# QLF Flow Chart — a visual map of the framework > **This is a navigation map of the [Quantum Logical Framework (QLF)](README.md)** — one substrate, five > families, twelve domains, and the documents that derive each. The **Jump to** and **Open** links are the > navigation; the visual diagrams live in the rendered version linked below. The taxonomy: **one substrate → five domain families → twelve domains → the individual results.** > **The diagrams don't render reliably on GitHub** (a known GitHub Mermaid/dark-theme issue), so this page is the text index. For the **rendered, clickable diagrams** and a **printable PDF** (with internal + external links), open the live version on GitHub Pages: **[rhovision-org.github.io/quantum-logical-framework/FlowChart.html](https://rhovision-org.github.io/quantum-logical-framework/FlowChart.html)** (or clone/pull and open the local `FlowChart.html`). Regenerate with `python3 tools/build_flowchart_html.py`. --- ## Master map — the substrate and its twelve domains **Jump to:** [1 Space, time and the continuum](#1-space-time-and-the-continuum) · [2 The fundamental constants](#2-the-fundamental-constants) · [3 Forces](#3-forces) · [4 Atoms and QED](#4-atoms-and-qed) · [5 Gravity and GR](#5-gravity-and-gr) · [6 Cosmology and the dark sector](#6-cosmology-and-the-dark-sector) · [7 Particles and the Standard Model](#7-particles-and-the-standard-model) · [8 Quantum-gravity / TOE pillars](#8-quantum-gravity--toe-pillars) · [9 The Millennium Prize program](#9-the-millennium-prize-program) · [10 Beyond the SM](#10-beyond-the-sm) · [11 Chemistry, molecules and folding](#11-chemistry-molecules-and-folding) · [12 Quantum field theory and renormalization](#12-quantum-field-theory-and-renormalization) The five families: **Foundations** (1-2, 12) · **Matter and forces** (3, 4, 7) · **Emergent matter** (11) · **Gravity and the cosmos** (5-6) · **Frontiers** (8-10). **Emergent matter** is the one family that is not fundamental physics. The other four ask what the substrate *is*; this one asks what it **assembles** once atoms exist — and it earns a family of its own precisely because nothing in it is a new law. Chemistry, polymers and folding all run on the rule domains 3–4 already established, which is the claim being made by putting them on the map at all. **Open:** [`README.md`](README.md) Root reading: **everything derives from the 8-twist substrate under Zero Free Action** — [`Philosophy.md`](Philosophy.md) (possibilist ontology, and §3a the *working method*: things happen every way that closes; ZFA decides what closes first, and what happens in the most ways happens most often), [`WHITE_PAPER.md`](WHITE_PAPER.md). **The method's limit, proven:** [`Law_Of_Exceptions.md`](Law_Of_Exceptions.md) — *a system with more states can always break a finite closure*, so every restrictive law has a real, constructed exception and no finite closure is final ([`QLF_LawOfExceptions`](lean/QLF_LawOfExceptions.lean)). Capacity turns out to be an **excursion** budget ([`QLF_ClosureDepthLaw`](lean/QLF_ClosureDepthLaw.lean)), which is why the proton dissolves at `T_c` and baryon number at `T_EW` while electric charge — whose proof carries no capacity — has no exception at any scale. Corollary for everything below: **construction proves possibility, not uniqueness.