Theorems
The framework's derivations prove 944 named results (theorems, propositions, and corollaries) across 99 derivations. Each traces back to the three axioms (and any declared structural postulates), with proof steps available on each derivation page.
944
Named results
99
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 Prop 5.4 Cauchy-slice conservation Prop 5.6 Cauchy-slice finiteness Prop 5.7 Cross-level coherence generation 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 Thm 0.5 Both realizations satisfy Axiom 3 Prop 2.3 Drift bound Prop 2.5 Exact closure gives persistence Thm 3.1 Persistence requires exact closure Cor 3.1 Memory-persistence tradeoff Prop 3.2 Minimal period Prop 3.3 Eternal observer exclusion Cor 3.2 Universal frequency grid 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
Thm 1.2 Axis 1 is exhaustive and mutually exclusive Thm 2.2 Axis 2 is exhaustive and mutually exclusive among ledgered observers Thm 3.1 Exhaustiveness Thm 5.1 Type-I quanta are intrinsically Internal-charge-carriers Thm 6.2 Substrate Co-Creation Thm 7.3 Existence: EWSB Higgs is the unique self-conjugate residue Prop 7.4 Multiplicity formula for single-irrep gauge-SSB
Prop 1.1 Formula substitution Prop 3.1 Per-axis $p_{\mathrm{phys}}^{\mathrm{eff}}$ Prop 3.2 Per-axis $d_{\mathrm{req}}$ Prop 4.1 Per-particle achieved distance Prop 4.2 Margin table at binding axis Prop 6.1 Neutrino floor structure Prop 6.2 Quark-lepton split at each generation Prop 6.3 Exponential generation hierarchy Prop 6.4 Bootstrap termination → no fourth generation Prop 8.1 What the audit establishes Prop 8.2 What the audit does NOT establish
Prop 2.1 Formation ≠ Preservation Prop 2.2 Preservation ≠ Detection Prop 2.3 Formation ≠ Detection Prop 3.1 Formation is necessary Prop 3.2 Preservation is necessary Prop 3.3 Detection is necessary Cor 3.4 Tripartite necessity Prop 4.1 Joint sufficiency Prop 5.1 Margin as exponential-over-logarithmic Prop 6.1 SM satisfies all three conditions Prop 6.3 Color-neutral hadrons satisfy all three
Prop 1.3 Framing from loop closure via the normal-bundle almost-complex structure Prop 1.5 Relational coherence = linking number Prop 2.4 CS level ratios from the division algebra chain Prop 2.5 Finite representation content at each level Prop 3.2 Topological sector of canonical 4D Yang-Mills is labeled by the spatial Chern-Simons functional Prop 3.3 $\theta = 0$ makes the topological sector maximally accessible Prop 3.4 Wilson-loop linking from the topological sector Prop 6.1 Möbius automorphism group of the horizon Prop 6.2 Chern–Simons modular equivariance
Prop 1.1 Spatial axis — null horizon integer content Prop 1.2 Temporal axis — loop closure integer content Prop 1.3 Algebraic axis — bootstrap homotopy integer content Thm 2.3 CPTP factorization Thm 3.2 Dual framing Cor 3.4 Theses A and A' as orthogonal slices Cor 3.5 Finite capacity as 4-volume of slack Prop 4.1 Spatial axis as HaPPY holographic code Cor 4.1.1 Spatial-axis code rate Prop 4.2 Algebraic axis as Kitaev topological code Prop 4.3 Temporal axis as continuous-time Floquet code Thm 5.1 Pairwise-commuting stabilizer product Thm 5.3 Cross-axis couplings make the product framework-specific Cor 5.4 Product structure with framework couplings is novel
Prop 1.1 Heisenberg-like resolution from coherence Prop 2.2 Explicit edge equation Thm 3.1 Same-mass edge at Compton Cor 3.2 Exact mass-independence in Compton units Prop 4.1 Asymmetric edge — analytic limits Cor 4.3 Massless mediators give formally infinite edge Prop 5.1 Symmetry of the edge condition for same-mass pairs Prop 6.1 Horizon-integer mechanism — symmetric Prop 6.2 Detection-threshold mechanism Thm 6.3 Mutual Opacity — same-level pairs Cor 6.4 Cross-level opacity Prop 7.1 Integer channel inventory
Prop 2.1 Holographic equivalence Prop 3.1 Space-like holographic equivalence Prop 3.2 Unitary equivalence of A and A' Prop 4.1 Null portions carry integer content Cor 4.2 Distinguished status of $\partial M_A$ Cor 4.5 Exterior cancellation Cor 4.6 Operational completeness of the sheaf-level description Cor 5.1 Horizon as canonical holographic surface Prop 6.1 Phase-resolution ladder Cor 7.1 Inter-level integer restriction Cor 8.1 Obstruction class is combinatorial Prop 4.1 null portions classification Cor 4.2 distinguished status of $\partial M_A$ Cor 4.5 exterior cancellation Cor 4.6 operational completeness of the sheaf
