Derivations

Domains: All Applied 7 Biology & medicine 1 Consciousness & ethics 11 Fundamental constants 23 Cosmology 37 Gravity 32 Mathematics & foundations 29 Particle physics 55 Quantum mechanics 27

Tiers: All Applied 23 Derived 148 Foundational 51

Recently published

Foundational THEOREM

Law of Logic Forces the Canonical Cost

The Law of Logic cost theorem: the canonical reciprocal cost J(x) = 1/2(x + x^-1) - 1 is the unique continuous positive-ratio cost satisfying reciprocity, normalization, the RCL...

Mathematics · v7
Foundational THEOREM

phi forcing complete

**COMPLETE PHI FORCING THEOREM**.

Mathematics · v6
Foundational THEOREM

The Golden Ratio as a Forced Self-Similar Scale

**THEOREM**: The unique positive closed ratio is φ.

Constants · v7
Derived MODEL

f gap

Anchored in `Constants.GapWeight`.

Constants · v6
Derived THEOREM

tau0 Calibrator from Constants

tau0 is calibrated against CODATA constants self-consistently.

Constants · v5
Derived THEOREM

Recognition Coupling Bridge

Recognition coupling encodes the bridge between micro and macro scales.

Mathematics · v5
Applied HYPOTHESIS

Sterile Neutrino Exclusion

Eg parametric exclusion of an eV-scale sterile neutrino.

Particles · v5
Foundational THEOREM

Count Law at D=3

Counting laws on the 3-cube reproduce the standard combinatorial bounds. If `CountLaw` holds for some encoding, then the family has cardinality exactly `2 ^ D - 1`.

Mathematics · v5

Applied

Engineering predictions involving phi-spacing, recognition bandwidth, and device-scale falsifiers.

Derivation	Tier	Status	Anchors
Phi-Ladder Plasmonic Modes Plasmon resonances align with the phi-ladder	Applied	HYPOTHESIS	1
Phi-Spaced Pulses Beat Uniform Pulses Phi-spaced excitation produces a measurable advantage over uniform spacing	Applied	HYPOTHESIS	1
Photonic Metamaterial from Phi Phi-spaced photonic crystals deliver the predicted bandgaps	Applied	HYPOTHESIS	1
Planet Strata From RS Planetary layering follows RS recognition octaves	Applied	HYPOTHESIS	1
Robotics Control from RS Information-limited control gains emerge for RS-aware robotic systems	Applied	MODEL	1
Critical Recognition Loading Phase transitions occur at canonical critical recognition loadings	Derived	THEOREM	1
Registry Predictions Proved The RS prediction registry forms a single proved bundle	Derived	THEOREM	1

Fundamental constants

Dimensionless constants and SI bridge quantities forced by J-cost, phi, and the RS unit system.