** **An application of that same capacity machinery:** [`Mpemba.md`](Mpemba.md) — anomalous relaxation, where relaxation time *is* the maximum excursion, giving a proven no-go (none for the imbalance measure), a proven enabler (no scalar determines relaxation), and proven **instances** (more energy closing strictly faster) — while the ensemble effect and the water phenomenon stay open ([`QLF_Mpemba`](lean/QLF_Mpemba.lean)). **Foundational logic & mathematics:** the substrate's *logic* is **quantum logic** — argued as the correct foundation of mathematics (bottom-up, sound vs. exploding) in [`Quantum_Logic_Foundations.md`](Quantum_Logic_Foundations.md), with the minimal quantum logic `MO2` machine-verified on the substrate (orthomodular + non-distributive, [`lean/QLF_QuantumLogic.lean`](lean/QLF_QuantumLogic.lean)); and ordinary mathematics *emerging* from it (ℕ, the ring, `μ₄`, su(2)/su(3), the continuum as completion) in the companion [`Mathematics_From_QLF.md`](Mathematics_From_QLF.md). The **category-theoretic** shape — the causal-set base category, ZFA as an equalizer, the capacity-horizon **bifibration** whose free prime-factorisation *forces* self-similarity (and closes the log-periodic channel, so the α-residual weight `w = 1/2` is structural), the Tannakian/anabelian layer, RhoQuCalc as a dagger SMC — is collected in [`Category_Theory_QLF.md`](Category_Theory_QLF.md) (a description, not an axiom). **Truth is objective across observers, and in two ways** ([`lean/QLF_MultiObserver.lean`](lean/QLF_MultiObserver.lean), no axioms): whether a history closes, and the phase it carries, are the *same in every observer's frame* (the discrete axis-permutation × gauge-swap group) — so two or more perspectives agree on the truth (`observers_agree`), witnessed by two independent invariants, the count verdict and the Pauli scalar (`truth_from_two_perspectives_in_two_ways`). The `/solve` "meeting of minds" consensus is a theorem. **It from bit:** the unit of information is the two-valued **½-spin closure** — one bit (`log 2` for the spinor alphabet `{+I,−I}`) vs *zero* for a single-valued vector `{+I}` — the `2π` double-valuedness reproven from the explicit rotation matrices and grounding the spinor **Cartan** discovered in 1913 as the carrier of information ([`lean/QLF_SpinorInformation.lean`](lean/QLF_SpinorInformation.lean), [`Mathematics_From_QLF.md`](Mathematics_From_QLF.md) §Rung 5a). **The 2026 discoveries, read on the substrate:** [`Discoveries_2026.md`](Discoveries_2026.md) takes 24 results of 2026: [Part I](Discoveries_2026.md#part-i), the twelve segments of Nap Theory's video [*The Most Insane Physics Discoveries Of 2026*](https://youtu.be/NwKzvBN9wh4), and [Part II](Discoveries_2026.md#part-ii), twelve results the video does not cover. For each it gives what QLF already said, what is now proved ([`QLF_Discoveries2026`](lean/QLF_Discoveries2026.lean), no axioms), and a prediction or kill condition. They land in domains 7, 10 and 11 below. One of them corrected QLF: STAR's baryon junction exposed that the windowed winding reads the neutron as `B = 0`, so baryon number now lives on the junction, the spatial projection that no gauge twist can move ([`QLF_BaryonWinding`](lean/QLF_BaryonWinding.lean)). **Harmonic-closure model:** reality and constructable truth are the *closing spectrum* of frequency-component closures — each frequency `f = 1/R` is one ZFA closure, i.e. a quantum-logical **computation** (a set of Feynman diagrams: path integral = generate, ZFA closure = the firebreak selecting the physical ones) ([`Frequency_Synchronization.md`](Frequency_Synchronization.md) §0). --- ## 1. Space, time, and the continuum **Connectors:** *causal order* → Closure-reachability (pre-geometric causal s… · *faithful 3-D render* → 3 spatial dimensions · *logical latency* → Time = per-event Planck tick · *synthesize* → Synthesized spacetime · *RCA_0 floor* → No continuum / no Choice **Open:** [`README.md`](README.md) · [`SpaceTime.md`](SpaceTime.md) · [`TheContinuum.md`](TheContinuum.md) `3` is the minimal dimension that renders any relational structure faithfully — and it reappears everywhere below ([`SpaceTime.md`](SpaceTime.md) §3a). --- ## 2. The fundamental constants The `6 spatial + 2 gauge` split (the `3` axes) fixes a family of constants. **α is the flagship.