PMMD — Maximal Mutual Determination

From a Qubit System to the Standard Model via E₈ Foam

Author: Gianluca Genovese

This repository accompanies the PMMD framework paper. It contains the full LaTeX source and compiled PDF of the manuscript together with all the code used to produce the framework's numerical results — percolation criticality, the Bombelli–Sorkin dimension d_BS, the 3-fold merger statistics, the V₁₁₂/Koide phenomenology, the bond-bias RG runs, the τ-demarcation / dark-sector statistics, and the E₈→H₄ cut-and-project geometry.

Status.
This is v6.0, a work in progress.
v5.2 is the latest version previously archived on Zenodo; v6.0 is the current release.
New in v6.0: the manifestly M-free discrete relational action (both couplings fixed: θ=π from spin-½, β→∞ from maximal mutual determination, no free continuous parameter); the continuum-limit bridge theorem (rigorous d=2, asymptotic-safety d=4); discrete chain-uniqueness with the SIC formalisation of determination (K₄ = the qubit SIC tetrahedron). The primordial value Ω_prim = ±2π/3 — the minimal Z₃ topological charge on the 3-cycle (k=1), the Pancharatnam/phase-balance datum the chain propagates and that carries the chirality — and every derived scale are unchanged. See src/phenomenology/verify_sic_geometry.py. The framework is developed under an explicit epistemic stratification (Stratum 1 = rigorous; Stratum 2 = theorem with distributed/numerical proof; Stratum 3 = structural articulation; Stratum 4–5 = heuristic/order-of-magnitude). The code below supports specific claims at the stratum stated in the paper; please read each result's stratum there before citing a number as "derived."

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Repository structure

pmmd-framework/
├── paper/                     Manuscript (LaTeX source + PDF)
│   ├── PMMD_v6.0.tex
│   └── PMMD_v6.0.pdf
├── src/                       ALL code, grouped by topic
│   ├── flavour/               Koide (HPC + extrapolation), V₁₁₂, generations, PMNS
│   ├── gauge/                 SU(5)/SO(10)/SM identification (X-charge, Berry split, spinor)
│   ├── e8_structure/          E₈ roots, percolation p_c, 3-fold merger, group theory
│   ├── dimension_dBS/         Bombelli–Sorkin dimension d_BS (cut-and-project, growth, BLS)
│   ├── phenomenology/         foam probes, FSS, bond-bias RG, v6.0 geometry checks, hpc/ soliton pipeline
│   ├── dark_sector/           τ-demarcation surface / bubble stats, Ω_DM/Ω_B
│   ├── dynamics/              growth dynamics + induced-gravity (foam rigidity)
│   ├── electromagnetism/      qubit-native photon / EM dynamics
│   └── analysis/              result merging and combined analysis
├── retired/                   superseded scripts (the d_BS-at-p_c partial-order probe) — see retired/README.md
├── scripts/                   distributed-run launchers (bash / PowerShell)
├── data/                      small result/parameter files (JSON, NPZ)
├── figures/                   generated figures (PNG) — see docs/SCRIPTS.md
└── docs/                      protocols, run plans — and SCRIPTS.md (full per-script catalogue)

Full per-script documentation — what each script computes, where it hooks into the paper, how to run it, and how to read its output — is in docs/SCRIPTS.md. The index below is a short overview.

Code index

src/e8_structure/

• e8_group_theory.py — E₈ / W(E₈) group-theoretic computations (GAP-style checks, V₁₁₂ decomposition support).
• w_e8_character_analysis.gap — GAP script (verified against the character-table library): the permutation character R_perm of W(E₈) on the 240 roots = 1 ⊕ 8 ⊕ 35 ⊕ 84 ⊕ V₁₁₂ with ⟨R_perm,R_perm⟩ = 5; χ of V₁₁₂ = [31,4,4,4] on the four order-3 classes; and the 72 roots fixed by the A₂-Coxeter element (= E₆), the remaining 168 in 56 size-3 orbits.
• option_G_W6_decomposition.g — GAP script (verified): W(E₆)-module decomposition of V₁₁₂ via PossibleClassFusions — the A₂-Coxeter order-3 class (centraliser 311040, size 2240) gives the Koide split 58 ⊕ 27 ⊕ 27̅ (the other three order-3 classes give 40 ⊕ 36 ⊕ 36), with V₁₁₂|_{W(E₆)} dimension multiset {1:4, 6:3, 20:3, 30:1} = 112. Companion to option_B_V112_koide.py (flavour/).
• e8_percolation_2color_v2.py — two-colour (matter/antimatter) percolation on the E₈ foam graph.
• e8_percolation_implicit_v42.py — the current percolation-criticality estimator p_c ≈ 1/175 (implicit-adjacency Newman–Ziff; cited in the paper). The earlier partial-order versions (v43, v44) were the d_BS-at-p_c probe and are now in retired/.
• e8_percolation_hpc.py — high-performance percolation driver (numba JIT + fork-based shared memory; Linux): finite-size-scaling scan of the foam-graph p_c.
• e8_percolation_winhpc.py — cross-platform (Windows/Linux/macOS) HPC percolation driver (numba + shared_memory): same FSS scan.
• e8_percolation_2color_v2.py — two-colour (matter/antimatter) percolation; analyze_pc_cycle_sum.py — the 4-cycle correction to 1/p_c.
• e8_3fold_merger_stats_v2.py — 3-fold cluster-merger event statistics (Σ⁺/Σ⁻ merging).