Derivation	Tier	Status	Anchors	Predictions
Explicit CODATA Handles in the Canon The canon ships concrete definitions of c, hbar, G as CODATA constants for cross-checks	Applied	MODEL	6	0
External Anchors: CODATA Bridges Empirical SI/CODATA anchors that the RS predictions are compared against	Applied	MODEL	4	0
alpha-G Score Card Cross-bracket consistency of alpha and G predictions	Derived	THEOREM	1	0
AlphaLock Window in the Unit Interval The locked alpha lives in (0,1); the certified numerical bounds reproduce the CODATA value	Derived	THEOREM	5	0
Boltzmann Constant from Recognition k_B is a single RS rung tied to the thermal recognition channel	Derived	THEOREM	4	1
CODATA Alpha is Inside the RS Band Explicit Lean check that the experimentally accepted alpha lies inside the certified band	Derived	THEOREM	2	1
Fermi Coupling Constant Inside CODATA Bracket The weak coupling G_F * (mu c)^2 / (hbar c)^3 lands inside the RS certified bracket	Derived	THEOREM	3	1
f gap	Derived	MODEL	1	0
Hartree and Rydberg Inside CODATA Brackets RS predicts the Hartree-to-rest-mass and Rydberg-to-rest-mass ratios within tight bands	Derived	THEOREM	4	0
Newton's Gravitational Constant G = phi^5 / pi in RS-native units; SI value reproduced via the unit-conversion bridge	Derived	THEOREM	4	1
Planck Constant in RS-Native Units hbar = phi^{-5} in (tick, voxel, coh) units; SI value follows from the calibration	Derived	THEOREM	3	1
Planck Length in RS Units The Planck length is reconstructible as a derived rung of the phi-ladder	Derived	THEOREM	4	1
Precision Band for the Fine-Structure Constant Lean-certified upper and lower bounds on alpha^{-1} place CODATA inside	Derived	THEOREM	7	1
Recognition Length lambda_rec The canonical recognition length is the unique balanced positive root of the curvature-cost equation	Derived	THEOREM	5	1
RS Lambda_rec Matches the Planck Length The recognition length lambda_rec coincides with the Planck length in SI	Derived	THEOREM	3	0
Strong Coupling alpha_s from the Gauge Sum alpha_s is forced by the RS gauge-sum prediction and its bounds	Derived	THEOREM	4	1
tau0 Calibrator from Constants tau0 is calibrated against CODATA constants self-consistently	Derived	THEOREM	1	0
The pi^5 Curvature Tuple is Forced The 5-dimensional config space forces a pi^5 angular factor	Derived	THEOREM	4	0
The RS Unit System is Self-Consistent tick, voxel, coh, and act compose consistently with derived c, hbar, G	Derived	THEOREM	4	0
All Fundamental Constants Proved Together A single Lean module collects the closed-form RS predictions of the core constants	Foundational	THEOREM	1	0
The Fine-Structure Constant α⁻¹ = 4π·11·exp(−w₈·ln(φ)/(4π·11)) is forced from J-cost uniqueness, φ self-similarity, and D=3; the certified band (137.030, 137.039) contains COD...	Foundational	THEOREM	15	2
The Golden Ratio as a Forced Self-Similar Scale	Foundational	THEOREM	4	1
The Speed of Light c is one voxel per tick by construction; the SI value 299,792,458 m/s is a unit conversion, not a measurement of the universe	Foundational	THEOREM	10	3

Cosmology

Large-scale observables including Omega_Lambda, H0, CMB structure, baryogenesis, and inflation.

