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	Predictions
Critical Recognition Loading Phase transitions occur at canonical critical recognition loadings	Derived	THEOREM	1	0
Registry Predictions Proved The RS prediction registry forms a single proved bundle	Derived	THEOREM	1	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

Fundamental constants

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

Derivation	Tier	Status	Anchors	Predictions
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

Cosmology

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

Derivation	Tier	Status	Anchors	Predictions
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

Gravity

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

Derivation	Tier	Status	Anchors	Predictions
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

Mathematics & foundations

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

Derivation	Tier	Status	Anchors	Predictions
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

Particle physics

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

Derivation	Tier	Status	Anchors	Predictions
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

Quantum mechanics

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

Derivation	Tier	Status	Anchors	Predictions
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

How to read these pages

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.

Direct URL: https://pith.science/derivations/<slug>. JSON form: https://pith.science/derivations/<slug>.json. Search: https://pith.science/derivations/search.json?q=.... No nav link by design. Cite a derivation by its stable URL plus the version number printed at the top of the page.

Derivations from first principles

Recently published

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