src/dimension_dBS/

• compute_dBS.py, compute_dBS_full.py — Myrheim–Meyer / Bombelli–Sorkin combinatorial dimension on causal sets.
• compute_BLS_rigorous.py — Bombelli–Lee–Meyer–Sorkin continuum-limit dimension estimator.
• compute_df_L_scaling.py — fractal-dimension finite-size scaling in L.
• analyze_d_BS.py — analysis/aggregation of d_BS runs.
• cutproject_e8_dBS.py — ideal E₈ cut-and-project d_BS (box-counting 3.85, longest-chain 3.75–3.85, FSS 4.06). The retired sim_c_cutproject_dBS.py (the d_BS-at-p_c probe, ≈3.6) is now in retired/.
• growth_e8_dBS.py, growth_e8_phase_dBS.py — d_BS on the framework's own growth-dynamics ensemble (the Stratum-2 support, converging to ≈ 3.97).

src/flavour/ (Koide, V₁₁₂, generations, PMNS)

• option_B_V112_koide.py — V₁₁₂ Z₃ decomposition → Q = 2/3, angle θ = 2/9; pmmd_koide_hpc.py + extrapolate_koide.py — the continuum Koide value Q_K → 0.674; v112_construction.py, pmns_chain_architecture_v1.py, cycle_classifier_v1.py, … (full list in docs/SCRIPTS.md).

src/gauge/

• SU(5)/SO(10)/SM identification: berry_and_su5_split.py, su5_xcharge_and_orthberry.py, spin10_split_test.py, sm_identification.py.

src/phenomenology/

• option_D_d4_orbit.py — D₄ orbit structure of the chain base.
• sim_bond_bias_RG_v42.py — bond-bias renormalisation-group run (ε_macro).
• option_A_FSS_extrapolation.py — finite-size-scaling extrapolation (e.g. f_c·p_c).
• verify_sic_geometry.py — (new in v6.0) self-contained checks of the primordial C₃/K₄ structure: the canonical value Ω_prim = 2π/3 (the Z₃ phase balance Σ e^{iφ}=0, the per-step phase of a once-wound 3-cycle, the datum the chain propagates and that carries the chirality), and the SIC tetrahedron geometry (pairwise fidelity 1/3, Σnᵢ=0; four faces tiling S² with per-face Berry phase π/2, charge 1/4, total winding Q=1). The geometric reading (π/2) and the canonical value (2π/3) are the two readings of the same C₃ content, related by the k=1 Wess–Zumino normalisation (ratio 4/3). Run it directly.
• verify_geometry_remarks.py — (new in v6.0) self-contained checks of the v6.0 geometric remarks: E₈→H₄ two-600-cell ratio φ (Moody–Patera), the φ-power scaling of δ_9, the charged-lepton triple as a Berry triple, and the H₄-protected Lorentz-suppression estimate. Run it directly.
• foam_energy_functional.py — (new in v6.0) the foam energy functional (CP¹ sigma model + WZ/Berry + Bloch potential): extracts the BPS lump scale α_E8⁻¹·m_quantum ≈ M_Pl, the marginal Higgs (λ(M_sub)≈0, consistent with SM near-criticality), and the demonstration that the flavour hierarchy is a distinct sector (Koide/SU(3), not the energy functional).
• orbit_type_geometry_probe.py — (new in v6.0) honest probe of whether the foam's three icosahedral orbit-types give the generation hierarchy; finds O(1)/φ geometric ratios (not ~100–3000), localising the hierarchy in the flavour/overlap sector rather than the bare geometry.

src/dark_sector/

• tau_correlated_growth.py — τ-correlated cluster growth (visible/dark allocation).
• tau_demarcation_surface_stats.py — statistics of the τ-demarcation (null) surface.
• tau_finite_bubble_distribution.py — finite-bubble distribution of the τ allocation.
• omega_dm_baryon_patch_merger.py — Ω_DM/Ω_B ≈ 5.4 as a stochastic merger outcome.

src/analysis/

• analyze_3fold_merger_combined.py, merge_*.py — aggregation of distributed-run outputs.

scripts/

• launch_*.sh, relaunch_*.sh — launchers for the distributed Sim B (bridge-claim Poisson-universality) and 3-fold-merger runs.

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Running

python3 -m venv venv && source venv/bin/activate
pip install -r requirements.txt

# Quick self-contained check (a few seconds):
python3 src/phenomenology/verify_geometry_remarks.py

Most physics scripts depend only on numpy/scipy; some plotting uses matplotlib and some graph code uses networkx. The large percolation, d_BS, and Sim B runs are computationally heavy and were executed as distributed jobs; the scripts/ launchers and docs/ protocols document how. The data/ and figures/ folders hold representative small outputs, not the full distributed result sets.

Reproducibility notes (honest)

• verify_geometry_remarks.py reproduces, in seconds, the φ-geometry, Koide, and Lorentz-suppression numbers cited in the v6.0 remarks.
• The percolation p_c ≈ 1/175, the growth-ensemble d_BS ≈ 3.97, and the 3-fold-merger statistics require longer runs; finite-size values differ from the asymptotic ones by documented amounts (see the paper and docs/).
• The originally-envisioned percolation "Sim B / Sim C" probe of d_BS (the partial-order measurement at the bare critical density p_c) is retired: measured there, the dimension is the fractal critical-cluster value ≈3.6, not the macroscopic order. The macroscopic d_BS = 4 is carried by the mature, over-percolated foam (ideal cut-and-project 4.06, growth ensemble 3.97). The retired code is kept in retired/ for transparency; p_c itself (e8_percolation_implicit_v42.py) is not retired.

Citation

If you use this work, please cite the manuscript (paper/PMMD_v6.0.pdf) and, once published, the corresponding DOI. See CITATION.cff.

License

Code is released under the MIT License (LICENSE). The manuscript text and figures are © Gianluca Genovese.