** **Connectors:** *N = 3^2* → alpha = 1/137 · *spatial 3/8* → sin^2theta_W = 3/8 · *gauge 2/8* → Omega_Lambda ~ log 2 (observed match) · *surface ~ r^2* → Newton 1/r^2 · *l = 3* → nuclear magic numbers · *6pi^5* → m_p/m_e = 6pi^5 · *2/alpha* → m_pi/m_e = 274 **Open:** [`SpaceTime.md`](SpaceTime.md) · [`Alpha.md`](Alpha.md) · [`Weak_Force.md`](Weak_Force.md) · [`Cosmological_Constant.md`](Cosmological_Constant.md) · [`Gravity_From_Delay.md`](Gravity_From_Delay.md) · [`Magic_numbers.md`](Magic_numbers.md) · [`Proton_Resonance_R_e.md`](Proton_Resonance_R_e.md) · [`Pion_QLF.md`](Pion_QLF.md) · [`Genesis.md`](Genesis.md) The census spectrum explorer [`Genesis.md`](Genesis.md) exercises the constants sector end-to-end: the exact `−p/2` census spectral exponent (Lean-anchored, `QLF_CensusWalk`), census → π, and `α⁻¹ = 128 + d²` (`d = 3 → 137`). **α's full story** (derivation, IR/3-D scale, the running, the no-drift theorem, 4D/5D over-determination): [`Alpha.md`](Alpha.md). **The `+0.036` residual is now healed to its honest shape** ([`Alpha_Residual.md`](Alpha_Residual.md) §9c): the bracket is two-sided and machine-checked both ends (`137.01587 < α⁻¹ < 137.04813`), the equal-weight prediction **`137.032` is structural** (`w = ½` forced by the free-monoid bifibration, not fitted), and `0.036` is **existence-proven** (reachable by a prime resummation) with its **multiplicity open** — which resummation = the continuum vacuum-polarisation running, the Standard Model's own frontier. Every substrate mechanism swing is closed; CODATA is an estimate of the missing count, not a target. --- ## 3. Forces One gauge-twist mechanism, seen from three projections of the 3-axis structure. **Connectors:** *abelian* → U(1) - electromagnetism · *non-abelian, chiral* → SU(2) - weak · *colour, confined* → SU(3) - strong · *+ one mass = scale* → dimensionless couplings · *projection* → one force, three projections **Open:** [`Forces_From_Three_Axes.md`](Forces_From_Three_Axes.md) · [`Electricity.md`](Electricity.md) · [`Weak_Force.md`](Weak_Force.md) · [`Alpha.md`](Alpha.md) · [`Forces_From_Alpha.md`](Forces_From_Alpha.md) --- ## 4. Atoms and QED Everything here is **downstream of the derived α** ([`Alpha.md`](Alpha.md) §10). **Connectors:** *1/2alpha^2m_e c^2* → Rydberg / Bohr · *~ alpha^2* → Dirac fine structure · *loop alpha* → Lamb shift · *alpha/2pi* → g-2 · *Z_0/2alpha* → von Klitzing R_K · *~ alpha^4* → hyperfine / 21 cm **Open:** [`Alpha.md`](Alpha.md) · [`Hydrogen.md`](Hydrogen.md) · [`Dirac_Correction.md`](Dirac_Correction.md) · [`Lamb_Shift.md`](Lamb_Shift.md) · [`g_minus_2.md`](g_minus_2.md) · [`Electricity.md`](Electricity.md) · [`Magnetism_Spatial_Dynamics.md`](Magnetism_Spatial_Dynamics.md) --- ## 5. Gravity and GR **Connectors:** *G = L_P^2c^3/hbar* → Newton's law + G · *43''/century* → Mercury perihelion · *deltaQ = T deltaS* → Einstein equations (equation of state) · *causal order -> metric* → Curvature · *spin-2, v = c* → Gravitational waves · *Hilbert: delta S = 0 at balance* → Einstein equations **Open:** [`Gravity_From_Delay.md`](Gravity_From_Delay.md) · [`Mercury_Perihelion.md`](Mercury_Perihelion.md) · [`Einstein_Equations.md`](Einstein_Equations.md) · [`Curvature.md`](Curvature.md) · [`Stationary_Action.md`](Stationary_Action.md) --- ## 6. Cosmology and the dark sector **Connectors:** *gauge 2/8* → Horizon count closes the 10^122 catastrophe; Omega_Lambda ~ log 2 matched · *high-V epoch* → Inflation (same field) · *denser logic* → Dark matter (no particle) · *residual w = -1* → Dark energy · *event rate* → Age ~ 13.8 Gyr · *freeze-out n/p* → ^4He fraction Y_p ~ 1/4 · *measured values* → Concordant with