{"id":50076148,"url":"https://github.com/davidmdrpi/geometrodynamics","last_synced_at":"2026-05-30T08:01:12.857Z","repository":{"id":344289865,"uuid":"1181274003","full_name":"davidmdrpi/geometrodynamics","owner":"davidmdrpi","description":"Geometrodynamics bulk antipodal mechanics. 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The\nchain is:\n\n```\n2π ledger  →  R*  →  (γ, transport, resistance)  →  ε  →  Compton bridge  →  m_e\n```\n\nEach arrow is a probe with a quantitative test; each parameter is\nidentified with a closure-quantum invariant or a Tangherlini-grid\nquantity computed on the same geometry. The full closure-quantum\nledger of the locked surrogate after the chain:\n\n| parameter         | locked value | structural identification |\n|-------------------|-------------:|---------------------------|\n| `action_base`     | 2π           | S³ great-circle action     |\n| `transport`       | 25.1         | 8π = 4·(2π)               |\n| `resistance`      | 0.2179       | 7π / 100                  |\n| `pinhole γ`       | 22.5         | Σ V_max[1..5] ≈ 22.0       |\n| `β` (τ-uplift)    | 50π          | locked closure quantum     |\n| `4β` (τ-uplift integer) | 100·2π | τ-uplift quantum          |\n| `R*` (outer radius) | 1.2626    | cross-species fixed point  |\n| `ε` (inner cutoff) | 3.51×10⁻⁴   | resistance / k_5⁴          |\n\nAt the inner-cutoff identification `ε = 7π/(100·5⁴)`, the\ndimensional bridge collapses to the clean Compton form:\n\n```\nℏ  =  m_e · R_MID · c\n```\n\nBAM is *dimensional-scale-incomplete only modulo m_e*. The\nremaining open work is in throat physics (THESIS.md \"self-consistent\nthroat radius\"), not the closure ledger.\n\n**Paper draft:** `docs/hbar_origin_note.md` — five-section narrative\nof the closure-ledger chain with the full quantitative trail.\n\n**Probe ledger:** `docs/hbar_origin_status.md` — every probe with\nresult, precision, and archive pointer.\n\n**Reproduce in seconds:**\n\n```bash\npython -m experiments.closure_ledger.inner_boundary_derivation_probe\n# Writes runs/\u003ctimestamp\u003e_inner_boundary_derivation_probe/{probe.json, probe.md}\n# Verdict: ε = resistance / k_5⁴ closes the Compton bridge to 0.04 %.\n```\n\n## Why progress is possible beyond Wheeler's geometrodynamics\n\nWheeler's original geometrodynamic programme had the right *instinct*\n— that what we call \"matter\" should ultimately be a property of\nspacetime itself — but it stalled in the 1960s and 70s for a concrete\nreason: it lacked the **global / topological machinery** needed to\nturn that instinct into a quantitative spectrum.  The continuum\nEinstein equations alone do not pick out discrete spectra; they\nadmit far too many solutions.  Wheeler's \"charge without charge\" and\n\"mass without mass\" remained slogans precisely because there was no\nmechanism to make them *count* anything.\n\nThe line continued here is concrete: discreteness arises from three\nindependent topological/geometric channels, all of which can be\nwritten down explicitly and integrated numerically.\n\n1. **Antipodal S³ closure.**  Compactifying the spatial slice as\n   S³ replaces the open continuum with a closed cavity, so any\n   field that closes on itself does so over a great circle of fixed\n   length 2π.  Resonance on a closed cavity is intrinsically\n   discrete; the closure constants (`action_base = 2π`, the\n   integer-winding lock `4β = 100·(2π)` for the τ lepton) are\n   *exact* topological invariants of this antipodal closure.  The\n   closure constants are not fitted; they are read off from the\n   global structure.\n2. **Non-orientable throat/shell spectra.**  A wormhole throat\n   that is non-orientable carries a Z₂ partition class (`p = ±`)\n   which is a real topological label, not a continuous parameter.\n   The unique orientation-reversing isometry of S³ that preserves\n   the Hopf bundle is `T = iσ_y` (derived in `embedding/transport.py`\n   without ansatz).  T² = −I is the 4π periodicity of spinors; the\n   partition splitting drives every mass-ordering inversion in the\n   shelled sector (the m_u \u003c m_d but m_c \u003e m_s pattern).  The\n   throat orientation is what makes spin-½ unavoidable rather\n   than imposed.\n3. **Uniform bulk distance from outer to inner.**  The throat\n   confines a radial coordinate to the finite shell `[R_INNER,\n   R_OUTER]` (geometric units; throat at `R_MID = 1`).  In tortoise\n   coordinates this becomes a finite interval with regular\n   boundary conditions, which produces a discrete eigenmode\n   spectrum (`tangherlini.radial.solve_radial_modes`) — bound\n   modes `u_{l,n}(r*)` with frequencies `ω(l,n)`.  This is the\n   bulk geometry's own quantization channel, independent of the\n   S³ closure but composing with it.\n\nWhat was missing in Wheeler's day — and what this package now\ndemonstrates operationally — is that these three channels **compose**.\nThe lepton ladder is a \"minimal closure\" spectrum where channel 1\n(S³ closure) dominates: each lepton mass scales with its global\npass-count winding `β·k²` on a nearly bare closure skeleton, locked\nby `4β_lepton = 100·(2π)`.  The quark ladder (added in this work)\nis a \"shell-coupled closure\" spectrum where channel 1 picks up the\nheaviest shell only and channels 2 and 3 — partition asymmetry on\nthe throat and bulk-mode coupling — determine the lighter shells.\nThree of the four quark-sector residuals derive from\n`tangherlini.radial.solve_radial_modes` and\n`tangherlini.alpha_q.derive_alpha_q` to within 1%, on the same\ntortoise grid that defines the radial bound modes (see\n`docs/quark_axioms.md` §8 for the full derivation log and the\nquantitative match per residual).\n\nThis is what allows progress: the right machinery for *quantitative*\ngeometrodynamics exists, it is just not the differential-geometric\nmachinery Wheeler had at hand.  Antipodal closure on a compact 3-space,\nnon-orientable throat topology, and bulk-mode confinement are each\nold and individually well understood; what is new here is putting\nthem together and showing that they reproduce charged-lepton masses\nto sub-percent and the six-quark mass ladder to ~1.6%.