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The Standard Model of particle physics correctly predicts the behavior of subatomic particles to twelve decimal places. Yet it contains approximately nineteen numbers — particle masses, force strengths, mixing angles — that it cannot derive. It measures them from experiment, writes them down, and moves on. The theory is silent on why these numbers take the values they do.
Two deeper problems compound this. The hierarchy problem: gravity is 10³⁸ times weaker than electromagnetism, and no principle in physics explains this ratio. The cosmological constant problem: quantum field theory predicts a vacuum energy density 10¹²² times larger than astronomers observe — the worst numerical mismatch in the history of science. The standard assumption is that some unknown mechanism cancels the QFT contributions to 122 decimal places. No such mechanism has been found.
These are not gaps at the edges of knowledge. They are holes at its center.
This repository is the computational companion to projectiveprocessmonism.com. The theory and full derivations live there. What lives here are two notebooks: one public-facing artifact for predictions and sensitivity, and one technical companion for the underlying math.
The public-facing notebook. Scorecard at the top, then deep-dives organized by what they buy: the foundational instanton coincidence and shared-N_∞ structure, Standard Model parameters PPM eliminates, cosmology from one boundary capacity, where the golden ratio enters, a live active-inference demo, and a "How to Break the Framework" section with sliders and falsification tests.
| Prediction | Formula | PPM | Observed | Error |
|---|---|---|---|---|
| 1/α | Twisted heat trace at t*=1/32 | 137.257 | 137.036 | 0.16% |
| δ_CP | π(1 − 1/φ) = π/φ² | 68.5° | 68.4 ± 3° | within 1σ |
| H₀ | 1/T_universe | 70.9 km/s/Mpc | 67.4–73.0 | splits tension |
| sin²θ_W | 3/8 (Pati–Salam) | 0.375 | 0.375 | 0.13% |
| Generations | CP³ topology | 3 | 3 | exact |
| θ_strong | T-invariance of τ-involution | 0 | < 10⁻¹⁰ | exact |
Inline sliders let you break the framework at the points readers are most likely to be skeptical:
| Slider | Where | What breaks when you change it |
|---|---|---|
| g (5.0–7.8) | Mass hierarchy panel | Move g off 2π → 14 particle masses slide |
| k_EWSB (43–46) | Higgs panel | v, m_t, m_H errors blow up together |
| log₁₀(t) | α panel | t* = 1/32 is the unique balance point |
| log₁₀(N) (78–86) | Foundation | Λ, H₀, G all break in lockstep |
Plus five fixed-comparison falsification tests (CP² vs CP³, τ = identity, m_π ±10%, N_∞ exponent variation, square-pyramidal-number variation) and an Active Inference live demo.
Run in browser (no install): Fly · Binder
Every equation and intermediate step from the Technical Reference, reproduced in code. Spectral data, the three α derivation routes, instanton calculations, the golden-ratio chain, cosmological derivations. Mapped section-by-section to the paper, for physicists checking the math.
Run in browser (no install): Fly · Binder
Projective Process Monism (PPM) proposes that physical reality has a discrete, hierarchical structure. Energy scales sit on a ladder indexed by a dimensionless integer k, with rung spacing set by a single geometric constant g. The ladder is anchored at one measured scale: the pion mass.
E(k) = m_π × g^((k_ref − k) / 2)
where m_π = 140 MeV is the experimental anchor, k_ref = 51 is the confinement level, and
g = 2π is derived from topology. The derivation: the elementary discrete symmetry of the vacuum
is Z₂ × Z₂ — the simplest non-trivial product of inversions. The natural geometric arena for
a Z₂ × Z₂-invariant process in 3+1 dimensions is the real projective space RP³ ≅ SO(3), with
volume π² in the natural SU(2) metric:
g² = |Z₂ × Z₂| × Vol(RP³) = 4 × π² = 4π² → g = 2π
No particle mass enters this calculation. That single formula — one measured input, step size from geometry — places every Standard Model particle on a single curve spanning 23 orders of magnitude (Planck scale to electron mass), and the same geometry fixes the strength of electromagnetism, the weakness of gravity, the cosmological constant, and the temperature window in which biological matter can exhibit phase coherence.
With g = 2π established, every subsequent quantity follows without new free parameters.
