prop 1.2 Inverse-square falloff Thm 2.1 Gravitational redshift from equivalence principle Cor 2.2 Gravitational time dilation Thm 3.1 Geodesic principle Thm 4.1 Weak equivalence principle Thm 4.3 Strong equivalence principle Thm 5.1 Schwarzschild solution Prop 5.2 Event horizon Prop 5.3 Newton's constant
Thm 0.1 Homogeneity — now a theorem Thm 2.1 Time dilation Prop 2.2 Structural interpretation Thm 3.1 Lorentz contraction Prop 3.2 Single effect Thm 4.2 Lorentz group as loop closure symmetry Prop 5.2 Boost as hyperbolic rotation Thm 6.1 Speed limit from loop closure Cor 6.2 Massless observers Cor 6.3 No tachyons Thm 7.1 Poincaré group from homogeneity Prop 7.2 Noether charges Prop 8.1 Discrete Lorentz symmetries Prop 9.1 Wigner rotation as observer loop Berry phase
Prop 1.2 Classical singularities are inevitable Prop 1.3 Singularities signal framework breakdown Thm 2.1 Planck-scale cutoff Cor 2.2 Minimum spacetime volume Thm 3.1 Bounded curvature Cor 3.2 Maximum energy density Thm 4.1 Cosmological singularity resolution Cor 4.2 Bounce conditions Thm 5.1 Black hole singularity resolution Prop 5.2 Penrose-Hawking theorem evasion Prop 6.1 Trans-Planckian resolution Cor 7.1 Complementary resolution of black hole information loss
Quantum
Thm 0.1 Amplitude–coherence identification, formerly S1 Thm 6.1 Uniqueness of the Born rule Cor 6.2 Born rule for mixed states Thm 7.1 Hilbert space from coherence conservation Prop 8.2 Dimension condition Cor 8.3 Logical chain Prop 9.1 Probability as coherence fraction Prop 9.2 Frequency interpretation
Thm 2.2 Von Neumann coupling Thm 3.1 Collapse as relational invariant generation Prop 3.2 Properties of "collapse" Thm 4.1 Observer-relative descriptions Prop 4.2 Simultaneous correctness Thm 5.2 Wigner's friend resolution Thm 6.1 Measurement as coherence domain expansion Prop 6.2 Second law compatibility Prop 7.1 Resolution without additional postulates Prop 7.2 Closest to Rovelli, but derived Prop 8.1 Structural resolution
Prop 1.1 All measurement outcomes are observer-relative Prop 1.2 Basis decomposition is observer-relative Thm 2.1 Relational consistency Cor 2.2 Wigner's friend consistency Thm 3.2 Total coherence is observer-invariant Thm 3.3 Conservation laws are observer-invariant Thm 3.4 Network topology is observer-invariant Thm 4.1 Non-fabrication Cor 4.2 Observer-relative ≠ subjective Thm 5.1 No universal definiteness Thm 6.1 Trichotomy Cor 6.2 Dissolution of the subjective/objective dichotomy Prop 7.1 Structural positioning
Prop 1.2 Logical priority Prop 2.2 Operator structure Thm 3.1 Basis selection Prop 4.3 Uniqueness of basis Thm 5.1 Interaction-dependence of basis Thm 6.2 Complementarity from relational structure Cor 6.3 Uncertainty from complementarity Prop 7.1 Structural decoherence Prop 7.2 Comparison with decoherence Prop 8.1 Degeneracy resolution
Prop 1.2 Monogamy constrains the nerve Prop 2.0 Functoriality Prop 2.1 Hierarchy Thm 2.2 Sheaf condition for $\mathcal{C}$ and $\mathcal{P}$ Thm 3.1 Temporal $H^1 = 0$ Cor 3.1 Unique coherence future Cor 3.2 Unique probability future Thm 4.1 Uniqueness of $\mathcal{C}$ global section Thm 4.2 Uniqueness of $\mathcal{P}$ global section Thm 5.1 Outcome sheaf admits multiple local sections Thm 5.2 Non-globalizability of outcome sections Thm 6.1 Sheaf-theoretic trichotomy Cor 6.1 The central question answered Cor 6.2 No duplication