Derivation	Tier	Status	Anchors	Predictions
Stellar Evolution Tracks Main sequence to white-dwarf transitions anchored to RS phi-ladder	Applied	HYPOTHESIS	1	0
Tensor-to-Scalar Ratio Bound RS predicts r within the BICEP/Keck-respecting band	Applied	HYPOTHESIS	1	0
XENON Detector Null Prediction RS predicts a null signal in XENON-style direct detection experiments	Applied	HYPOTHESIS	2	0
Baryogenesis Trajectory Detailed baryogenesis trajectory through the early universe	Derived	THEOREM	2	0
Baryon Asymmetry from Phi-Ladder Trajectory Baryon-to-photon ratio eta_B emerges without imposed CP violation	Derived	THEOREM	7	1
CMB Acoustic Peak Ratios First-to-second and second-to-third CMB peak ratios match RS BBN-anchored predictions	Derived	THEOREM	1	0
CMB Temperature From RS The 2.725 K CMB temperature is fixed by RS thermal anchors	Derived	THEOREM	2	1
Cosmic Void Topology Voids follow a Q3-driven topological distribution	Derived	THEOREM	1	0
Cosmological Constant Density Omega_Lambda = 11/16 The dark-energy density parameter equals exactly 11/16 in RS	Derived	THEOREM	4	1
Dark Energy Equation of State w(z) for dark energy is predicted; w_0 = -1 with small RS-driven evolution	Derived	THEOREM	5	0
Dark Energy Evolution Structure Time evolution of dark-energy equation of state across z	Derived	THEOREM	1	0
Dark Matter as Topological Recognition Defect DM phenomenology emerges from topological defects, no new particle required	Derived	THEOREM	3	0
Early Universe Snapshot Key parameters of the radiation-dominated era anchored to RS rungs	Derived	THEOREM	1	0
eta_B Prefactor Phi-rung prefactor determining the baryon-to-photon ratio	Derived	THEOREM	2	0
Flatness and Horizon Problems Dissolved Both classical inflation paradoxes are resolved by RS without fine-tuning	Derived	THEOREM	2	0
Hubble Tension Pipeline End-to-end pipeline reconciling local vs CMB H_0	Derived	THEOREM	1	0
Hubble Tension Resolution Z-aging shifts reconcile CMB and SH0ES H_0 within RS bounds	Derived	THEOREM	6	1
Inflationary Slow-Roll from J-Cost Slow-roll inflation drops out of an explicit J-cost inflaton potential	Derived	THEOREM	1	0
Inflation from the J-Cost Inflaton Inflation parameters emerge from a J-cost potential without ad hoc tuning	Derived	THEOREM	8	2
Inflation Models from Configuration Dimension RS limits the space of viable inflation models given the configuration dimension	Derived	THEOREM	1	0
Large-Scale Structure from BIT Kernel Sigma8 and structure formation parameters predicted by the BIT kernel	Derived	THEOREM	4	0
Light-Element Abundances from BBN H, He, Li primordial abundances reproduced	Derived	THEOREM	1	0
Primordial Nucleosynthesis Ratios BBN H/He/Li ratios reproduced from RS thermal anchors	Derived	THEOREM	2	0
Primordial Power Spectrum Scalar and tensor primordial spectra are computed from the J-cost inflaton	Derived	THEOREM	1	0
Recognition Generates Perpetual Complexity The cost-conserving dynamics cannot collapse to a trivial steady state	Derived	THEOREM	1	0
Reionization History The reionization optical depth and redshift are forced by RS thermodynamics	Derived	THEOREM	1	0
RS Baryogenesis (Gravity Tree) Baryogenesis from a gravity-induced symmetry	Derived	THEOREM	1	0
Scale-Invariance Selection Certificate The CMB scale-invariance signature is a structural RS prediction	Derived	THEOREM	1	0
Sigma8 Suppression RS structure-formation suppression brings sigma8 into observational consistency	Derived	THEOREM	1	0
Spectral Index n_s from J-Cost RS predicts a slightly red-tilted spectral index consistent with Planck	Derived	THEOREM	1	0
Supernova Classification Type Ia/Ib/Ic/II partitions match RS recognition channels	Derived	THEOREM	1	0
Thermodynamic Selection in the Early Universe The early-universe phase selection is fixed by RS thermodynamics	Derived	THEOREM	1	0
Vacuum Uniformity The RS vacuum is statistically uniform across cosmological scales	Derived	THEOREM	1	0
8-Tick and 45-Gap Synchronize at 360 The cosmic recognition cycle has period 360 = lcm(8, 45)	Foundational	THEOREM	4	0
All Cosmological Predictions Proved Together RS cosmology is shipped as one consolidated certificate	Foundational	THEOREM	1	0
Configuration Space is Five-Dimensional Three spatial + one temporal + one balance dimension forces 5	Foundational	THEOREM	5	0
Why Space Has Exactly Three Dimensions The recognition cycle's 8-tick structure forces spatial D = 3	Foundational	THEOREM	4	1

Gravity

Spacetime, curvature, black holes, gravitational propagation, and Regge-limit derivations.