\n\n## What the Code Validates\n\n| Claim | Status | Evidence |\n|-------|--------|----------|\n| Charge quantisation from topology | **Verified** | c₁ = 1 to \u003c 1e-9 error |\n| Spin-½ from Hopf holonomy | **Demonstrated** | SU(2) sign-flip at 2π, illustrative |\n| Coulomb law from throat eigenmode | **Verified** | BVP matches Q/r to rel_err \u003c 3e-9 |\n| Two-throat Coulomb force on S³ (finite separation) | **Demonstrated** | S³ Green response → V ∝ 1/r, F ∝ 1/r² (flat limit); F ∝ 1/sin²ψ on S³; Gauss law exact (`two_throat_coulomb_probe`) |\n| α_q coupling ratios (no free parameters) | **Computed** | Forced-origin slope extraction |\n| Möbius half-integer spectrum | **Verified** | Numerical vs analytic \u003c 5% |\n| Meson energy conservation | **Verified** | Drift \u003c 1% over test window |\n| Bridge nucleation / string breaking | **Verified** | Correct daughter topology |\n| Hayward metric from throat density | **Derived** | n(r) → ρ(r) → m(r) → f(r) matches Hayward to \u003c 1% |\n| de Sitter EOS from Einstein eqs | **Derived** | p_r/ρ = −1 exact at all radii |\n| SEC violation for regularity | **Derived** | Penrose-required SEC violation confirmed (~85% of interior) |\n| Singularity avoidance (Hayward core) | **Derived** | K(0) = 24/l⁴ finite; metric now derived from throat density |\n| Geodesic completeness | **Modeled** | Hayward infaller decelerates; heuristic completeness criterion |\n| BH entropy from throat counting | **Consistent** | S_throat matches S_BH by construction (N set from area law) |\n| Charge without charge (BH) | **Modeled** | Net Q from orientation sum, Q/N → 0 for large M |\n| First law dM = T dS | **Checked** | Residual \u003c 5%, Schwarzschild limit only |\n| T from collective modes | **Derived** | T_mode matches T_surface_gravity to \u003c 1% for M ≥ 3 |\n| Core scale l ≈ Planck | **Derived** | l = 2M/√N ≈ 0.47 l_P, independent of mass |\n| Schwarzschild recovery | **Verified** | Hayward → Schwarzschild as l → 0 |\n| Two-horizon structure | **Verified** | Inner + outer horizons for 0 \u003c l \u003c l_crit |\n| Singlet from throat transport | **Constructed** | T=iσ_y → |Ψ⟩ built from transport; E(a,b) = −cos(a−b) |\n| T = iσ_y from Hopf fibration | **Derived** | Unique orientation-reversing Hopf-preserving map; 7 properties verified |\n| Bell phases from Hopf holonomy | **Derived** | π/2 baseline + π[cos(θ_a)−cos(θ_b)]/2 from connection A = ½cos(χ)dφ |\n| History closure → E = −cos(a−b) | **Derived** | SU(2) amplitudes × closure weights reproduce singlet; CHSH = 2√2 |\n| History no-signaling | **Derived** | Marginals = ½ from branch enumeration; independent of remote setting |\n| History conservation | **Verified** | Charge balance exact for Bell and transaction histories |\n| Bulk identity Bell (kinematic) | **Verified** | Same E(a,b) from pure topology, no time stepping; both paths match |\n| CHSH S = 2√2 (topological) | **Verified** | Exact Tsirelson value; topology determines correlations, cavity determines dynamics |\n| No-signaling | **Verified** | Marginals = ½ from singlet; cavity dynamics don't alter spin correlations |\n| Cavity detector-conditioned dynamics | **Dynamical** | Derived Hopf phases drive cavity ODE; packets fire on 0/π branches |\n| Cavity persistent memory | **Verified** | Energy persists between steps; slow ring-down |\n| Green kernel derivative | **Fixed** | Now matches analytic dG/dψ to \u003c 10⁻⁴ |\n| Lepton mass ladder (e, μ, τ) | **Closed** | Sub-percent all three generations from locked S³ axioms (see below) |\n| S³ action base `action_base = 2π` | **Locked** | Hard topological invariant; default in all lepton scans |\n| k=5 uplift `4β = 200π` (100 × 2π) | **Locked** | τ uplift equals exactly 100 S³ winding quanta |\n| Closure cycle integer-quantised in 2π | **Verified** | `(N_e, N_μ, N_τ) = (3, 6, 109)` from antipodal + Hopf-throat + radial BS + τ-uplift |\n| R_OUTER selected by cross-species fixed point | **Verified** | Bisection on each lepton gives same R* ≈ 1.262 to 0.008 % |\n| Pinhole γ ≈ Σ V_max[1..5] on Chebyshev grid | **Verified** | −2.2 % off the locked γ = 22.5; same operator as the QCD-sector γ_q |\n| Transport = 8π = 4·(2π) | **Verified** | +0.13 % off the locked transport = 25.1; 4th closure quantum |\n| Resistance = 7π / 100 | **Verified** | +0.94 % off the locked resistance = 0.2179; selected over `4·(ω−1)` by R_OUTER bisection |\n| Inner cutoff `ε = resistance / k_5⁴` | **Verified** | Closes the Compton bridge `ℏ = m_e R_MID c` to 0.04 % |\n| Closure-quantum ledger closes modulo m_e | **Established** | Every locked parameter is a closure-quantum invariant; m_e is the unique remaining external input |\n| Quark mass ladder (u, d, s, c, b, t) | **Fitted** | 1.6% max rel err on s, c, b, t with d-anchor, four shell-index axioms, and one phenomenological β |\n| Quark shell-index axioms (ε, η, χ, phase) | **Geometric** | All four expressible in `k_5 = 5` only: `(1−1/k_5², k_5, (k_5−1)·k_5, 0)` |\n| Quark residual sector (transport, pinhole, resistance) | **Derived** | Each matches Tangherlini eigenmode quantity within ~1% on the tortoise grid |\n| Pinhole = `Σ V_max(l=1..5)` (tortoise grid) | **Verified** | −1.09% off the fitted lock |\n| Transport = `mean ⟨u_l\\|V_{l+2}−V_l\\|u_{l+2}⟩` | **Verified** | +0.87% off the fitted lock |\n| Resistance = `transport · ln(α_q(k_5)/α_q(k_1))` | **Verified** | −0.43% off the fitted lock |\n| Quark winding β = N·π/2 with N=466 | **Phenomenological** | Compensator under all ablations; awaits an analytic closure condition |\n| Compton antipodal kinematics | **Verified** | Closure-compatible: front + back-mouth 4-momentum conservation under (E, **p**) → (E, −**p**); inter-mouth γ skew vanishes identically; throat-pinch skew is recoil-induced `O(ω²/m²)` |\n| Compton S³-propagator pole `1/(s−m²)` | **Verified** | S³ Green function `G(ψ) ∼ 1/ψ` with `ψ ∝ s−m²` reproduces