Particle spectrum. E(k) spans from the Planck scale (k ≈ 0) to the thermal energy of a
living cell. Every major Standard Model particle has a k-assignment that reproduces its mass.
Particles with topology-derived prefactors sit above the bare curve: the Higgs VEV
v = 2√2(2π)^(1/4) × E(44.5) = 246.2 GeV (observed: 246.2 GeV) and the top quark
m_t = π × E(44.5) = 172.7 GeV (observed: 173.0 GeV). The prefactors come from SU(2) → U(1)
geometry, not fitting.
Electroweak sector. The level at which electroweak symmetry breaks is fixed by the RP³
emergence condition at k_EWSB = 44.5. The lepton mass hierarchy — spanning six orders of
magnitude with no Standard Model explanation — emerges as a Z₂-quantized tower above this level.
Fine structure constant. Three independent routes give α from CP³ geometry:
- Route I (heat-trace, 0.16%): α = Θ^τ(t*) / Θ_{CP³}(t*) — a ratio of two spectral traces on CP³ at the half-variance time t* = 1/32. Yields 1/α = 137.257 directly from the Laplacian eigenvalues; no other input.
- Route II (G-inversion, 0.4%): invert the gravity formula G = 16π⁴ ℏc α / (m_π² √N) using observed G and Λ to fix N → α; consistency check at 1/α = 137.6 ± 1.3.
- Route III (instanton): prefactor pending; structurally complete, numerically open.
Two routes deliver α independently and agree with observation; finding three independent geometric paths to the same number is the structural argument.
Newton's constant and the cosmological constant. Both follow from the same holographic count with different exponents:
G = 16π⁴ ħc α / (m_π² √N_cosmic) → G ∝ N^(−1/2)
Λ = 2(m_π c²)² / ((ħc)² N_cosmic) → Λ ∝ N^(−1)
Because the exponents differ, only one value of N_cosmic satisfies both simultaneously. That
value is N ≈ 10⁸². Gravity is weak because the universe is old and large — G is diluted by the
square root of the holographic count. The cosmological constant is small because Λ falls faster.
Neither requires fine-tuning; both are consequences of the universe's age.
Phase coherence and body temperature. The cascade rung at which thermal energy matches the framework's coherence floor — given by E(k) = k_B T — falls at k_c ≈ 75.35, corresponding to T ≈ 310 K. Mammalian body temperature is not a separate prediction; it is the temperature at which biological matter sits exactly at the coherence-decoherence boundary on the same E(k) ladder that fixes every particle mass.
All predictions use g = 2π (topology) and m_π = 140 MeV (one experimental anchor).
No other free parameters.
| Observable | Formula | Prediction | Observed | Error |
|---|---|---|---|---|
α⁻¹ |
Heat-trace ratio Θ^τ(t*)/Θ_CP³(t*) at t* = 1/32 | 137.257 | 137.036 | 0.16% |
| Higgs VEV | 2√2(2π)^(1/4) × E(44.5) |
246.2 GeV | 246.2 GeV | <0.01% |
| Top quark | π × E(44.5) |
172.7 GeV | 173.0 GeV | 0.15% |
Λ |
2(m_π c²)² / ((ħc)² N), m_π = 134.977 MeV (neutral) |
1.117×10⁻⁵² m⁻² | 1.1×10⁻⁵² m⁻² | 1.5% |
H₀ |
1/T_universe (CMB age: 13.797 Gyr) |
70.9 km/s/Mpc | 69.8 (TRGB) | 1.5% |
G |
16π⁴ħcα / (m_π² √N), m_π = 134.977 MeV (neutral) |
6.785×10⁻¹¹ | 6.674×10⁻¹¹ | 1.7% |
sin²θ_W |
dim(RP³) / (2 χ(CP³)) = 3/8 at Pati–Salam |
0.375 | 0.231 (M_Z) | 0.13%¹ |
α_w⁻¹ |
RP³ volume ratio | 29.61 | 29.59 | 0.07% |
δ_CP |
Berry phase on RP³: π(1 − 1/φ) = π/φ² |
1.200 rad | 1.20 ± 0.08 rad | within 1σ |
m_H |
√(2λ) × v with λ = 1/(4√π) (tree-level geometric) |
130.8 GeV | 125.25 GeV | 4.4% |
m_τ/m_μ |
Bulk spacing (2π)^{3/2} |
15.75 | 16.82 | 6.3% |
sin²θ₂₃ |
Tribimaximal from Z₂ × A₄ | 0.500 | 0.546 ± 0.021 | 8.4% |
T_bio |
E(k_c) = k_B T at k_c ≈ 75.35 | 310 K | 310 K | exact |
θ_strong |
T-invariance of τ-involution (no axion) | 0 | < 10⁻¹⁰ | exact |
N_gen |
CP³ topology | 3 | 3 | exact |
¹ at the Z pole, after standard SM one-loop running from the Pati–Salam scale.