Derivation	Tier	Status	Anchors	Predictions
8-Tick Resonance in Gravity Gravitational waves exhibit an 8-tick resonance signature	Applied	HYPOTHESIS	1	0
Acoustic Phase Levitation Engineered phi-locked acoustic field induces measurable phase levitation	Applied	HYPOTHESIS	1	0
Black-Hole Echoes from RS Bounce Predicted echo amplitudes in post-merger gravitational wave data	Applied	HYPOTHESIS	4	0
Gravitational Wave Amplitudes Predicted GW amplitudes for BH/NS mergers anchored to RS energy budget	Applied	HYPOTHESIS	2	0
Rotation Curves Without Dark Matter Galactic rotation curves emerge from RS ILG kernel without invoking dark matter halos	Applied	HYPOTHESIS	3	0
SPARC Galaxy Rotation Falsifier RS predictions for ~150 SPARC galaxies fall inside a falsifier band	Applied	HYPOTHESIS	1	0
Spin Foam Realization RS realization of loop quantum gravity spin foams	Applied	MODEL	1	0
Backreaction Audit Gravitational backreaction effects audited and bounded	Derived	THEOREM	3	0
Baryonic Tully-Fisher Relation The BTFR slope and intercept emerge from the RS gravity kernel	Derived	THEOREM	1	0
Black-Hole Entropy from the Ledger Bekenstein-Hawking S = A/4 ell_P^2 is a direct ledger count	Derived	THEOREM	3	1
Black-Hole Recognition Bandwidth Horizons saturate recognition bandwidth; entropy and temperature follow	Derived	THEOREM	1	0
Coercive Projection on the Recognition Substrate Existence and uniqueness of a coercive projection that anchors gravity to RS	Derived	THEOREM	1	0
Coherence Collapse and Fall Information-theoretic coherence collapse drives gravitational fall	Derived	THEOREM	3	0
Einstein-Hilbert Action from RS The classical gravitational action emerges from discrete Regge calculus on the ledger	Derived	THEOREM	2	0
Gravitational Lensing from RS Lensing deflection formula recovered from the RS curvature derivation	Derived	THEOREM	2	0
Hawking Temperature as a Rung T_H = hbar c^3 / (8 pi G M k_B) sits on a canonical phi-ladder rung	Derived	THEOREM	1	0
Information is Preserved at the Horizon Recognition flow preserves information; the firewall paradox is resolved	Derived	THEOREM	4	0
Information-Limited Gravity (ILG) Asymptotic Enhancement The galaxy rotation flat-curve emerges from phi-ladder corrections without dark matter	Derived	THEOREM	5	0
Newton's G Runs with Recognition Scale Effective G varies between galactic and lab scales by a calibrated factor	Derived	THEOREM	2	0
Nonlinear Regge Convergence Discrete Regge converges to GR at high resolution, with explicit error bounds	Derived	THEOREM	2	0
Radial Acceleration Relation The McGaugh RAR is a structural prediction of RS gravity	Derived	THEOREM	1	0
Schwarzschild Radius from RS r_s = 2GM/c^2 reproduced from the RS gravitational kernel	Derived	THEOREM	1	0
Stress-Energy Tensor from RS T_{mu nu} is a derived field on the recognition lattice	Derived	THEOREM	1	0
Weak-Field Conformal Regge Calculus Conformal Regge calculus reproduces post-Newtonian limits	Derived	THEOREM	2	0
Equivalence Principle Inertial and gravitational mass equal as a structural identity	Foundational	THEOREM	1	1
Five Canonical Causal Relations Two events relate in exactly five ways; the cone structure is finite	Foundational	THEOREM	2	1
Gravity Propagates at the Speed of Light Both ride the same recognition substrate; the ratio is exactly 1	Foundational	THEOREM	4	1
Quantum-Gravity Octave Duality Recognition octave structure dualizes gravity and quantum theory	Foundational	THEOREM	2	0
Regge Calculus on the Recognition Lattice Discrete simplicial curvature reproduces general relativity in the continuum limit	Foundational	THEOREM	6	0
Spacetime is Emergent Spacetime arises as a coarse-grained projection of the recognition substrate	Foundational	THEOREM	1	1
Special Relativity Emerges Lorentz invariance is a structural consequence of the recognition substrate	Foundational	THEOREM	1	0
Zero-Parameter Gravity All gravitational predictions are made with zero adjustable parameters	Foundational	THEOREM	3	0

Mathematics & foundations

The formal forcing chain: logic, cost, phi, discreteness, dimension, arithmetic.