Prop 1.2 Hilbert space description Thm 2.2 Bell basis expansion Lean 4 Prop 2.3 Measurement generates a classical relational invariant Thm 3.1 Post-measurement state Prop 3.2 Coherence accounting Thm 4.1 Teleportation completion Cor 4.2 Classical communication is necessary Thm 5.1 Teleportation as coherence channel transfer Cor 5.2 Teleportation fidelity Prop 6.1 Higher-dimensional teleportation Prop 6.2 Entanglement swapping as iterated transfer
Particles
Prop 1.2 Properties of $C$ Prop 2.2 Parity and spin Prop 2.3 Parity of the coherence Lagrangian Prop 3.2 Properties of $T$ Prop 3.3 Time reversal of the coherence Lagrangian Thm 4.1 CPT is an exact symmetry Cor 4.2 CPT implies equal masses and lifetimes Cor 4.3 Individual C, P, T can be violated Prop 5.1 The three pillars
Prop 2.2 Level structure Thm 3.1 Dimensional transmutation Cor 3.2 Logarithmic naturalness Prop 4.1 Identification with physical scales Thm 5.1 Hierarchy stability Cor 5.2 No supersymmetry required Prop 6.1 Inter-level emptiness Prop 7.2 Mass-information reversal Prop 7.3 Exhaustion drives the transition
Gauge
Prop 2.1 Confining potential in quark-antiquark channel Prop 2.2 Scalar channel is most attractive Thm 3.1 Chiral condensate forms Prop 3.2 Condensate scale Thm 4.1 Goldstone bosons from chiral breaking Prop 4.2 Gell-Mann–Oakes–Renner relation Cor 4.3 Light pion mass Prop 5.2 Chiral condensate and color singlet structure
Prop 1.1 Bootstrap levels map to Cayley-Dickson steps Thm 1.2 Hurwitz ceiling Cor 1.3 The gauge hierarchy terminates at $\mathbb{O}$ Prop 2.2 Automorphism group of $\mathbb{O}$ Prop 4.2 Eight gluon fields Prop 4.3 Color charge Thm 5.2 QCD Yang-Mills equations Cor 5.3 Gluon self-interaction Prop 6.1 Non-associativity of octonions and confinement Prop 7.1 Asymptotic freedom from the bootstrap ceiling
Prop 1.2 Phase redundancy Thm 2.1 Local phase independence Prop 2.2 Gauge-invariant observable algebra Prop 2.3 Uniqueness of the gauge implementation Prop 3.3 Gauge transformation law Prop 4.2 Gauge invariance Prop 4.3 Holonomy interpretation Prop 4.4 Bianchi identity — homogeneous Maxwell equations Thm 5.2 Charge conservation Thm 6.1 Maxwell equations from uniqueness Thm 8.1 Lorentz force from coherence cost Prop 9.1 Electromagnetic waves
Thm 1.3 Crystallization is energetically necessary Prop 2.2 Goldstone's theorem and gauge boson masses Thm 3.1 W and Z masses from crystallization Cor 3.2 Mass ratio Prop 4.1 Higgs boson as radial mode Thm 5.1 Higgs mass is protected by the bootstrap hierarchy Prop 6.1 Fermion masses from crystallization
Thm 1.1 Gauge group completeness Cor 1.2 No GUT gauge bosons Thm 2.1 Coupling constants do not unify Cor 2.2 No proton decay from threshold effects Thm 3.1 Exact baryon number conservation Prop 3.2 Non-perturbative effects Prop 4.1 Gravitational $B$ violation Cor 4.2 Observational prediction Prop 5.1 GUT predictions vs. framework Prop 5.2 Falsifiability
Thm 1.1 Summary of the gauge hierarchy Thm 2.1 No fourth gauge factor Prop 2.2 Sedenions violate coherence conservation Lean 4 Thm 3.1 The gauge group is a product, not a simple group Cor 3.2 No grand unification Prop 4.1 Division algebra origin of fermion quantum numbers Prop 5.1 Gauge anomaly cancellation Lean 4 Thm 6.1 The Standard Model from division algebras