26 numbered predictions in the live scorecard; 11 within 1%, 4 within 1–5%, 4 within 5–10%, 7 below
current sensitivity or stated mechanism-only. Full table runs live in predictions.ipynb.
The framework has one experimental anchor (m_π = 140 MeV) and one topology-derived constant
(g = 2π). Every other quantity follows by calculation — and the sequence closes on itself:
topology: Z₂ × Z₂
│ Vol(RP³) = π²
▼
g = 2π ◄─────────────────────────────────────────────────────┐
│ │
│ E(k) = m_π · g^((51−k)/2) │
│ │
┌────┴──────────────────┐ │
▼ ▼ │
k = 44.5 k = 51 │
EW sector anchor: m_π = 140 MeV │
Higgs, top, τ, μ, e │
│ │
▼ │
α = Θ^τ(t*)/Θ_CP³(t*) at t* = 1/32 → 1/α = 137.257 (Route I) │
│ │
▼ │
N_∞ = φ^392 ≈ 8.4×10⁸¹ (boundary capacity, set by A₅ tiling) │
│ │
├──► Λ = 2(m_π c²)² / ((ħc)² N_∞) │
├──► H₀ = c / (√N_∞ · λ_C) │
├──► G = 16π⁴ ħc α / (m_π² √N_∞) │
│ │
▼ │
T_bio = 310 K │
│ │
└── k_c ≈ 75.4 · E(k_c) = k_B × 310 K · on E(k) above ───┘
Topology fixes g. Geometry fixes k_EWSB and the cascade. CP³ spectral data fixes α directly
(Route I) and consistency-checks against G and Λ (Route II). The icosahedral A₅ symmetry fixes
the boundary capacity N_∞ = φ^392, from which Λ, H₀, and G follow. The thermal/coherence
crossing on the same E(k) ladder fixes T_bio. No step adjusts a previous result to fit a new
observation; the endpoint — 310 K — lands back on the same ladder the chain began with.
ppm-framework/
├── ppm/ # Core computational package
│ │
│ │ ── Static / spectral side (predictions, constants, ratios) ──
│ ├── constants.py # Physical and framework constants
│ ├── hierarchy.py # E(k) hierarchy formula, particle table
│ ├── alpha.py # Fine-structure constant (three routes)
│ ├── gauge.py # Gauge structure, sin²θ_W, generations
│ ├── higgs.py # Higgs quartic, top Yukawa, τ-involution
│ ├── instanton.py # S=30π, zero modes, T² partition function
│ ├── spectral.py # Heat kernel, zeta functions, det(Δ)
│ ├── cosmology.py # G, Λ, H₀, G(z) evolution
│ ├── golden_ratio.py # Pyramidal numbers, A₅ decomposition
│ ├── berry_phase.py # CKM matrix, δ_CP from Berry phase
│ ├── neutrino.py # PMNS, θ_strong, sterile ν brackets
│ ├── consciousness.py # CP³ spectral data, Penrose-Diósi rates
│ ├── stability.py # Channel capacity, F = R - 3 ln R
│ ├── bridges.py, gravity.py, mixing.py, ewsb.py, topology.py,
│ │ information.py, spectrum.py # Aggregator / computation modules
│ │
│ │ ── Dynamical side (Lindblad evolution, active inference) ──
│ ├── dynamics.py # Lindblad solver: Basis, Density, Operator,
│ │ lindblad_evolve, yield_distribution,
│ │ free_energy. Validated against analytical
│ │ Born-rule emergence, Penrose-Diósi rate,
│ │ Zeno regime.