Derivation	Tier	Status	Anchors	Predictions
Dimensional-Quantity Type System Definitions for L, T, M dimensions and dimensioned positive quantities	Applied	MODEL	3	0
Euler-Mascheroni Constant Band gamma is bracketed by phi-arithmetic; current bound 0 < gamma < 2/3	Derived	THEOREM	6	1
Recognition Coupling Bridge Recognition coupling encodes the bridge between micro and macro scales	Derived	THEOREM	1	0
Recognition Coupling Law The coupling law relating recognition variables across scales	Derived	THEOREM	1	0
Arithmetic is Categorical Across Realizations All admissible realizations of recognition extract the same arithmetic up to unique iso	Foundational	THEOREM	4	1
Calibration is Forced by a Unique Fixed Point The lambda=1 calibration is the unique fixed point of substitutivity	Foundational	THEOREM	2	0
Count Law at D=3 Counting laws on the 3-cube reproduce the standard combinatorial bounds	Foundational	THEOREM	3	0
Discreteness of Recognition States Continuous configuration spaces cannot host a stable recognition; the ledger is forced to be discrete	Foundational	THEOREM	3	1
J-Cost is Additive on Independent Events Cost composes additively whenever recognition events are independent	Foundational	THEOREM	3	0
J-Cost is Quadratic near the Identity The log-coordinate J satisfies a calibrated quadratic bound around zero	Foundational	THEOREM	3	0
Law of Logic Forces the Canonical Cost J-cost uniqueness from the Law of Logic	Foundational	THEOREM	1	0
Logic from Cost and Distinction How classical logic is reconstructed from the recognition primitive	Foundational	THEOREM	2	0
Modus Ponens as a Derived Operation The single inference rule of classical logic emerges from substitutivity in the recognition ledger	Foundational	THEOREM	1	0
Multi-Level Composition Forces the Golden Ratio Uniform scaling across hierarchical recognition layers reduces to phi	Foundational	THEOREM	3	0
Natural Numbers Emerge from Logic Peano arithmetic is reconstructed inside the recognition realization	Foundational	THEOREM	5	1
Nontrivial Linking Requires D = 3 Topological linking is the unique three-dimensional signature in the canon	Foundational	THEOREM	5	0
Observers Are Cooper-Paired Recognition References A coherent observer is forced as a zero-cost persistent reference state	Foundational	THEOREM	8	0
phi forcing complete	Foundational	THEOREM	1	0
Recognition Event Cost is Nonnegative Every recognition event has nonnegative cost; self-recognition costs zero	Foundational	THEOREM	3	0
Recognition is the Unique Stable Cost Minimum Every J-cost minimum is a recognition event; nothing else achieves stability	Foundational	THEOREM	7	1
The 45-Gap as 9 x 5 Structure The 45-rung gap factorizes into 9 and 5; both feed phi-arithmetic	Foundational	THEOREM	2	0
The Arrow of Time from Recognition Time orientation is forced by the asymmetry between balanced ledger postings	Foundational	THEOREM	2	0
The d'Alembert Recognition Composition Law J satisfies J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y), uniquely	Foundational	THEOREM	1	1
The Gray Cycle Realizes the 8-Tick Cycle A 3-bit Gray code with single-bit transitions gives the minimal recognition cycle	Foundational	THEOREM	4	1
The Recognition Cycle Has Period 2^3 The 8-tick cycle is the only power-of-two period consistent with the three-dimensional cube	Foundational	THEOREM	3	1
Two Limits Discharged The classical and quantum limits of the recognition framework are both formally discharged	Foundational	THEOREM	1	0
Unified Recognition Hub The umbrella module that ties forcing chain, constants, cosmology, and gauge predictions together	Foundational	THEOREM	1	0
Uniqueness of the Recognition Tracker Any recognition is recoverable from a unique extraction mechanism	Foundational	THEOREM	2	0
Why Reality Posts a Ledger Recognition events must close in a conservation-respecting ledger; balance and reciprocity are forced	Foundational	THEOREM	9	1

Particle physics

Standard-Model masses, gauge couplings, mixing matrices, and representation counts.

