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Pith / Derivations

Derivations from first principles

Every entry below begins from the Recognition Science primitives (ledger, tick, voxel, cost composition law) and ends at a measurable consequence. Each step is a clickable Lean theorem; every empirical claim carries a named falsifier. 222 published of 246 tracked.

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

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

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.