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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

Consciousness & ethics

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

Derivation Tier Status Anchors Predictions
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

Fundamental constants

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

Derivation Tier Status Anchors Predictions
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
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
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
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
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
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

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.