Derivation	Tier	Status	Anchors	Predictions
Sterile Neutrino Exclusion Eg parametric exclusion of an eV-scale sterile neutrino	Applied	HYPOTHESIS	1	0
String Theory from J-Cost If strings exist, RS forces specific compactification structure	Applied	MODEL	3	0
Supersymmetry is Spontaneously Broken at the RS Scale RS predicts a specific SUSY breaking pattern; sparticle masses lie above the EW scale	Applied	MODEL	1	0
W-Mass Anomaly Structure RS prediction for the W-mass anomaly relative to Standard-Model expectations	Applied	HYPOTHESIS	1	0
alpha_QED Running with Energy The QED coupling runs predictably from low to high energy	Derived	THEOREM	3	0
alpha_s Two-Loop Running The strong coupling runs through canonical RS thresholds at two loops	Derived	THEOREM	4	0
Baseline Numerical Mass Predictions Concrete numerical predictions for the full Standard-Model spectrum	Derived	THEOREM	1	0
Baseline RS Mass Predictions Closed-form mass predictions and their consistency lemmas	Derived	THEOREM	1	0
Charged Lepton Ladder Electron, muon, tau placed at rungs 27, 32, 37 on phi-ladder	Derived	THEOREM	2	3
CKM Quark Mixing Matrix All four CKM angles and the CP phase emerge from the cube topology	Derived	THEOREM	2	1
Coherence Exponent for the Mass Yardstick The exponent that ties the yardstick to phi^5 emerges from the cost-composition law	Derived	THEOREM	1	0
Electron Mass on the Phi-Ladder m_e is forced at rung 27 of the phi-ladder with proved positivity	Derived	THEOREM	4	1
Electroweak Symmetry Breaking Higgs vacuum expectation value emerges; no fine tuning	Derived	THEOREM	2	0
Electroweak VEV is Not a Free Parameter v lives on a specific phi-ladder window; the Higgs mechanism dissolves the hierarchy puzzle	Derived	THEOREM	6	1
Family Mass Ratios Are Proved Inter-generation mass ratios are exact phi-power identities	Derived	THEOREM	1	0
Gap-Function Forcing The Z-dependent gap correction is structurally forced; no free parameter	Derived	THEOREM	1	0
Gauge Boson Verification Cross-check of W, Z masses against RS phi-ladder	Derived	THEOREM	1	0
Higgs EFT Bridge Higgs effective field theory parameters live on canonical phi-rungs	Derived	THEOREM	3	0
Higgs EFT Low-Energy Limit The Higgs EFT reduces to canonical SM-Higgs in the low-energy limit	Derived	THEOREM	1	0
Higgs Observable Skeleton All Higgs observable predictions in one structure	Derived	THEOREM	1	0
Higgs-Yukawa Bridge Yukawa couplings sit on phi-ladder positions that reproduce fermion masses	Derived	THEOREM	2	0
J-Cost Perturbation Closes the Mass Sub-Leading Terms The 88-lemma perturbative expansion sets sub-leading mass corrections	Derived	THEOREM	2	0
Longitudinal Vector Scattering Unitarity RS preserves unitarity in WW->WW scattering at high energy	Derived	THEOREM	1	0
Mass Anchor Policy Rules that fix mass anchors and their gap windows	Derived	MODEL	3	0
Muon/Electron Mass Ratio The dimensionless ratio m_mu/m_e is bracketed by the RS phi-ladder spacing	Derived	THEOREM	1	1
Neutrino Mass Hierarchy Normal hierarchy is forced; inverted hierarchy is excluded	Derived	THEOREM	1	0
Neutrino Yardstick and Hierarchy Neutrino masses share a yardstick that fixes the normal hierarchy	Derived	THEOREM	3	0
Particle Physics Depth from RS Hierarchical particle physics emerges from RS recognition layers	Derived	THEOREM	1	0
Pion Mass Spectrum Charged and neutral pion masses sit at canonical light-quark rungs	Derived	THEOREM	1	0
PMNS Neutrino Mixing Matrix Three mixing angles plus CP phase forced by RS Q3 representations	Derived	THEOREM	4	1
Proton Mass QCD binding plus quark rungs give the proton mass within the RS gap window	Derived	THEOREM	2	1
Quark Coordinate Reconciliation Quark masses and mixing reconcile across reference frames	Derived	THEOREM	1	0
Quark Mass Ladder Up, down, strange, charm, bottom, top quarks at canonical phi-rungs	Derived	THEOREM	3	0
Radioactive Decay Types Alpha, beta, gamma decays fall out as RS recognition channels	Derived	THEOREM	1	0
Renormalization Group Transport Couplings, masses and operators transport along the RG flow consistently	Derived	THEOREM	2	0
Standard Model Group Structure SU(3)xSU(2)xU(1) emerges as the unique gauge structure from RS	Derived	THEOREM	2	0
Standard-Model Mass Hierarchy The qualitative ordering m_e < m_u < m_d < ... is forced	Derived	THEOREM	1	0
Strong CP and the Theta Vacuum The CP-violating theta_QCD is forced to be statistically zero	Derived	THEOREM	1	0
Strong Force from RS QCD confinement at low energy emerges as an RS color-binding bound	Derived	THEOREM	3	0
Symmetry Breaking Pattern All Standard-Model symmetry breakings reduce to RS recognition steps	Derived	THEOREM	1	0
Three Gauge Couplings Run Consistently alpha_QED, alpha_W, alpha_s run with calibrated thresholds	Derived	THEOREM	2	0
Topological Defects RS classifies stable topological defects in field theory	Derived	THEOREM	3	0
Top Quark MSbar Mass Top-quark MSbar mass with phi-ladder + RGE corrections inside CODATA bracket	Derived	THEOREM	1	1
Torsion Forcing in the Mass Sector The torsion-driven correction is the canonical fix for heavy quark closure	Derived	THEOREM	1	0
Weak Force from RS Charged- and neutral-current weak interactions emerge with the canonical mediator masses	Derived	THEOREM	2	0
Weak Gauge Coupling g_2 emerges from the RS gauge sum; alpha_W is bracketed	Derived	THEOREM	1	0
Weinberg Angle sin^2 theta_W The electroweak mixing angle is fixed by the RS gauge ratio	Derived	THEOREM	3	1
Why Three Fermion Generations The Q3 partition forces exactly three families	Derived	THEOREM	2	1
W, Z, H Masses on the Phi-Ladder Electroweak gauge and Higgs masses sit at canonical phi-rungs	Derived	THEOREM	2	3
W/Z Mass Ratio The W to Z mass ratio is the cosine of the Weinberg angle	Derived	THEOREM	1	1
All Three Gauge Couplings Proved alpha_em, alpha_W, alpha_s emerge from one recognition framework	Foundational	THEOREM	1	0
Mass Spectrum on the Phi-Ladder All Standard-Model masses follow m = yardstick * phi^{rung - 8 + gap(Z)}	Foundational	THEOREM	4	2
Q3 Representation Structure All Standard-Model representations sit in the Q3 partition lattice	Foundational	THEOREM	1	0
Spinor Structure Forces D = 3 Two-component spinors are the unique RS-compatible structure	Foundational	THEOREM	6	0
Yang-Mills Mass Gap A mass gap exists for the RS-derived Yang-Mills theory	Foundational	THEOREM	2	2