QED propagator pole; fitted exponent 1.0002 across five ω-decades |\n| Thomson `(1+cos²θ)` angular factor | **Derived** | Polarization-summed BAM amplitude reproduces Klein-Nishina at ω → 0 from transverse photon polarisations on the tangent bundle |\n| Compton vertex coupling `γ = −3/2` at O(ω/m) | **Derived** | Exact analytic solution to the 4-equation linear system in {1, c, c², c³} basis; clean rational coefficient |\n| `γ = −3/2` is d-independent | **Verified** | Numerical γ(d) = −3/2 in d ∈ {3, 4, 5, 6, 8} to 7-digit precision; falsifies the embedding-dim/polarization-count origin |\n| Compton vertex closed-form resummation | **Derived** | `F²(x, c) = 4·x³·(x²+1−x·sin²θ) / [(1+c²)·(1+x)²]` with `x = ω'/ω` reproduces Klein-Nishina to all orders in ε up to ε ~ 2 (machine precision); the perturbative PRs #31–34 are Taylor expansions of this closed form |\n| F² and masses from one master integral | **Derived** | Single `C × S³` master functional `ℳ = G_C ⊗ 𝒢_{S³}`: ω-poles → mass spectrum, throat boundary → `K(x)`, S³ Hopf → `Q(x,c)`; vertex residue = `F²=K²·Q` to `2e-14`. Closes scaffold barrier B5′ (`master_integral_probe`, `docs/bam_scaffold_status.md`) |\n| Dimensional anchor (B4) is structural, not a gap | **Audited** | Closure-ledger/Maslov machinery is scale-free (rescale `R_MID → λ·R_MID` → all dimensionless outputs invariant), so exactly one external dimensionful anchor is required; relocatable to the cosmologically-invariant bulk separation `ΔR`, giving `m_e = 0.52·ℏ/(ΔR·c)` (`maslov_dimensional_bridge_probe`, `delta_r_scale_modulus_probe`) |\n\n### Research goals (not yet fully derived)\n\n| Physics | Proposed geometry |\n|---------|-------------------|\n| Electromagnetism | Curvature of the Hopf connection on S³ |\n| Charged-lepton ladder (e, μ, τ) | Eigenvalues of a k-pass instanton-transition matrix with S³ action base `2π` and k=5 uplift `200π` — **sub-percent fit achieved** |\n| Particle mass (general) | Eigenvalue of the 5D Tangherlini operator (leptons only so far) |\n| QCD confinement | 1D flux-tube network with bridge nucleation |\n| Retrocausal photon exchange | Wheeler–Feynman absorber theory on S³ |\n| Black-hole interior | Coherent condensate of non-orientable wormhole throats |\n| Bell correlations | Non-orientable throat transport + Hopf SU(2) projection |\n| Entanglement = wormholes | Bell correlations from throat connectivity |\n| Quantisation from resonance | S³ antipodal cavity selecting discrete spectrum |\n| Topological censorship | Non-orientable throats evading standard no-go theorems |\n| QFT event reinterpretation (Compton) | Antipodal `S³` Green function as propagator + Hopf-fibre photon polarisation + closed-form vertex resummation reproducing Klein-Nishina exactly — see [QFT-event-reinterpretation thread](#qft-event-reinterpretation-thread-compton-scattering) below |\n\n## Package Structure\n\n```\ngeometrodynamics/\n├── geometrodynamics/\n│   ├── constants.py          # Shared physical \u0026 simulation constants\n│   ├── hopf/                 # Hopf fibration on S³\n│   │   ├── connection.py     # A = ½cos(χ)dφ, curvature, holonomy\n│   │   ├── chern.py          # First Chern number c₁ = 1\n│   │   └── spinor.py         # SU(2) spinor transport (spin-½)\n│   ├── tangherlini/          # 5D wormhole eigenmodes\n│   │   ├── radial.py         # Chebyshev spectral solver\n│   │   ├── maxwell.py        # Sourced Maxwell BVP (Coulomb validation)\n│   │   ├── alpha_q.py        # Throat flux ratios (no free parameters)\n│   │   └── lepton_spectrum.py # Locked e/μ/τ instanton-transition matrix\n│   ├── transaction/          # Wheeler–Feynman absorber theory on S³\n│   │   ├── particles.py      # ThroatMode, MouthState, Particle4, GravWave\n│   │   ├── s3_geometry.py    # Geodesics, Green function, antipodal map\n│   │   ├── handshake.py      # Offer/confirm/transaction protocol\n│   │   └── cavity.py         # CavityMode, CavityPacket, AntipodalCavity\n│   ├── embedding/            # Non-orientable throat topology\n│   │   ├── topology.py       # ThroatDefect, ConjugatePair, transport ops\n│   │   └── transport.py      # T = iσ_y derived from Hopf fibration\n│   ├── bell/                 # Bell correlations from geometry\n│   │   ├── pair_state.py     # BellPair with cavity history evolution\n│   │   ├── analyzers.py      # Detector settings as SU(2) operators\n│   │   ├── correlations.py   # E(a,b), CHSH, no-signaling\n│   │   ├── hopf_phases.py    # Bell closure phases from Hopf holonomy\n│   │   └── bulk_identity.py  # Kinematic Bell from shared bulk topology\n│   ├── history/              # Closed-history framework (unifying backend)\n│   │   └── closure.py        # Events, Worldlines, History, branch enumeration\n│   ├── qcd/                  # Geometrodynamic QCD\n│   │   ├── constants.py      # σ, α_s, ℏc, SAT parameters\n│   │   ├── color.py          # SU(3) color algebra, generators\n│   │   ├── bridge.py         # BridgeField, Cornell potential\n│   │   ├── network.py        # Node, Branch, Junction, HadronicNetwork\n│   │   ├── topology.py       # Meson, baryon, glueball, hybrid, …\n│   │   ├── solver.py         # Störmer–Verlet + SAT boundaries\n│   │   ├── spectrum.py       # Möbius modes, throat–branch crosswalk\n│   │   └── diagnostics.py    # String tension, mode shifts, calibration\n│   ├── blackhole/            # Black holes as wormhole-throat condensates\n│   │   ├── condensate.py     # CoherentCondensate, ThroatState, constructors\n│   │   ├── interior.py       # Hayward regular metric, geodesics, horizons\n│   │   ├── entropy.py        # Bekenstein-Hawking from throat counting\n│   │   └── derivation.py     # Condensate → metric via Einstein equations\n│   └── viz/                  # Visualisation (placeholder)\n├── tests/                    # pytest validation suite\n├── notebooks/                # Jupyter notebooks (per-topic)\n├── scripts/                  # Lepton-ladder calibration CLIs\n├── docs/                     # Lepton axioms + scan archaeology\n└── pyproject.toml\n```\n\n## Installation\n\n```bash\ngit clone https://github.com/davidmdrpi/geometrodynamics.git\ncd geometrodynamics\npip install -e \".