Prop 1.2 Observer sources the coherence field Prop 2.1 Coherence field satisfies Klein-Gordon Prop 2.2 Static form Prop 3.1 Green's function of the static Klein-Gordon operator Prop 4.1 Pattern signal for a point-source observer Cor 4.3 Dimensionless form Prop 6.1 Gauge-channel contributions Prop 6.2 Massless mediator limit — Coulomb tails Prop 6.3 Composite pattern signal Prop 7.1 Color channel effective Yukawa for color-neutral composites
Prop 2.1 Four intrinsic constraints on $M_A$ Thm 3.1 Minimal-observer projection Prop 3.2 Immediate consequences of Theorem 3.1 Prop 4.1 Level-indexed projection Prop 5.2 Presheaf structure on $\mathbf{Obs}$ Prop 6.1 Level-mismatched observers cannot glue to a single de Sitter background Prop 6.3 The $\Lambda$ hierarchy is the obstruction class
Prop 1.1 Composite edge equation Prop 2.1 Short-range dominance Prop 3.1 Long-distance edge determined by EM Prop 4.1 Color channel has no Yukawa-like isolated-source form Prop 5.1 Edge equation fails for isolated color-charged source Cor 5.3 Free quark is not a viable isolated observer Prop 6.1 Color-neutral composites have terminating flux tubes Prop 6.2 Exterior color signal Yukawa-screened at $\Lambda_{\mathrm{QCD}}$ Prop 6.3 Hadron edge is at the hadron scale Thm 7.1 Confinement from edge-viability Prop 8.1 Yukawa range formula Thm 8.2 Observed range hierarchy matches framework prediction
Prop 2.1 BKM pullback on spinor field configurations Prop 2.2 Ostrogradsky exclusion applies Prop 2.3 Spin-statistics forces linear-in-$\nabla\psi$ Thm 2.4 Uniqueness of the Dirac kinetic term Prop 2.5 Clifford factorization of the BKM pullback Prop 3.1 Mass from rest-cycle coherence content Thm 3.2 Mass term Prop 4.2 Weyl decomposition Prop 4.3 Chirality decomposition of the kinetic term Prop 4.4 Chirality decomposition of the mass term Cor 4.5 Massless-to-massive transition Prop 4.6 Gauge-coupling chirality from Chirality Selection Prop 5.1 Pseudo-real structure of the weak doublet Prop 5.2 Majorana condition Thm 5.3 Majorana mass term Prop 6.1 Generation replication from winding classes Prop 7.1 Spin-statistics from energy positivity Prop 7.2 Pauli exclusion Prop 7.3 CPT invariance Prop 7.4 Stress-energy tensor
Prop 1.1 Poisson density and per-cell displacement variance Prop 1.2 Gaussian per-cell distribution Prop 1.3 Per-cell bit-flip probability Prop 1.6 Below-threshold operation Prop 2.1 Spatial-axis achieved distance Prop 2.2 Algebraic-axis achieved distance Prop 2.3 Temporal-axis achieved distance Prop 2.5 Per-axis required distance Prop 3.1 Gauge interactions as Type III carrier exchanges Prop 3.2 Per-Planck-cell per-Planck-tick exchange rate Prop 3.3 Bit-flip per Type III event Prop 4.1 Channel independence at substrate level Prop 4.2 Additive composition at low rate Cor 4.3 Additive form of $p_{\mathrm{phys}}^{\mathrm{eff}}$ Prop 5.1 EM channel — spatial axis only Prop 5.2 Weak channel — spatial and algebraic axes Prop 5.3 Color channel — spatial, algebraic, and temporal-via-confinement Cor 6.1 Massless gauge mediators Cor 6.2 Confinement as noise threshold
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
Thm 1.1 Relational invariants are observers Prop 2.1 Iteration closure Cor 2.3 Integer action across the bootstrap closure Thm 3.1 Necessity of hierarchy Thm 4.1 Irreducibility Cor 4.2 Ontological irreducibility Prop 5.1 Bootstrap as categorical construction Prop 6.1 Floor Prop 6.2 Ceiling Cor 7.3 No boundary
Prop 2.1 Elementary creation is local phase coupling Thm 3.1 Chain requirement Prop 4.1 The redistribution class creates nothing Prop 4.2 Composite formation is currency-coupled Cor 4.3 Inheritance bottoms out in currency Prop 5.1 Finite inheritance chains Prop 5.2 Terminal characterization Thm 6.1 Currency Completeness Cor 6.2 The currency sector never idles
Prop 1.1 Phase values live in a group-like structure Prop 2.2 Unit spheres in composition algebras are Moufang loops or Lie groups Thm 3.1 Hurwitz, 1898 Cor 3.2 Unit-sphere closure classification Prop 4.1 Explicit sedenion zero divisor Prop 4.2 Axiom 3 inconsistent at sedenion level Thm 5.1 Bootstrap termination at $\mathbb{O}$ Prop 6.1 Integer invariants from homotopy Prop 6.2 No integer invariant at sedenion level Cor 6.3 Bootstrap-scale integer content for timelike surfaces