LCDM data · *interpretive pillars* → Divergent on LCDM interpretation · *equation of state* → Thermodynamic gravity **Open:** [`Cosmological_Constant.md`](Cosmological_Constant.md) · [`Curvature.md`](Curvature.md) · [`DarkMatter.md`](DarkMatter.md) · [`SPARC.md`](SPARC.md) · [`AgeOfUniverse.md`](AgeOfUniverse.md) · [`Fusion.md`](Fusion.md) · [`Mysteries_Of_Physics.md`](Mysteries_Of_Physics.md) §3a Dark matter is the closure-balance RAR, blind-tested parameter-free on 147 SPARC galaxies (`a₀ = cH₀/2π`, the `2π` derived; [`SPARC.md`](SPARC.md)). `a₀` is constant locally (`a₀/(cH) = 1/2π`) and scales with `H(z)` across redshift; the MUSE-DARK data prefer this expansion clock over the age clock and a constant `a₀` ([`DarkMatter.md`](DarkMatter.md) §5c). On that clock the dark-matter fit votes for the local `H₀ ≈ 72.9`, a conditional vote on the **Hubble tension**, not a resolution (§5a). The dynamical `ρ_Λ ∝ H²` reading is excluded, so QLF is no longer in the early-dark-energy class. **No epoch is special, in the own-clock sense:** every observer sees the same rate profile, faster behind and slower ahead (`QLF_InflationObserver`, pre-registered; a constant `Λ` would break it). **Named tension:** `ρ_Λ ∝ H²` then keeps `Ω_Λ = log 2` at recombination and BBN, far above early dark-energy bounds ([`Curvature.md`](Curvature.md) §8a). **Convergence with accepted cosmology (the ledger, [`Mysteries_Of_Physics.md`](Mysteries_Of_Physics.md) §3a).** QLF is **concordant with ΛCDM's observational core** — the hot Big Bang, CMB, BBN (`Y_p = 1/4`), the ≈13.8 Gyr age, `Ω_Λ = log 2 ≈ 0.69`, and the `w≈−1` accelerating expansion are all reproduced or left intact (*a Big-Bang-singularity alternative, not a hot-Big-Bang-observation alternative*). It **diverges only on ΛCDM's two interpretive pillars** — particle cold dark matter (→ the RAR/MOND reading above) and a static `Λ` (→ dynamical `ρ_Λ ∝ H²`), i.e. the open, contested questions. Its **deeper convergence** is with accepted **thermodynamic/emergent gravity** (Jacobson 1995, Bekenstein–Hawking, holography), from which it *derives* the Einstein `8πG` coefficient and `Λ = log 2` (`QLF_EinsteinEquations`). **Named exception:** `ρ_Λ ∝ H²` keeps `Ω_Λ = log 2` at early epochs, in tension with CMB/BBN early dark-energy bounds ([`Curvature.md`](Curvature.md) §8a). --- ## 7. Particles and the Standard Model **Connectors:** *axis count* → 3 fermion generations · *Q = 2/3* → Koide -> m_tau · *3 angles + CP* → CKM / PMNS mixing · *self-conjugate* → neutrino is Majorana · *DeltaL = 2* → beta-decay / 0nubetabeta · *one scale x ratios* → mass spectrum · *m = 1/R fold delay* → mass (Higgs mechanism) · *log 2 = one bit* → spin-1/2 = one bit (it from bit) · *spatial projection (STAR 2026)* → baryon number = junction winding · *doubly charmed (LHCb 2026)* → charge ladder 3Q = 6 - 3k · *Hermitian conjugate (BASE 2026)* → antiparticle = time mirror **Open:** [`Standard_Model.md`](Standard_Model.md) · [`Beta_Decay_Neutrino_Nature.md`](Beta_Decay_Neutrino_Nature.md) · [`Per_Qubit_Mass_Quantum.md`](Per_Qubit_Mass_Quantum.md) · [`Spin_QLF.md`](Spin_QLF.md) · [`Higgs.md`](Higgs.md) · [`Mathematics_From_QLF.md`](Mathematics_From_QLF.md) · [`Discoveries_2026.md`](Discoveries_2026.md) **2026: baryon number moved to the junction.** QLF's own neutron word `>^+/+` reads `B = 0` on the windowed winding, because each down quark's gauge twist breaks the windows beside it. Read on the **junction**, the word with its gauge twists removed, both nucleons have `B = 1`, and no gauge-twist insertion (a W emission, `u ↔ d`) can change it (`junction_insert_gauge`). That is STAR's 2026 claim that the carrier is the gluon junction, not the valence flavours, reached on the substrate; it agrees on the carrier, not on STAR's factor-of-two stopping excess. On the same junction LHCb's doubly charmed baryons sit on `3Q = 6 − 3k` with `B = 1` on every rung, and the antiparticle is the time mirror with the same length and opposite charge and baryon number, for every word ([`Discoveries_2026.md`](Discoveries_2026.md) §1, §2, V1; [`QLF_BaryonWinding`](lean/QLF_BaryonWinding.lean)). **It from bit — information is the ½-spin closure.