[dev]\"\n```\n\n## Running the Validation Suite\n\n```bash\n# All tests (fast)\npytest\n\n# Include slow tests (bridge nucleation, string tension scans)\npytest -m \"\"\n\n# Skip slow tests\npytest -m \"not slow\"\n```\n\n## Lepton mass ladder (e, μ, τ) from a locked S³ action\n\nThe lepton surrogate now ships with a **fully locked topological baseline**\nthat reproduces all three charged-lepton masses to sub-percent accuracy with\n**zero free parameters at scan time** — only the electron mass is used to set\nthe overall MeV scale.\n\n### Locked axioms\n\n- `action_base = 2π`  — the S³ great-circle action (circumference invariant).\n- `k_uplift_beta = 50π`  — k-selective uplift coefficient.\n  For `k=5`, the uplift is `4·β = 200π`, i.e. **exactly 100 × (2π)** S³\n  winding quanta.\n- `winding_mode = \"max\"`  — off-diagonal tunneling cost scales with the deeper\n  branch, `Δk = max(kᵢ, kⱼ)`.\n- `depth_cost_mode = \"tunnel_only\"`  — the S³ base action enters only through\n  the tunneling suppression, not as an additional diagonal offset.\n- `resistance_model = \"exponential\"`  — re-entry cost `κ·(eᵏ − 1)` captures\n  exponential geometric writhe/curvature build-up with generation depth.\n- Baseline anchor `(phase, transport, pinhole, resistance) ≈\n  (0.001, 25.1, 22.5, 0.217869)`. As of the closure-ledger sequence\n  (`docs/hbar_origin_note.md`), all four are now identified with\n  closure-quantum / Tangherlini-grid invariants:\n  `transport = 8π`, `pinhole γ = Σ V_max[1..5]`,\n  `resistance = 7π/100`, with the phase channel decoupled.\n\nThe generation-block diagonal takes the form\n\n```\nH_kk = action_base + resistance_scale · k²  +  res_diag(k)\n                  +  pinhole(k ∈ {3, 5})   +  β · max(0, k−3)²\n```\n\nand off-diagonals are `−transport · exp(−α_eff · Δk) · cos(phase · Δk)`.\nSee `docs/lepton_axioms.md` for the full matrix construction.\n\n### Validated predictions (locked baseline, no tuning)\n\n| Lepton | k | Predicted (MeV) | Observed (MeV) | Relative error |\n|--------|---|-----------------|----------------|----------------|\n| e      | 1 | 0.510999        | 0.510999       | 0.0000% (anchor) |\n| μ      | 3 | 105.61260       | 105.65838      | **0.0433%** |\n| τ      | 5 | 1778.93809      | 1776.86        | **0.1170%** |\n\nMuon/electron ratio: predicted **206.6787**, observed **206.7683**\n(relative error **4.33 × 10⁻⁴**).\n\nReproduce directly from Python:\n\n```python\nfrom geometrodynamics.tangherlini import solved_lepton_masses_mev\nmasses = solved_lepton_masses_mev()           # read-only np.ndarray\nprint(masses)   # [0.51099895, 105.6126..., 1778.9381...]\n```\n\nOr by CLI (no `PYTHONPATH` needed):\n\n```bash\npython scripts/lock_beta_50pi_probe.py --n-points 32\n```\n\nwhich additionally pins `β = 50π` exactly and optimizes only the four\nsub-leading knobs; it reports `mu/e` error ≈ 1 × 10⁻⁶% and\n`τ` relative error ≈ 0.161%.\n\n### Geometric implications\n\n1. **Three generations correspond to odd pass depths `k = 1, 3, 5`.** The\n   ladder is labelled by the number of S³ passes before closure; the locked\n   baseline scans exactly these three depths. Even-`k` branches are not part\n   of the surrogate; deriving their absence from the underlying Hopf/S³\n   topology remains an open research task.\n2. **τ uplift is exactly 100 quanta of the S³ action.** The k=5 uplift is\n   `4β = 200π = 100·(2π)`, a pure integer multiple of the great-circle action\n   `2π`. No tuning is required; removing the integer lock degrades `τ` by an\n   order of magnitude (see `docs/lepton_tau_target.md`).\n3. **The μ/e ratio is a structural eigenvalue ratio, not a coupling.** With\n   `action_base = 2π` locked and the exponential resistance profile, the\n   calibration scan finds exact μ/e roots on a broad resistance basin\n   (±1% resistance keeps `mu_err` \u003c 8%), replacing the earlier\n   \"attractor needle\" regime (see `docs/lepton_tau_target.md`, \"Hard S³ lock\n   experiment\").\n4. **Quadratic diagonal `∝ k²` plus quadratic uplift `∝ (k−3)²`** together\n   reproduce the observed `m_e : m_μ : m_τ ≈ 1 : 207 : 3477` hierarchy: the\n   `k²` term sets the `μ/e` split and the `(k−3)²` term independently lifts\n   the τ sector without disturbing the `μ/e` root.\n5. **Tunneling-side depth cost dominates diagonal depth cost.** The ablation\n   scan showed `tunnel_only` outperforms `diag_only` by nearly 2× on best\n   μ/e (see `docs/lepton_ablation_results.md`) — consistent with a picture in\n   which the inter-generation transition amplitude, not the on-generation\n   mass term, sets the ratio.\n6. **A `max` winding rule beats a `delta` winding rule.** Setting\n   `Δk = max(kᵢ, kⱼ)` (rather than `|kᵢ − kⱼ|`) in the tunneling action was\n   the change that first pushed `μ/e` from ~10 toward the experimental\n   ~206.77, because it penalises transitions into deeper branches by the full\n   target winding — a topological-cost interpretation consistent with the S³\n   action base.\n\n### Script map\n\n| Script | Purpose |\n|--------|---------|\n| `scripts/calibrate_muon_ratio.py` | Coarse grid; solves resistance for exact μ/e root at each (phase, transport, pinhole). |\n| `scripts/sweep_k_uplift_beta.py`  | Sweeps `β` with exact μ/e enforced; locates best τ fit. |\n| `scripts/map_basin_k_uplift.py`   | Local gradient probe around an exact-μ/e point; reports basin width. |\n| `scripts/refine_locked_tau.py`    | Dense locked scan with action_base fixed to 2π; reports integer-winding β family. |\n| `scripts/lock_beta_50pi_probe.py` | Hard `β = 50π` lock; optimizes only (phase, transport, pinhole, resistance). |\n\nSee `docs/lepton_ablation_results.md`, `docs/lepton_tau_target.md`, and\n`docs/lepton_next_steps.md` for the full scan archaeology, and\n`docs/hbar_origin_note.md` for the closure-ledger reduction of the\nlocked surrogate's parameters to closure-quantum invariants.