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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

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

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.