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 Null-trajectory carriers 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 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 1.2 Enumeration of candidate final states Thm 2.1 Charge conservation excludes all-neutral channels Thm 2.2 Lightest-charged-lepton condition excludes charged-lepton channels Thm 2.3 Mass-gap condition excludes charged-non-lepton channels Cor 2.4 Decay-immunity Thm 3.1 Electron lifetime is dissolution-limited Prop 3.2 Type III rate in the present epoch Cor 3.3 Practical immortality in the present epoch
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 Prop 7.1 Massive vectors as Type II composites Cor 7.2 Photon is a Type-I quantum, not on the ledger
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 1.5 Mutually exclusive Prop 2.1 Adjacent ambient elements form relational invariants Prop 2.3 Boundary invariants drain channel coherence Cor 2.4 Embedding ⇒ partial decoherence Thm 3.1 Strict irreducibility forbids ambient embedding Cor 3.3 Partial irreducibility ⇒ partial handle Prop 4.1 Wormhole manifold construction Prop 4.2 Wormhole sprinkling matches the framework's combinatorial pattern Thm 4.3 Handle existence for the framework's channel Thm 4.4 No faithful flat-space embedding Cor 4.6 Handle topology forced Thm 4.7 Spanning inextendible antichain Prop 5.1 Discrete cardinality identification Thm 5.5 Sprinkling realization for each $N_{\min}$ Thm 5.8 Quantitative channel-sprinkling match
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
Prop 1.1 Covering Prop 1.2 Level-indexed scope Prop 2.1 Three regimes Prop 2.2 Null character of the horizon carrier Prop 3.1 The horizon is a shell of null relational carriers Prop 3.2 Permanence Prop 4.1 The horizon shell's necessary kinematic properties Prop 5.1 The holographic bound counts horizon shell modes
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 1.1 Relational invariants are permanent Prop 1.2 Complete thermalization is axiomatically forbidden Prop 2.1 Minimal observers have duration but not memory Prop 3.1 Geometry is constituted and deconstituted by observer level Prop 4.0 Coherence redistributes to the minimal network Prop 4.1 Constraint saturation replaces epistemic equivalence Prop 5.1 Constraint saturation Prop 5.2 The DAG–state distinction Prop 5.3 Level-indexed saturation Prop 5.4 Markovianity of the dynamics Prop 6.1 State identity of initial and final configurations Prop 7.2 The four-phase cycle Prop 7.3 The second law is universally preserved Prop 7.4 The arrow of time is level-indexed Prop 8.1 The floor phase has no well-defined duration Prop 8.2 Scale is undefined at the floor Prop 8.3 "First cycle" is not well-posed Prop 9.1 Commensurable frequencies Prop 9.2 Exact periodicity Prop 9.3 The cosmological history is a closed loop Prop 9.4 Conditional status and the fixed-point connection
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 endpoint 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 Thm 8.10 Strict positivity of $\Lambda$ Cor 8.11 Numerical lower bound
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 1.1 Recap: persistence requires exact closure Prop 2.1 Relational invariant absorption expands the state space Prop 2.2 Re-closure at a nearby fixed point Prop 2.3 Only Type III interactions generate memory Prop 3.1 Minimal observer has zero memory capacity Prop 3.2 Complex observer has finite memory capacity Thm 4.1 Memory-Persistence Tradeoff Thm 5.1 Type II clock-pause Cor 5.2 Composite-level decay channel replaces constituent decay channel
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
Thm 1.3 Petz classification Prop 1.5 BKM is the entropy Hessian Prop 1.6 BKM is the Kubo–Mori metric Thm 3.1 Hessian-identity narrowing Thm 4.1 KMS/Gibbons–Hawking narrowing Prop 5.1 Spin-statistics is non-selective at this layer Prop 5.2 Bootstrap hierarchy reinforces but does not alone select Prop 6.1 SLD role Prop 6.2 Role separation Thm 7.1 Classical limit recovery Thm 7.2 Pure-state limit
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