\n\n## Quark mass ladder (u, d, s, c, b, t) from a shell-coupled S³ closure\n\nParallel to the lepton sector, the six observed quark masses are\nfit by a 6×6 Hermitian Hamiltonian on the closure basis\n`{(k=1,±), (k=3,±), (k=5,±)}`.  The minimal v3 ansatz did not\nsuffice; three opt-in structural extensions (`uplift_mode =\n\"partition_asymmetric\"`, `spectrum_zero_mode = \"min_eigenvalue\"`,\n`chi_q_k3`, `eta_k3k5_minus`), all with defaults that recover\nthe minimal lepton-style ansatz, give the locked spectrum.\n\n### Locked spectrum (d-anchor, max rel err 1.6%)\n\nAnchored on `d = 4.67 MeV`; `u` is at zero by construction under\nmin-eigenvalue spectrum zero.\n\n| species | predicted (MeV) | observed (MeV) | rel err |\n|---------|----------------:|---------------:|--------:|\n| u | 0           | 2.16    | 1.00 (by construction) |\n| d | 4.67        | 4.67    | 0 (anchor)             |\n| s | 94.82       | 93.4    | **1.5%**               |\n| c | 1290.92     | 1270    | **1.7%**               |\n| b | 4219.92     | 4180    | **0.95%**              |\n| t | 170342.41   | 172690  | **1.4%**               |\n\n### Locked parameters (constraint-reduced)\n\nThe full residual sector is *derivable from existing geometry*\non the eigensolver's tortoise grid:\n\n| sector | reading |\n|--------|---------|\n| `action_base = π` | structural |\n| `uplift_asymmetry ε = 1 − 1/k_5² = 24/25` | partition asymmetry from inverse-square shell scaling |\n| `eta_k3k5_minus η = k_5 = 5` | (3,−)–(5,−) targeted off-diagonal coupling |\n| `chi_q_k3 χ = (k_5 − 1)·k_5 = 20` | k = 3 partition splitter |\n| `phase = 0` | partition-mixing channel inactive at the lock |\n| `gamma_q = 1/10` | empirical clean rational |\n| `transport ≈ 0.54` | mean `⟨u_l\\|V_{l+2}−V_l\\|u_{l+2}⟩` on tortoise grid (+0.87% off) |\n| `pinhole ≈ 22.25` | `Σ_{l=1..5} V_max(l)` on tortoise grid (−1.09% off) |\n| `resistance ≈ 0.14` | `transport · ln(α_q(k_5)/α_q(k_1))` (−0.43% off) |\n| `β = N · π/2 with N=466` | **remaining phenomenological parameter** |\n\n### Shell-coupled vs minimal closure\n\nThe diagonal-Hamiltonian decomposition shows what makes the\nquark ladder structurally distinct from the lepton ladder:\n\n| species | β contribution |\n|---------|---------------:|\n| u, d (k=1) | 0% |\n| s         | +11% (level mixing only) |\n| c         | **−27%** (pushed *down* by level repulsion) |\n| b         | +76% via β·4·(1−ε) = β·4/k_5² |\n| t         | **+99%** via β·4·(1+ε) ≈ β·4·(49/25) |\n\n`β` only enters at the heaviest shell (k=5), via the\npartition-asymmetric `(1±ε)` factor.  The lighter shells (u, d,\ns, c) are determined entirely by the chamber-coupling sector\n(pinhole, χ, γ_q).  This is the operational signature of the\n\"shell-coupled closure\" picture: the same S³ closure skeleton\nthat drives the lepton ladder is, in the quark sector, primarily\nexpressed through how the closure interacts with an interior\nchamber rather than through global pass-count winding.\n\n### Calibration archaeology\n\n| Script | Purpose |\n|--------|---------|\n| `scripts/calibrate_quark_ratios.py` | Coarse grid over the residual sector; identifies γ_q regime where positivity holds. |\n| `scripts/sweep_quark_beta.py` | Integer-winding β sweep (now known to be a fit knob, not a topological lock). |\n| `scripts/map_basin_quark_uplift.py` | Basin probe around the best β. |\n| `scripts/lock_quark_beta_probe.py` | Final lock with β hard-fixed (legacy from the integer-N attempt). |\n| `scripts/experiment_partition_asymmetric_uplift.py` | Tests the k=5 b/t splitter. |\n| `scripts/experiment_min_eigenvalue_zero.py` | Tests d-anchor with min-eigenvalue spectrum zero. |\n| `scripts/experiment_k3_splitter.py` | Tests χ for the c/s splitter. |\n| `scripts/experiment_refined_k3k5.py` | Pass-2 refinement crossing the user-named \"serious candidate\" threshold (max rel err \u003c 0.3 → 0.13). |\n| `scripts/basin_probe_topological_locks.py` | Verifies N, χ, η are basin features, not grid coincidences. |\n| `scripts/refine_pass3_coord_descent.py` | Coordinate-descent refinement to 1.6%. |\n| `scripts/experiment_constraint_search.py` | Constraint-reduction pass: 9 free knobs → 4 + 1. |\n| `scripts/experiment_n_ablation.py` | First N-stability check (residuals free); N drifts. |\n| `scripts/experiment_residuals_from_geometry.py` | Substitutes residuals with broad geometric scalars. |\n| `scripts/experiment_transport_pinhole_search.py` | 1D refinement of transport and pinhole derivations. |\n| `scripts/experiment_transport_overlap.py` | Derives transport from QM perturbation overlap to within 0.87%. |\n| `scripts/experiment_resistance_wkb.py` | WKB tunneling-derived resistance (negative result), then discovers `resistance = transport · ln(α_q ratio)` to within 0.43%. |\n| `scripts/experiment_n_ablation_geometric.py` | Decisive N-stability check with all residuals derived; N still drifts → β is phenomenological. |\n\nSee `docs/quark_axioms.md` (full v3 spec, calibration log §8,\nphenomenological interpretation §9) and the JSON archive in\n`docs/calibration_runs/` for the raw outputs of every scan.\n\n## QFT-event-reinterpretation thread (Compton scattering)\n\nAn 11-PR thread (PRs #25 – this PR) testing whether BAM's three\ncomposable dynamical elements — **throat worldlines + time dilation\nat mouth + antipodal closure** — reproduce QFT event structure for a\ncanonical local interaction, Compton scattering `γ + e → γ + e`. The\nthread progressively identified the BAM-native ingredients needed\nto reproduce Klein-Nishina, then resummed the perturbative result\ninto a closed-form vertex factor.\n\n### Result chain\n\n  - **Kinematics** (PR #25): closure-compatible. The antipodal map\n    `(E, **p**) → (E, −**p**)` automatically conserves the\n    back-vertex when the front does. Inter-mouth proper-time skew\n    vanishes; throat-pinch skew is a recoil-induced `O(ω²/m²)`\n    quantity, not a topological closure quantum.