Prop 1.2 Three independent phase channels Thm 2.1 Quaternionic closure is forced Cor 2.2 Division algebra hierarchy Prop 3.2 Local quaternionic phase independence Prop 3.3 Covariant derivative Prop 3.4 Gauge transformation law Prop 4.2 Gauge covariance Prop 4.3 Self-interaction Thm 5.1 Yang-Mills equations Cor 5.2 Gauge boson self-coupling Cor 6.2 Spinor representations Prop 7.1 Topological basis for chirality Prop 8.1 Electroweak structure Thm 9.2 Weak isospin conservation Prop 9.3 Weak boson spectrum
Holography
Prop 1.2 Network is a causal set Prop 1.3 Poisson sprinkling at Planck density Prop 2.2 Geodesic variance Thm 2.3 Holographic bound fixes $\alpha_H = 1/4$ Cor 2.4 Strain power spectral density Prop 3.2 Poisson density statistics Thm 4.2 Quantum Jeans scale Thm 5.1 Gaussian cutoff Thm 6.1 Cross-prediction
Prop 1.2 Channel properties Prop 2.1 Entanglement from the channel Prop 2.2 No-signaling from relational invariants Prop 3.1 Coherence concentration curves spacetime Thm 3.2 Wormhole geometry from coherence channel Prop 3.3 Throat area from area scaling Thm 4.1 The wormhole is non-traversable Cor 4.2 Consistency with no-signaling Thm 5.1 ER=EPR is exact Prop 6.1 Thermofield double as maximal ER=EPR
Thm 1.2 Loop breaking Prop 1.3 Axiom conflict Thm 2.1 Pair production at the horizon Thm 3.1 Hawking temperature Cor 3.2 Inverse mass dependence Thm 4.1 Planck spectrum Cor 4.2 No interior information Prop 5.1 Mass loss rate Cor 5.2 Evaporation timescale Prop 5.3 Planck-scale endpoint Prop 6.1 Unruh equivalence
Flavor
Prop 1.4 Mixing arises from basis mismatch Prop 2.2 Coherence cost on $\mathcal{W}$ Thm 3.1 Discrete symmetry of the coherence cost Prop 3.2 Residual symmetries select bases Prop 4.1 Two-sector architecture Prop 5.1 Small CKM mixing from strong mass hierarchy Prop 6.1 CP phases from complex $A_5$ representation
Cosmology
Prop 1.1 Coherence-dual pairs are symmetric Prop 3.1 Maximal C violation from chirality Prop 3.2 CP violation from complex mixing phases Lean 4 Prop 4.1 Bootstrap crystallization as phase transitions Thm 5.1 Qualitative baryogenesis Prop 5.2 Order-of-magnitude estimate Cor 6.1 Baryogenesis requires $d = 3$
Thm 1.1 Weinberg angle from division algebra embedding Prop 2.1 Algebraic ratio constraint on couplings Prop 3.1 Self-consistent determination of $\Lambda_{\text{EW}}$ Thm 4.1 No gauge coupling unification Cor 4.2 Prediction: no proton decay from gauge boson exchange Prop 5.1 Estimate of $\alpha_{em}$ Prop 6.1 Status of $\alpha_s$ Prop 7.1 Two-loop stability of the electroweak sector
Prop 1.3 Horizon area and equation of state Cor 1.4 Coherence flux and equation of state Thm 2.1 No phantom energy Cor 2.2 Null energy condition for dark energy Thm 3.1 Uniqueness of the $w = -1$ fixed point Prop 4.1 Gibbons-Hawking characterization of non-self coherence Prop 4.2 Distinguishability of horizon-dominated interactions Prop 4.3 Minimum distinguishability requirement Prop 4.4 Two-sided bound on $\Lambda$ Prop 5.1 Bounded eternal expansion Prop 5.2 Connection to holographic noise Prop 5.3 Falsifiability