Quantum mechanics

Quantum mechanics as the small-tick limit of recognition dynamics.

Derivation	Tier	Status	Anchors	Predictions
Superconducting Qubit Coherence Predicted coherence times for transmon and fluxonium qubits	Applied	HYPOTHESIS	3	0
Bell Inequality Violation RS predicts the Tsirelson bound for Bell violations	Derived	THEOREM	1	1
Caldeira-Leggett Friction Quantum friction reproduced from RS recognition dissipation	Derived	THEOREM	1	0
Classical Limit Emergence Pointer-state decoherence reproduces classical mechanics at large recognition counts	Derived	THEOREM	2	0
Double-Slit Interference Pattern Classical double-slit fringes recovered from RS recognition flow	Derived	THEOREM	1	0
Entanglement Entropy Area Law Entanglement entropy scales with area, not volume	Derived	THEOREM	2	0
Holographic (Bekenstein) Bound Information content of any region is bounded by horizon area	Derived	THEOREM	1	0
Nonlocality Without Signaling RS preserves no-signaling under entangled recognition events	Derived	THEOREM	1	0
Page Curve of Black-Hole Evaporation Entropy follows the canonical Page curve; information is preserved	Derived	THEOREM	1	0
Path Integral Measure Feynman's path-integral measure is the RS recognition trajectory weight	Derived	THEOREM	1	0
Photoelectric Effect Einstein's photoelectric relation falls out of RS quantization	Derived	THEOREM	2	0
Photon Statistics Bose-Einstein photon statistics from recognition counting	Derived	THEOREM	1	0
Quantum Decoherence from J-Cost Decoherence rates are predicted by J-cost dissipation channels	Derived	THEOREM	2	0
Quantum Error Correction Bounds RS predicts logical error-rate scaling for surface codes	Derived	THEOREM	3	0
Quantum Field Operators from RS Standard quantum field operator algebra emerges from recognition operators	Derived	THEOREM	3	0
Quantum Hall Effect Integer and fractional Hall plateaus from RS phi-quantization	Derived	THEOREM	1	0
Quantum Mechanics at the Planck Scale Spacetime discreteness and quantum mechanics merge at the Planck scale	Derived	THEOREM	2	0
Quantum Tunneling from J-Cost Tunneling rates predicted from the J-cost barrier	Derived	THEOREM	1	0
Quantum Zeno Effect Frequent recognition events suppress dynamical evolution as predicted	Derived	THEOREM	1	0
Recognition Hamiltonian Spectrum The Hamiltonian spectrum is forced by recognition-coupling rules	Derived	THEOREM	3	0
Solid-State Physics from RS Band structure, semiconductor physics from RS configuration space	Derived	THEOREM	3	0
Statistical Mechanics from RS Boltzmann distribution and partition function recovered from recognition counts	Derived	THEOREM	3	0
Superconductivity and Superfluidity Cooper-pair recognition coherence drives both phenomena	Derived	THEOREM	2	0
Thermal Fixed Point Universality RS universality classes for critical phenomena	Derived	THEOREM	2	0
Dirac Equation from RS The Dirac equation arises as the RS spin-1/2 recognition kernel	Foundational	THEOREM	3	0
Schroedinger Equation Emerges from RS i hbar partial_t psi = H psi appears as the small-tick limit of recognition dynamics	Foundational	THEOREM	4	0
Wavefunction Collapse as Recognition Measurement-induced collapse is the RS recognition event in the small-tick limit	Foundational	THEOREM	2	0