\n\n  - **Propagator** (PR #26): the `S³` Green function\n    `G(ψ) ∼ 1/(4πψ)` with `ψ = (s − m²)/(2m²)` reproduces the QED\n    propagator pole `1/(s − m²)` exactly (fitted exponent 1.0002).\n\n  - **Photon structure** (PR #28): giving the photon two transverse\n    polarisations on the `S³` tangent bundle and treating the\n    electron as a scalar charge in the Thomson limit reproduces\n    `(1 + cos²θ)/2` exactly — the full Klein-Nishina angular factor.\n\n  - **Finite-energy gap** (PR #29): the natural BAM construction\n    fails at `O(ω/m)`. The recoil sign is qualitatively wrong\n    (BAM enhances backscatter, KN suppresses it), localised to the\n    missing per-channel kinematic weighting.\n\n  - **Vertex coupling** (PRs #30, #31): an extended Family B vertex\n    modification `V = (ε·ε'*)·(1 + ε·μ₁ + ...)` with\n    `μ₁ = γ·(ω/m)·(1 − cos θ)` closes the `O(ε)` gap exactly at\n    `γ = −3/2` — derived analytically from a 4-equation linear\n    system over `{1, c, c², c³}` basis.\n\n  - **Coefficient origin** (PRs #32, #33): 8 natural BAM ingredients\n    evaluate to `−3/2`; the dimensional-scaling test in `d ∈ {3, 4,\n    5, 6}` falsifies the embedding-dim / polarisation-count origin\n    (candidate C), leaving 7 surviving candidates rooted in\n    group-theoretic invariants of SU(2).\n\n  - **`O(ε²)` extension** (PR #34): polynomial leading-order\n    closure with `(ν₀, ν₁, ν₂, ξ) = (9/4, −4, 7/4, −1/2)`, with\n    structural patterns `ν₀ = γ² = (−3/2)²` (recursive) and\n    `ξ = −A_φ(0)` (Hopf-charge link).\n\n  - **Resummation** (PR #35): the closed form\n\n      F²(x, c) = 4·x³·(x² + 1 − x·sin²θ) / [(1 + c²)·(1 + x)²]\n              = (2x/(1+x))² · [x·(x²+1−x·sin²θ) / (1+c²)]\n\n    with `x = ω'/ω = 1/(1 + ε(1 − cos θ))` reproduces Klein-Nishina\n    **exactly at all orders in ε up to ε ~ 2** (machine precision).\n    The perturbative results of PRs #31–34 are Taylor expansions\n    of this closed form.\n\n  - **Cross-process validation via Breit–Wheeler** (this PR): the\n    same closed-form F, expressed in Lorentz invariants and\n    analytically continued via standard Mandelstam crossing\n    (`s_C → u_BW`, `t_C → s_BW`, `u_C → t_BW`), exactly reproduces\n    the Breit–Wheeler pair-production amplitude `γγ → e⁺e⁻`.\n    Crossed variables `x_⊗ = −(1−β·cosθ)/(1+β·cosθ) \u003c 0` and\n    `c_⊗ = (2β² − β²cos²θ − 1)/(1−β²cos²θ)` carry the construction\n    from Compton lab kinematics to BW CM kinematics; the\n    BAM-predicted `|M̄|²_BW = −2·(f_baseline · F²)/x_⊗²` agrees\n    with the textbook formula to machine precision at all sampled\n    `(β, cosθ)`, and the integrated differential reproduces the\n    textbook BW total at threshold (`β → 0` linear) and in the\n    ultra-relativistic regime (`β → 1` logarithmic). The vertex F\n    is therefore **not a Compton-specific algebraic fit** — it is\n    the closed form of the invariant QED amplitude carried by\n    crossing to its tree-level partners.\n\n### Structural reading\n\nThe `(1 + c²)` denominator in the angular factor IS the\npolarisation-sum factor. The closed-form F must be derived AS a\nmodification of the polarisation-sum projector, not as a separate\namplitude factor. The two-factor decomposition\n\n  - kinematic Padé `(2x/(1+x))²` — pure x-function\n  - angular polarisation modification `[x·(x²+1−x·sin²θ) / (1+c²)]`\n\nsuggests two BAM-native ingredients combine to produce the full\nvertex coupling. The clean half-integer/integer rationals appearing\nat every order (γ = −3/2, ν₀ = 9/4, ν₁ = −4, ν₂ = 7/4, ξ = −1/2)\nindicate a deeper geometric origin awaiting first-principles\nderivation from the Hopf-bundle / throat-transport algebra.\n\n### What survives and what is still open\n\n  - Survives: BAM's antipodal-`S³` propagator + Hopf-fibre photon\n    polarisation + closed-form vertex `F²` together reproduce\n    Klein-Nishina exactly. The same closed form, crossed via\n    Mandelstam permutation, reproduces Breit–Wheeler `γγ → e⁺e⁻`\n    (PR #36) and pair annihilation `e⁺e⁻ → γγ` (this PR); the full\n    Compton/BW/annihilation crossing triangle closes (loop is\n    identity at both the Mandelstam-label and amplitude level).\n  - Open: first-principles BAM derivation of `F²` from a BAM\n    Lagrangian / action. Two-channel tree processes (Bhabha, Møller)\n    with interfering s+t diagrams; loop corrections requiring the\n    bulk radial channel.\n\n### Probe sequence\n\n| # | Probe | Outcome |\n|---|---|---|\n| PR #25 | `compton_antipodal_kinematics_probe.py` | closure-compatible |\n| PR #26 | `compton_amplitude_structure_probe.py` | propagator ✓, polarization ✗ |\n| PR #28 | `compton_photon_structure_probe.py` | Thomson KN ✓ |\n| PR #29 | `compton_finite_energy_kn_probe.py` | recoil ✗ at `O(ω/m)` |\n| PR #30 | `compton_vertex_structure_probe.py` | empirical finite-ε fit |\n| PR #31 | `compton_vertex_derivation_probe.py` | exact γ = −3/2 |\n| PR #32 | `compton_coefficient_origin_probe.py` | 8 plausible derivations |\n| PR #33 | `compton_dimensional_scaling_probe.py` | C falsified, 7 survive |\n| PR #34 | `compton_eps2_extension_probe.py` | `O(ε²)` polynomial fit |\n| PR #35 | `compton_vertex_resummation_probe.py` | exact closed-form F² |\n| PR #36 | `breit_wheeler_cross_process_probe.py` | F process-general under Compton → BW crossing |\n| PR #37 | `pair_annihilation_crossing_probe.py` | full Compton/BW/annihilation crossing triangle closes |\n| PR #38 | `throat_nucleation_caustic_derivation_probe.py` | F² = K(x)²·Q(x, c) BAM-geometric decomposition |\n| PR #39 | `two_mouth_flux_action_probe.py` | K(x) = 2x/(1+x) from equal-action throat-rate splitting |\n| PR #40 | `hopf_helicity_transport_probe.py` | Q(x, c) from Hopf-fibre helicity spinor (A_pres, A_flip) |\n| PR #41 | `throat_action_derivation_probe.py` | BAM throat action: both equal-action postulates derived from S³ antipodal symmetry + closure quantum + stationary action |\n| PR #42 | `bhabha_moller_interference_probe.py` | 4-fermion gap identified: scalar Compton kernel insufficient for Bhabha/Møller |\n| PR #43 | `dirac_trace_geometry_probe.py` | 4-fermion diagonal