** The unit of information is the two-valued spinor closure: one bit (`log 2`) for `{+I,−I}` vs *zero* for a single-valued vector `{+I}` (a single-valued object cannot express a distinction). The `2π` double-valuedness is reproven from the explicit rotation matrices (`spinor_double_valued_vector_blind`: `+I` on the vector `SO(3)` rep, `−I` on the spin-½ `SU(2)` rep), grounding **Cartan**'s 1913 spinor as its carrier. Priority runs *abstraction → physical*: information **is** the distinction, the ½-spin closure its minimal realization ([`lean/QLF_SpinorInformation.lean`](lean/QLF_SpinorInformation.lean), [`Mathematics_From_QLF.md`](Mathematics_From_QLF.md) §Rung 5a). --- ## 8. Quantum-gravity / TOE pillars QLF meets the three TOE-candidate programs — and reproduces their wins from the substrate. **Connectors:** *j = 1/2 spin network* → Loop Quantum Gravity · *C(2n,n) modes* → String theory · *Q = half-spin shift* → Supersymmetry · *closure floor mu^2=1/2* → Planck scale **Open:** [`README.md`](README.md) · [`LQG_QLF.md`](LQG_QLF.md) · [`StringTheory.md`](StringTheory.md) · [`SUSY_QLF.md`](SUSY_QLF.md) · [`Planck_Scale.md`](Planck_Scale.md) --- ## 9. The Millennium Prize program The thesis: *the continuum and Choice are mathematics' UV catastrophe* — each problem = a constructive RCA₀ core + one explicit continuum/Choice boundary axiom. **Connectors:** *critical line* → Riemann hypothesis · *log 2 gap quantum* → Yang-Mills mass gap · *rank = ord* → Birch-Swinnerton-Dyer · *balanced => algebraic* → Hodge conjecture · *generate != verify* → P vs NP · *no blow-up* → Navier-Stokes **Open:** [`Continuum_Choice_Fallacy.md`](Continuum_Choice_Fallacy.md) · [`Riemann-Conjecture-Proof.md`](Riemann-Conjecture-Proof.md) · [`YangMills_MassGap_QLF.md`](YangMills_MassGap_QLF.md) · [`BSD_QLF.md`](BSD_QLF.md) · [`Hodge_QLF.md`](Hodge_QLF.md) · [`P_vs_NP_QLF.md`](P_vs_NP_QLF.md) · [`NavierStokes_QLF.md`](NavierStokes_QLF.md) Overview: [`Millennium.md`](Millennium.md). --- ## 10. Beyond the SM What QLF derives that the SM treats as free input, and the falsifiable predictions it makes. **Connectors:** *not free* → derived: alpha, Koide, theta-bar=0, Omega_La… · *test now* → Majorana neutrino -> 0nubetabeta · *scale-free by construction* → no cosmological drift of alpha(0) · *soft* → dark matter is not a particle · *kill test (2026)* → Th-229 nuclear clocks · *test pending* → LZ single event · *GIM: no new particle* → B-meson tension · *nulls predicted* → CPT null; no objective collapse, no time jitter **Open:** [`Beyond_Standard_Model.md`](Beyond_Standard_Model.md) · [`Beta_Decay_Neutrino_Nature.md`](Beta_Decay_Neutrino_Nature.md) · [`Alpha.md`](Alpha.md) · [`DarkMatter.md`](DarkMatter.md) · [`Discoveries_2026.md`](Discoveries_2026.md) **The 2026 tests.** Each 2026 result that QLF could fail has its kill condition written down before the data settle it ([`Discoveries_2026.md`](Discoveries_2026.md)): a nuclear-clock drift of `α` (`alpha_cannot_drift`, since `α = 1/(128 + d²)` has no time argument), a reproducible proton–antiproton difference at BASE, a confirmed dark-matter signal at LZ, a new particle behind the `B → Kπμμ` tension, and a collapse-model signal in large superpositions or a random walk in time. All are predicted null. --- ## 11. Chemistry, molecules and folding The **Emergent matter** family: matter in bulk, from the same one rule — **a bond is a shared closure**, no forces and no orbitals put in by hand ([`Chemistry.md`](Chemistry.md)). Nothing here is a new law; everything is domains 3–4 assembled. **Connectors:** *saturate* → shared closure = bond · *H2O, CO2, graphite, rust* → molecules and carbon allotropes · *Pauli holds them apart* → crystals and condensates · *(v-2)/2 each* → closure count · *b1 counts them* → double bond = ring = one closure · *only k is free* → reaction class · *free valence for water?