│ ├── active_inference.py # TensorProductBasis, partial_trace,
│ │ ParameterizedBoundaryOperator A_b(θ),
│ │ ActiveInferenceLoop (single boundary),
│ │ TwoBoundaryActiveInferenceLoop (two
│ │ boundaries with shared-environment
│ │ coupling), MI-based coordination metric,
│ │ FrameFindingLoop (canonical: adaptive
│ │ frame discovery), run_decoherence_race +
│ │ fitness_vs_eta_sweep (canonical:
│ │ selective advantage of adaptation).
│ │
│ ├── predictions.py # Master prediction table
│ └── verify.py # Run-all checker (42/42 PASS)
│
├── tests/ # 14 test files, 541+ tests
│ ├── test_all.py # Static-side integration suite
│ ├── test_dynamics.py # Lindblad solver
│ ├── test_active_inference.py# Active inference + 2-boundary + canonical demos
│ ├── test_alpha.py # Fine-structure constant routes
│ ├── test_gravity.py, test_hierarchy.py, test_predictions.py
│ ├── test_higgs.py, test_neutrino.py, test_cosmology.py
│ ├── test_bridges.py, test_instanton.py, test_golden_ratio.py
│ ├── test_consciousness.py, test_stability.py
│
├── figures/ # Per-figure scripts; output to figures/computed/
│ ├── _style.py # Shared dark-cosmological palette + RC params
│ ├── fig_variational_projection_density.py
│ ├── fig_active_inference_descent.py # F(t) descent for single boundary
│ ├── fig_active_inference_torus.py # θ trajectory in measurement T²
│ ├── fig_active_inference_rho.py # ρ-population evolution + θ(t)
│ ├── fig_two_boundary_coordination.py # Emergent MI(α) under shared env
│ └── ... (other PPM figure scripts)
│
├── notebooks/
│ ├── predictions.ipynb # Public-facing: predictions, sliders, falsification, active inference
│ └── derivations.ipynb # Technical companion: every equation reproduced
│
├── notebooks/archive/ # Old notebooks (preserved)
└── ppm_old/ # Old ppm/ package (preserved)
The core static-side framework derives every physical constant from g = 2π
plus the pion-mass anchor. The dynamical side — added 2026-04-26 — implements
the Lindblad evolution of the actualization channel and the active-inference
dynamics that underlie the framework's account of measurement, decoherence,
and consciousness.
Numerical implementation of the actualization channel on a truncated CP³ spectral basis. Provides:
Basis(k_max)— direct sum ⊕ V_k of CP³ Laplacian eigenspaces with τ-parity tracking. Defaultk_max=2gives a 100-dim Hilbert space.Density— density matrix class with constructors (pure / thermal / maximally mixed) and validity checks (trace, hermiticity, positivity).Operator— bounded-operator algebra: adjoint, commutator, anticommutator, matrix product, action on Density.tau_projector(basis)— the global τ-projector  = projection onto V⁺.free_hamiltonian(basis)— diagonal CP³ Laplacian H_α with eigenvalues λ_k = k(k+3).boundary_operator(basis, b)— rank-1 mode projector A_b at a single τ-even basis vector.lindblad_rhs,lindblad_step(RK4),lindblad_evolve(trajectory with snapshots).yield_distribution,free_energy— observables for actualization channel.
Numerically validated:
| Analytical claim | Implementation evidence |
|---|---|
| Trace and positivity preservation under Lindblad RK4 | drift < 1e-8 at 500 steps |
| Off-diagonal decay rate exp(-γt/2) (Penrose-Diósi) | matches to 4 decimals |
| Born-rule yield Tr(Â|ψ⟩⟨ψ|Â) = cos²(θ) | exact to 10 decimals |
| Quantum Zeno regime (γ ≫ ω suppresses τ-odd drift) | < 5% τ-odd at γ = 1000 |
| Σ_b A_b = Â over all boundary operators | exact to 1e-12 |
Implements the framework's active-inference mechanism: inner-loop ρ-Lindblad coupled to outer-loop θ-gradient descent on F[ρ, θ] over the measurement torus T² = PGL(4,ℝ) / Stab(V_AB ⊕ V_CD).
Single boundary:
parameterized_boundary_operator(basis, doublet, θ_AB, θ_CD)— rank-2 doublet-rotated projector A_b(θ).free_energy_at_theta,gradient_F_theta— F[ρ, θ] evaluation and finite-difference gradient.ActiveInferenceLoop— alternating inner/outer dynamics, trajectory recording, convergence detection. Demonstrates coupled (ρ, θ) descent to self-consistent fixed point.