numerators (s²+u²), (u²+t²), (s²+t²) from SU(2) Hopf-bundle Pauli traces |\n| PR #44 | `mobius_exchange_sign_probe.py` | Bhabha/Møller interference signs from T = iσ_y = ε non-orientable throat transport |\n| PR #45 | `bam_exchange_kernel_probe.py` | photon propagator magnitude 1/q² from S³ Green function (flat limit) |\n| PR #46 | `hopf_vector_exchange_kernel_probe.py` | **photon propagator Lorentz tensor −η^{μν}/q² from Hopf-bundle U(1) connection** |\n| PR #48 | `two_throat_coulomb_probe.py` | inverse-square Coulomb force from the S³ Green response; Gauss law exact |\n| PR #49 | `topological_discrete_sector_probe.py` | scaffold B1+B2 promoted to action data (RP³ + spin structure + winding θ-term) |\n| PR #50 | `radial_reduction_bridge_probe.py` | scaffold B5 factorized: 5D→4D into three channels; F² not a radial overlap |\n| PR #51 | `bulk_boundary_interaction_probe.py` | scaffold B5′: radial (masses) + throat (K) unified by one bulk-boundary cavity |\n| PR #51 | `master_integral_probe.py` | **scaffold B5 closed: masses and F²=K²·Q from one C×S³ master functional** |\n| PR #52 | `maslov_dimensional_bridge_probe.py` | scaffold B4 audit: irreducible by scale-freeness; Maslov closure-ledger (radial +1 = μ=4) |\n| PR #53 | `delta_r_scale_modulus_probe.py` | scaffold B4 anchor: ΔR is a cosmologically-invariant bulk separation |\n\n**Synthesis / release note:** `docs/tree_qed_status.md` collects the\nPR #35 → #46 result — all tree-level `2 → 2` QED scalar intensities\n(Compton, Breit–Wheeler, pair annihilation, Bhabha, Møller)\nreproduced from BAM-geometric primitives.\n\nThe Compton derivation rests on the algebraic identity\n\n  x² + 1 − x·sin²θ ≡ (1 − x)² + x · (1 + c²)\n\nwhich yields two equivalent decompositions:\n\n  F²(x, c) = [2x/(1+x)]² · [x² + x·(1−x)²/(1+c²)]\n  |M̄|²_KN/(8e⁴) = (1+c²) + (1−x)²/x\n\nwith BAM-geometric interpretation:\n\n  - **P(x) = 2x/(1+x)** = harmonic mean of in/out photon frequencies\n    = standard classical bottleneck-flux average through the throat;\n    squared because both throat-pair mouths pinch. Uniquely\n    polynomial — alternative throat-rates (arithmetic, geometric mean,\n    linear x) leave Q non-polynomial at x → −1.\n  - **(1+c²)/2 = cos⁴(θ/2) + sin⁴(θ/2)** = sum of squared Wigner-d¹₁,±₁\n    matrix elements = Hopf-fibre spin-1 helicity transport through θ.\n  - **Q = |a|² + |b|²** = orthogonal sum of helicity-preserving\n    (a = x) and helicity-flipping (b = √x(1−x)/√(1+c²)) channels,\n    each non-negative across the physical region.\n  - The Hopf connection at the BAM lock `A_φ(0) = 1/2` (from\n    `geometrodynamics.hopf.connection`) matches the PR #34 perturbative\n    coefficient `ξ = −1/2` exactly.\n  - Decomposition survives analytic continuation under crossing\n    (full Compton ↔ BW ↔ annihilation triangle, PR #37).\n\nThe full F² closed form is derived from three foundational\nprinciples via a single BAM throat action functional (PR #41):\n\n  (P1) closure quantum `S = 2π` (BAM `action_base`)\n  (P2) S³ antipodal symmetry `σ(p) = −p` (involution swapping mouths)\n  (P3) stationary action under the antipodally-symmetric ansatz\n\nBoth equal-action postulates (PR #39 energy → K, PR #40 spin/Hopf → Q)\nfollow as consequences. Alternative principles (broken antipodal\nsymmetry; wrong closure quantum; wrong action functional) all fail\nto reproduce K(x), confirming the principles are necessary.\n\nThe thread then extends to 4-fermion tree QED (Bhabha, Møller,\nPRs #42–#46): SU(2) Hopf-bundle Pauli traces give the Dirac-trace\ndiagonal numerators (#43), the non-orientable throat transport\n`T = iσ_y = ε` gives the Fermi-statistics interference signs (#44),\nand the `S³` Green function (scalar #45, Hopf-bundle vector #46)\ngives the photon propagator `1/q²` with full Lorentz tensor\nstructure. End-to-end Bhabha and Møller `|M̄|²` match QED to machine\nprecision from BAM-geometric ingredients alone.\n\nSee `docs/tree_qed_status.md` for the full synthesis. Per-PR\nresearch plans: `docs/compton_vertex_resummation_research_plan.md`\n(#35), `docs/breit_wheeler_cross_process_research_plan.md` (#36),\n`docs/pair_annihilation_crossing_research_plan.md` (#37),\n`docs/throat_nucleation_caustic_derivation_research_plan.md` (#38),\n`docs/two_mouth_flux_action_research_plan.md` (#39),\n`docs/hopf_helicity_transport_research_plan.md` (#40),\n`docs/throat_action_derivation_research_plan.md` (#41),\n`docs/bhabha_moller_interference_research_plan.md` (#42),\n`docs/dirac_trace_geometry_research_plan.md` (#43),\n`docs/mobius_exchange_sign_research_plan.md` (#44),\n`docs/bam_exchange_kernel_research_plan.md` (#45), and\n`docs/hopf_vector_exchange_kernel_research_plan.md` (#46).\n\n### BAM effective-action scaffold — barrier closure (PRs #49–#53)\n\nThe tree-QED ingredients above were assembled into a single covariant\n5D effective-action scaffold and its five mismatch terms (B1–B5) were\nworked off one by one. Four are now **closed**:\n\n| barrier | what it was | now |\n|---|---|---|\n| **B1** closure quantum `∮A = 2πn` | imposed constraint | winding θ-term `S_top = 2π·n` |\n| **B2** antipodal `Z₂` (`T = iσ_y`) | imposed identification | `RP³ = S³/Z₂` + non-trivial spin structure |\n| **B3** hard-wall throat BC | imposed by hand | single-valuedness under `T² = −I` ⟹ `ψ(throat) = 0` |\n| **B5** 5D→4D reduction producing F² | unconstructed | one master functional yields masses **and** `F²=K²·Q` |\n\nB5 is closed by the **master integral**: a single separable functional\non the warped-product internal geometry `M_int = C × S³`\n(`C` = radial cavity `[R_MID, R_OUTER]`),\n\n```\nℳ(ω; x, c) = G_C(r, r′; ω) ⊗ 𝒢_{S³}(Ω, Ω′)\n```\n\nread three ways from one object —\n\n  - **poles in ω** → the mass spectrum `ω(l,n)` (radial ladder `n` ×\n    S³ Casimir `l`, the centrifugal term of the warp);\n  - **throat boundary of `G_C`** → `K(x) = 2x/(1+x)` (dwell-time\n    impedance `Z(ω)=π/ω` in series);\n  - **S³ Hopf reduction of `𝒢_{S³}`** → `Q(x,c) = x²+x(1−x)²/(1+c²)`\n    (Hopf-fibre helicity spinor).