Prop 1.1 Planck-scale observers constitute geometry Prop 1.2 Horizon entropy is an observer census Prop 2.1 Substrate homogeneity from constitutive universality Prop 2.2 Cosmological homogeneity is inherited Prop 3.1 Particles are substrate resonances Prop 3.2 Crystallization depletes the substrate Prop 3.3 Two crystallization regimes Prop 4.1 Cosmological fractions are crystallization fractions Prop 4.2 Dark energy is substrate coherence Prop 5.1 The great desert is substrate-filled Prop 5.2 First crystallization scale Prop 6.1 Substrate perspective on double saturation Prop 6.2 Holographic bound on crystallization Prop 7.1 Holographic noise as substrate signature Prop 7.2 Unified source of dark energy constraints and holographic noise Prop 7.3 Cross-prediction with dark matter
Prop 1.1 CKM insufficiency Prop 1.2 Framework predicts Majorana neutrinos Cor 1.3 Leptogenesis is available Prop 2.2 Mass scale from framework Thm 3.2 CP asymmetry parameter Prop 3.3 Resonant enhancement Prop 4.2 Efficiency factor Thm 5.1 Lepton-to-baryon conversion Thm 6.1 Baryon asymmetry from leptogenesis Prop 6.2 Electroweak-scale leptogenesis is viable Cor 7.1 Heavy neutrino signatures Cor 7.2 Connection to $0\nu\beta\beta$
Prop 1.5 Epistemic horizons are observer-specific Prop 1.6 Viability uses the causal horizon; tightening uses the epistemic horizon Thm 2.1 Geometric bound Prop 3.1 Coherence budget Thm 4.1 Observer dissolution at Planck density Prop 4.2 Re-formation after bounce Prop 5.1 Coherence ontology Thm 5.2 Coherence conservation excludes the bounce Prop 5.3 Type II fusion reinforces the prohibition Thm 5.4 Sign prediction Prop 6.1 The bound does not explain the hierarchy Prop 6.2 Conditions for a tighter bound Prop 7.2 Cross-level geometric consistency Prop 7.3 Coherence absorption at bootstrap levels Prop 7.4 Coherence partition within a horizon volume Prop 7.5 The hierarchy is the second law Prop 8.3 Cross-level consistency Prop 8.4 Reframing the hierarchy Prop 8.5 Breakdown of the level-independence approximation Prop 8.6 The hierarchy from level 0's perspective Prop 8.7 Numerical consistency check Prop 8.8 The hierarchy requires an independent constraint
Thermo Extensions
Thm 2.1 Conservation of distinguishability Cor 2.2 Relational coherence is conserved Cor 2.3 Isometry of the coherence geometry Prop 3.2 Monotonicity of distinguishability Prop 4.1 Čencov uniqueness from conservation of distinguishability Thm 5.1 No-cloning Thm 6.1 No-deleting Cor 6.2 Coherence is a conserved resource Prop 7.1 Entropic restatement Prop 8.1 Čencov monotonicity of the coherence Hessian Thm 5.1 No-cloning
Prop 1.1 Observer states form a statistical manifold Prop 2.1 Coherence divergence properties Cor 3.2 Uniqueness of coherence geometry Prop 4.1 Metric identification Cor 4.2 Coherence cost as information distance Prop 5.1 ℏ as the coherence-information bridge Prop 5.2 Entropy as Fisher volume Prop 6.1 Fisher curvature and state space geometry
Prop 1.2 Hierarchy barriers Prop 2.1 Matching-rule confinement Prop 3.1 Correlated submanifold Prop 3.2 Dimension deficit Prop 4.1 Independence Thm 5.2 Conditional ergodicity Thm 6.2 ETH within a hierarchy level Thm 6.3 ETH failure across hierarchy levels Thm 7.2 Ultrametric phase-space structure Cor 7.3 Aging and history-dependence Cor 7.4 Stretched exponentials
Prop 1.3 Scale-dependent coherence conservation Thm 2.1 Exact coherence flow Cor 2.2 Effective coupling flow Prop 3.1 Effective coherence action Thm 3.2 Coherence flow equation — analog of Wetterich-Morris Thm 4.1 Fixed-point correspondence Cor 4.2 Asymptotic behavior Thm 5.2 Coherence c-theorem Cor 5.3 Connection to entropy Prop 6.1 No Landau poles from coherence conservation Cor 6.2 Asymptotic safety conjecture