* → hydrophobic / polar · *valence 2 is neutral* → divalent chains · *backbone carries none* → folding · *only H-H pays* → folding · *two liquids (2026)* → water = two closure inventories **Open:** [`Chemistry.md`](Chemistry.md) · [`Protein_Folding.md`](Protein_Folding.md) · [`lean/QLF_Unsaturation.lean`](lean/QLF_Unsaturation.lean) · [`lean/QLF_Folding.lean`](lean/QLF_Folding.lean) · [`hydrocarbon_census.py`](hydrocarbon_census.py) · [`protein_census.py`](protein_census.py) **The counting layer.** An atom's valence is what it contributes to a molecule's **closure count**, `(valence − 2)/2`, and the cycle rank `b₁ = E − V + 1` counts the closures the molecule carries. That identification does three things at once: the textbook **degree of unsaturation** *is* that count (with oxygen absent from the formula because its contribution is **zero**); a **ring and a double bond are one phenomenon**, so `C₆H₁₂` is a single census class; and since a balanced reaction pins `V` and `E`, a reaction's change in closure count **is** its change in molecule count — addition, elimination and substitution are that one number ([`QLF_Unsaturation`](lean/QLF_Unsaturation.lean), no axioms). **Up to folding.** Valence 2 is the neutral element of the count, which is why a divalent monomer makes a **chain** and why the peptide bond — closure-neutral, 2 molecules in and 2 out — leaves a polypeptide backbone carrying no closure of its own. So **every closure a folded chain has is a contact**, and a contact is a ZFA closure in the literal sense: zero net displacement, count-balanced, Pauli-closed by the keystone ([`QLF_Folding`](lean/QLF_Folding.lean)). The lattice-protein parity rule and the `log 2` contact quantum follow; the **mirror no-go** — counting cannot select a handedness — bounds what the census can ever answer ([`Protein_Folding.md`](Protein_Folding.md)). **Water as two liquids (2026).** The two interconverting structures reported in 2026 read as two closure inventories: four shared closures in a tetrahedron, or a crowded fifth neighbour. In the two-state model the density maximum is proved (`density_maximum`, and none without the open structure). It gives [`Mpemba.md`](Mpemba.md) a concrete hidden census; the census that reproduces 4 °C is still open ([`Discoveries_2026.md`](Discoveries_2026.md) V2). --- ## 12. Quantum field theory and renormalization A **Foundations** domain: the continuum thesis of domains 1 and 9, applied to physics' most quantitative framework. **QFT is the continuum limit of the substrate's discrete ZFA-event combinatorics** — it works because the substrate really is doing sum-over-histories; its infinities appear only where that sum is pushed to a *continuum* of modes ([`QFT_QLF.md`](QFT_QLF.md)). **Connectors:** *grade by closure* → expansion order · *IsDiagram order 0* → tree level = the closure census · *single binding* → one loop = the 2/3pi coefficient · *listen at capacity R* → renormalization = the capacity horizon · *A_R+1 minus A_R* → counterterm, one finite term · *Q(R) = Q0 2^R* → running: QED log = census octave count · *prefix-free code* → Kraft bound: the series converges · *no mode continuum* → Dyson divergence = continuum artefact **Open:** [`QFT_QLF.md`](QFT_QLF.md) · [`Perturbation_Theory_QLF.md`](Perturbation_Theory_QLF.md) · [`lean/QLF_ExactRG.lean`](lean/QLF_ExactRG.lean) · [`lean/QLF_FractalDiagram.lean`](lean/QLF_FractalDiagram.lean) · [`lean/QLF_VacuumPolarization.lean`](lean/QLF_VacuumPolarization.lean) · [`lean/QLF_RunningCouplings.lean`](lean/QLF_RunningCouplings.lean) · [`TheContinuum.md`](TheContinuum.md) **The perturbation series is the possibility tree**, graded not by an external coupling but by **closure structure** — continuation length, diagram order (`IsDiagram`, the closure↔Feynman-diagram map), or closure depth ([`Perturbation_Theory_QLF.md`](Perturbation_Theory_QLF.md)). Two orders are proven against QED: the **tree level is exactly the closure census** (`order_zero_iff_closure`), and the **one loop is the `2/(3π)` vacuum-polarization coefficient** (`orderOneWeight_eq`), census-sourced and committed before comparison ([`QLF_FractalDiagram`](lean/QLF_FractalDiagram.lean), [`QLF_VacuumPolarization`](lean/QLF_VacuumPolarization.lean)). **Renormalization is Wilsonian and finite.** The regulator is the **capacity horizon** `R` — a listener of capacity `R` hears the closures of depth `≤ R`; the counterterm for the step `R → R+1` is a single finite term `A_{R+1} − A_R`, never a subtracted infinity; the running is `Q(R) = Q₀·2^R`, so the QED logarithm **is** the census octave count ([`QLF_VacuumPolarizationTower`](lean/QLF_VacuumPolarizationTower.lean)). And the substrate's series **converges absolutely** in its cylinder measure — `twist_kraft` plus `|A| ≤ W` — so no Borel resummation, and the Dyson divergence is exposed as an artefact of the mode continuum the substrate does not have ([`QLF_ExactRG`](lean/QLF_ExactRG.lean), the exact-RG recursion + finiteness + convergence, no axioms). **The α residual is the worked calibration of the bridge line** ([`Alpha_Residual.md`](Alpha_Residual.md) §9c): QLF proves the census space (two-sided bracket) and proves `w = ½ → 137.032` structural; the open piece is the *multiplicity* — which continuum resummation — not a census-truncation rule. Existence proven, multiplicity open — the exact shape of every Millennium bridge, run to convergence in a case with a measured answer. Every substrate mechanism swing is closed; the turbulence swing produced a side-derivation (She–Léveque `C₀ = 2` from `/solve` axis-minimality). **Contrast, once:** the rigorous continuum QFT (a Wightman / Osterwalder–Schrader theory on ℝ⁴) is not constructed here — that step is the Yang–Mills boundary axiom of domain 9. What the reformulation proves is that the discrete substrate the continuum is the *limit of*, plus the one explicit crossing, carries QFT's empirical content. --- ## See also - [`README.md`](README.md) · [`lean/README.md`](lean/README.md) — project overview + the full Lean module table. - [`Open_Problems.md`](Open_Problems.md) — the honest gap registry (closed / principled-boundary / open). - [`Beyond_Standard_Model.md`](Beyond_Standard_Model.md) — the derived / predicted / open scorecard. - [`Alpha.md`](Alpha.md) — one result mapped end to end, as a worked example. - [`Discoveries_2026.md`](Discoveries_2026.md) — the 2026 physics results read on the substrate: what is proved, what is predicted null, and the one correction (baryon number on the junction). - [`Chemistry.md`](Chemistry.md) · [`Protein_Folding.md`](Protein_Folding.md) — domain 11 end to end: one rule (a bond is a shared closure) up through the closure count to a fold as a closure census. - [`Fredkin_QLF.md`](Fredkin_QLF.md) — conservative logic on the substrate: Fredkin's conservation law **is** ZFA count balance ([`QLF_Fredkin`](lean/QLF_Fredkin.lean), no axioms), so a reversible computer runs **free** and only erasure is charged. Run it: [`fredkin_machine.html`](fredkin_machine.html).