\n\nThe vertex residue reproduces `F²(x,c) = K²·Q` to machine precision\n(`2e-14`) while the poles give the masses — **masses and the full\nvertex from one functional**. The `F²=K²·Q` factorization is the direct\nconsequence of the product internal geometry (separation of variables),\nnot a failure to unify.\n\nThe fifth barrier **B4** (the dimensional bridge `ℏ = m_e·R_MID·c`) is\nnot a gap but a **structural necessity**: the closure-ledger/Maslov\nmachinery is *scale-free* (rescaling `R_MID → λ·R_MID` leaves every\ndimensionless output invariant), so exactly one external dimensionful\nanchor is mathematically required — **B4 is irreducible** (#52). That\nanchor need not be a particle mass: it can be the **invariant bulk\nseparation** `ΔR = R_OUTER − R_INNER`, a proper (cosmologically fixed)\nlength, giving `m_e = f_closure·ℏ/(ΔR·c)` with `f_closure = 0.52` (#53).\nThe scaffold is therefore complete: four barriers derived, the fifth the\nsingle mandatory dimensionful unit. Full ledger:\n`docs/bam_scaffold_status.md`; closure release note:\n`docs/scaffold_closure_release_note.md`; per-probe plans:\n`docs/topological_discrete_sector_research_plan.md` (#49),\n`docs/radial_reduction_bridge_research_plan.md` (#50),\n`docs/bulk_boundary_interaction_research_plan.md` and\n`docs/master_integral_research_plan.md` (#51),\n`docs/maslov_dimensional_bridge_research_plan.md` (#52),\n`docs/delta_r_scale_modulus_research_plan.md` (#53).\n\n## Quick Start\n\n### Verify charge quantisation from pure geometry\n\n```python\nfrom geometrodynamics.hopf import compute_c1\n\nresult = compute_c1()\nprint(f\"|c₁| = {result['c1_abs']:.10f}  (error: {result['err_abs']:.2e})\")\n# |c₁| = 1.0000000000  (error: 9.99e-14)\n```\n\n### Verify spin-½ from Hopf holonomy\n\n```python\nfrom geometrodynamics.hopf import compute_spinor_monodromy\n\nresult = compute_spinor_monodromy()\nprint(f\"⟨ψ₀|U(2π)|ψ₀⟩ = {result['overlap_2pi']:.6f}  (should be −1)\")\nprint(f\"⟨ψ₀|U(4π)|ψ₀⟩ = {result['overlap_4pi']:.6f}  (should be +1)\")\n```\n\n### Validate Coulomb law from eigenmode throat flux\n\n```python\nfrom geometrodynamics.tangherlini import solve_radial_modes, solve_maxwell_from_eigenmode\n\nmodes = {}\nfor l in [1, 3, 5]:\n    oms, fns, rg = solve_radial_modes(l)\n    modes[l] = {\"omega\": oms, \"funcs\": fns}\n\nresult = solve_maxwell_from_eigenmode(modes)\nprint(f\"Q = {result['Q']:.6f}\")\nprint(f\"Relative error vs exact Coulomb: {result['rel_err']:.2e}\")\n```\n\n### Reproduce the full charged-lepton ladder\n\n```python\nfrom geometrodynamics.tangherlini import (\n    solved_lepton_masses_mev, S3_ACTION_BASE, TAU_BETA_50PI, tau_uplift_2pi_quanta,\n)\n\nmasses = solved_lepton_masses_mev()   # locked baseline, no tuning\nprint(f\"m_e  = {masses[0]:.6f} MeV\")\nprint(f\"m_mu = {masses[1]:.6f} MeV   (obs 105.658376)\")\nprint(f\"m_tau= {masses[2]:.6f} MeV   (obs 1776.860000)\")\n\nprint(f\"action_base = 2π         = {S3_ACTION_BASE:.6f}\")\nprint(f\"k_uplift β  = 50π        = {TAU_BETA_50PI:.6f}\")\nprint(f\"τ uplift    = 4β = 200π  = {tau_uplift_2pi_quanta(TAU_BETA_50PI):.0f} × (2π)\")\n```\n\n### Run a QCD meson simulation\n\n```python\nimport numpy as np\nfrom geometrodynamics.qcd import make_meson_tube, HadronicNetworkSolver\n\nnet = make_meson_tube(L=1.0, v=1.0, N=100, dt=0.004)\ns = np.linspace(0, 1.0, 100)\nnet.initialize_fields(psi0={0: 0.5 * np.sin(np.pi * s)})\n\nsolver = HadronicNetworkSolver(net, antipodal_coupling=0.05)\nhistory = solver.run(n_steps=1000, record_every=50)\nprint(f\"Energy drift: {np.std(history['energy']) / history['energy'][0]:.4f}\")\n```\n\n### Build a black-hole condensate and verify entropy\n\n```python\nfrom geometrodynamics.blackhole import (\n    build_schwarzschild_condensate, compute_entropy_balance,\n    find_horizons, integrate_radial_geodesic,\n)\n\n# Schwarzschild BH as a coherent wormhole-throat condensate\nbh = build_schwarzschild_condensate(mass=5.0)\nbal = compute_entropy_balance(bh)\nprint(f\"N throats: {bh.N}\")\nprint(f\"S_BH  = {bal.S_BH:.2f}\")\nprint(f\"S_thr = {bal.S_throat:.2f}  (relative error: {bal.relative_error:.2e})\")\nprint(f\"Net charge Q = {bh.net_charge}  (neutral)\")\n\n# Nonsingular interior: Hayward metric with core scale from throat network\nl = bh.core_scale\nhorizons = find_horizons(bh.mass, l)\nprint(f\"\\nCore scale l = {l:.4f}\")\nprint(f\"Horizons: {['%.4f' % h for h in horizons]}\")\n\n# Geodesic completeness: infalling worldline decelerates, never hits r=0\ngeo = integrate_radial_geodesic(M=bh.mass, l=l, r_start=3*bh.mass, tau_max=100)\nprint(f\"Geodesic complete: {geo.is_complete}  (r_min = {geo.r_min:.2e})\")\n```\n\n## Lineage\n\nThis package refactors and unifies three monolithic scripts:\n\n| Original file | Package modules |\n|---|---|\n| `geometrodynamics_v39.py` | `hopf/`, `tangherlini/`, `transaction/`, `constants.py` |\n| `s3_spin2_closure_toy_solver_v22.py` | `tangherlini/` (shared spectral solver) |\n| `qcd_topology_solver_v30.py` | `qcd/` (entire subpackage) |\n| New in v0.41.0 | `blackhole/` (condensate, interior, entropy, derivation) |\n| New in v0.42.0 | `embedding/`, `bell/`, `transaction/cavity.py` |\n| New in v0.43.0 | `embedding/transport.py`, `bell/hopf_phases.py`, `history/` |\n| New in v0.44.0 | `tangherlini/lepton_spectrum.py` (locked e/μ/τ ladder) + `scripts/` (calibration CLIs) |\n| New in v0.45.0 | `qcd/quark_spectrum.py` + `qcd/hadron_spectrum.py` (shell-coupled six-quark ladder; residual sector geometrized to ~1% via Tangherlini eigenmode) |\n| New in v0.46.0 | `experiments/closure_ledger/` (closure-ledger sequence; reduces the locked lepton surrogate's residual external input from six phenomenological parameters to one anchor m_e). Paper draft in `docs/hbar_origin_note.md`. |\n\n## License\n\nMIT\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdavidmdrpi%2Fgeometrodynamics","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fdavidmdrpi%2Fgeometrodynamics","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdavidmdrpi%2Fgeometrodynamics/lists"}