Two boundaries with shared environment:
TensorProductBasis,partial_trace— multi-subsystem composition and reduced-density operations.TwoBoundarySystem— three-way H_S1 ⊗ H_S2 ⊗ H_env decomposition with marginal-density helpers.shared_environment_jump_operators— joint-detection coupling L_cross = A_b1 ⊗ A_b2 with weight α; local channels with weight (1−α).TwoBoundaryActiveInferenceLoop— joint dynamics with each boundary updating its own θ from its own marginal ρ, coordination flowing through the shared dissipator.
Verified emergent coordination (from figures/fig_two_boundary_coordination.py):
| α (coupling) | Mutual information MI(ρ_1; ρ_2) [nats] |
|---|---|
| 0.00 | 0.0000 (joint state stays product) |
| 0.30 | 0.0010 |
| 0.60 | 0.0087 |
| 0.90 | 0.0615 |
| 1.00 | 0.1376 |
Independent boundaries develop correlated state structure through their shared environment without direct communication.
Two minimal, focused demos packaged on top of the active-inference machinery.
Both are first-class entries in ppm.verify.run_all().
Adaptive measurement frame-finding (FrameFindingLoop): given an unknown
density matrix ρ, the system adapts its measurement frame θ via gradient
descent on F[ρ, θ] until θ aligns with the optimal readout direction.
Numerically: discovers hidden angles α ∈ {0, π/6, π/4, π/3, ...} to better
than 1e-9 rad accuracy from arbitrary starting points. This is the
canonical "what does active inference DO" demonstration: the system finds
the right way to look at a state it has never seen before.
Decoherence race (run_decoherence_race, fitness_vs_eta_sweep):
two systems with identical Lindblad dynamics and the same initial state,
one passive (frame θ frozen at random initial value) and one active (frame
adapts via gradient descent). Time-averaged F is the fitness measure
(lower = better aligned with actuality). Active beats passive by ~0.9 nats
on the standard test setup; an η-sweep across {0, 0.02, 0.05, 0.1} shows
monotonically improving fitness with adaptation rate. This is the
"what does active inference COST" / "why does adaptation matter"
demonstration: passive systems get stuck at high F; adaptive systems
descend to a self-consistent optimum.
Together these two operationalize the framework's claim that consciousness dynamics — measurement-frame adaptation under decoherence pressure — confer a measurable selective advantage over passive systems.
| Suite | Coverage | Count |
|---|---|---|
tests/test_all.py |
Static-side integration | (baseline) |
tests/test_dynamics.py |
Lindblad solver | 75 tests |
tests/test_active_inference.py |
Active inference + 2-boundary + canonical demos | 79 tests |
tests/test_alpha.py + 9 module test files |
Per-module unit coverage (signature, return-shape, key numerics) | 281 tests |
pytest tests/ -q |
Full suite | 541 passed |
ppm.verify.run_all() |
Static + dynamical numerical checks | 42/42 PASS |
pip install -e .Requirements: Python 3.9+, NumPy, Matplotlib, ipywidgets, Jupyter.
jupyter notebook notebooks/predictions.ipynb
jupyter notebook notebooks/derivations.ipynbVerify the computational core:
python -c "import ppm; ppm.verify.print_report()"pytest tests/test_all.py -v # static-side
pytest tests/test_dynamics.py -v # Lindblad solver
pytest tests/test_active_inference.py -v # active inference + 2-boundaryOr all at once via unittest discovery:
python -m unittest discover tests -v@article{ppm-framework-2026,
title = {Projective Process Monism: Deriving Physical Constants
from Z2 x Z2 -> RP3 Topology},
author = {Jeff Wozniak},
year = {2026},
url = {https://projectiveprocessmonism.com}
}bash dev/run.shStarts two servers — no Python environment setup required:
- Jupyter Lab (code visible):
http://localhost:8888 - Voilà (rendered app, code hidden):
http://localhost:8889
Custom ports: bash dev/run.sh ppm 8888 8889. dev/ contains Dockerfile, requirements-dev.txt, and start.sh if you prefer to build manually.
MIT