Biology & medicine

Biological partitions and phi-ladder structures inherited from the recognition substrate.

Derivation	Tier	Status	Anchors	Predictions
The 1024-Tick Breath Cycle Eight octaves of recognition form the breath cycle 2^10 = 1024	Derived	MODEL	1	0

Consciousness & ethics

Sigma-equilibration, ethics, virtue, consent, and awareness as recognition-cost consequences.

Derivation	Tier	Status	Anchors	Predictions
Consciousness Bandwidth Awareness throughput is a recognition-bandwidth quantity	Derived	THEOREM	1	0
Consent Interface from J-Cost Consent is the J-cost-minimizing protocol for joint recognition	Derived	THEOREM	1	0
Extraction is Thermodynamically Unstable Net-extraction strategies cannot persist; equilibration always wins	Derived	THEOREM	1	0
Love as the Unique Equilibrating Virtue Among virtues that drive sigma to zero, love is the unique stable target	Derived	THEOREM	1	1
Moral Debt as Accumulated Imbalance Unredressed recognition imbalance forms a ledger of moral debt	Derived	THEOREM	1	0
Recognition Bandwidth Saturation Every physical system has a finite recognition bandwidth; saturation drives observable effects	Derived	THEOREM	3	0
Stake Graph The graph of who has a stake in a recognition event is forced by event topology	Derived	MODEL	1	0
Virtue Generators from J-Cost The classical virtues are J-cost generators of recognition equilibration	Derived	THEOREM	4	0
Virtues Form a Finite Lattice All possible virtue combinations live in a finite, enumerable lattice	Derived	THEOREM	4	0
Recognition Cost Model for Ethics All ethical evaluations reduce to a recognition cost J(stakes)	Foundational	THEOREM	1	0
Sigma Equilibration as Drive The drive to equilibrate recognition imbalance is the basis of ethical action	Foundational	THEOREM	2	0

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Pith derivations are a hybrid of narrative explanation and machine-checked proof. Anything tagged THEOREM is backed by a Lean 4 proof in the canonical Recognition library; the corresponding declarations are linked inline. A tag of HYPOTHESIS means the claim is empirically falsifiable but not yet formally derived; a MODEL tag means the claim is a definitional choice. OPEN is reserved for active gaps.

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Derivations from first principles

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Law of Logic Forces the Canonical Cost

phi forcing complete

The Golden Ratio as a Forced Self-Similar Scale

f gap

tau0 Calibrator from Constants

Recognition Coupling Bridge

Sterile Neutrino Exclusion

Count Law at D=3