Open_Problems.md

Open Problems — QLF gap registry

A single map of what is closed, what is a principled boundary, and what is genuinely open, with a pointer to the document that owns each item. This is an index, not a re-derivation: the substance lives in the linked docs. Its mirror is Beyond_Standard_Model.md (what QLF forces / predicts past the SM). It complements Experimental_Consistency.md (which tracks per-result precision) by collecting the forward work in one place.

Why a registry. Open items were scattered across ~25 documents, each with its own "Open work" section. When a status changes, the claim can drift out of sync across docs (as the Hadronic-Depth attribution did before its correction). This file is the canonical status list; when an item moves, update it here and in its owning doc.

Status legend

Tag	Meaning
✅ Closed	Derived / machine-verified; no longer open.
🧱 Boundary	A principled limit (like an explicit axiom), not a gap to be closed by more work. The structural form is fixed; a number or bridge is inherited.
🔵 Open — quantitative	Mechanism identified; a specific number is not yet derived from substrate.
🟣 Open — structural / Lean	Result holds in prose/numerics; a clean theorem or Lean anchor is not yet written.
⚪ Future work	Out of current scope by specification (not by doubt about the mechanism).

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Recently changed (this is the live edge)

Item	Status	Where
Neutrino is Majorana → 0νββ	✅ Machine-verified — neutrino_majorana (lean/QLF_Majorana.lean): the antiparticle is the Hermitian conjugate (conjugate-and-reverse), and the ^v loop is a fixed point of it (the electron is not — electron_not_majorana, so it is Dirac). The neutrino is the unique self-conjugate fermion ⟹ lepton number violated ⟹ neutrinoless double-beta decay is the signature (LEGEND/nEXO). Conditional on the ^v assignment + antiparticle = Hermitian-conjugate	lean/QLF_Majorana.lean, Beta_Decay_Neutrino_Nature.md §1, Experimental_Consistency.md §9–10
Pauli closure — count balance ⟹ Pauli scalar, all twist histories (incl. cross-axis interleaving)	✅ Closed — count_balanced_pauli_closed	lean/QLF_TwistAlphabet.lean, Experimental_Consistency.md §2.1
Bethe constant k(n,0) (Lamb shift)	🧱 Boundary — continuum-dominated (I_1S ≈ 19.77 Ry, all bound ΔE < 1 Ry); free-electron sector above the RCA₀ floor	Lamb_Shift.md §6.1, bethe_log_demo.py

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Principled boundaries (not gaps)

Item	Why it is a boundary	Where
Riemann critical line	Substrate bridges proven; RH reduced to one explicit boundary, spectral_hilbert_polya (RCA₀ → WKL₀ crossing). Now constructively scaffolded by the MRE bridge: Z_QLF concrete, MRE saturation grounded in binary_kl and located at the critical-line prior 1/2, refining the boundary to MRE_bridge (riemann_hypothesis_in_qlf_via_MRE). The residual Mellin↔ζ step is the analytic sector where ZFC is proven to fail — ZFC's defect, not a QLF gap (rh_proof_in_progress)	lean/QLF_Riemann.lean, lean/QLF_RiemannMRE.lean
Yang–Mills mass gap	Gap proven on the substrate (lightest non-vacuum closure = one log 2 quantum; mass_gap_quantum_pos, gaugeMassGap = log 2 > 0); only the continuum-QFT reconstruction remains, carried by the explicit boundary axiom yang_mills_continuum_gap — the continuum sector where ZFC is proven to fail (mass_gap_proven_constructively)	lean/QLF_MassGap.lean, YangMills_MassGap_QLF.md
P vs NP	Lean-anchored: the realized (verifiable) set IS the O(n) verify-filter of the generated candidates (realized_is_verify_filter) with cardinality the real C(2n,n) (realized_count_eq_central_binomial, reusing find_stable_states_length_even). The formal separation is the boundary axiom generate_not_reducible_to_verify, over an infinite computational model — ZFC's proven-defective sector, not a QLF gap	lean/QLF_PvsNP.lean, P_vs_NP_QLF.md
Navier–Stokes smoothness	Lean-anchored: realized flows achieve ZFA (realized_flow_achieves_zfa, reusing encode_is_zfa) and are stable closures (realized_flow_is_stable, reusing qlf_universality) — no realized history blows up; blow-up = non-terminating history pruned by full_zeno_prune. Only continuum-PDE inheritance remains — the boundary axiom navier_stokes_continuum_limit, the continuum sector where ZFC is proven to fail	lean/QLF_NavierStokes.lean, NavierStokes_QLF.md
Birch–Swinnerton-Dyer	Self-dual central point s=1 proven (bsd_central_point_self_dual), grounded in the same H↔H† involution as Riemann (bsd_riemann_shared_involution); the elliptic-curve→closure encoding is built — concrete EllipticCurveQLF with computed Frobenius traces (frobeniusTrace, Ecn1_frobenius_two). Rank = ord (bsd_rank_equals_order) is now a theorem, discharged through the modularity mirror (Perspective/modularityMirror/centralMultiplicity); the single boundary is modularity_mirror_invariant — the mirror preserves the central multiplicity at its self-dual fixed point. Qualitative BSD bsd_in_qlf derived. Why the mirror is multiplicity-preserving (the uncomputable rank content) is the continuum-sector remainder (bsd_proof_in_progress)	lean/QLF_BSD.lean, BSD_QLF.md, Langlands.md
Hodge conjecture	The Hodge conjugation = the adjoint involution H↔H† (conj_involutive); Hodge classes = its balanced fixed points (conj_fixed_of_isHodge). A (p,q) class encodes to a history count-balanced iff p=q (encode_countBalanced), so hodge_class_is_algebraic is a theorem (Hodge ⟹ count-balanced ⟹ Pauli-closed via count_balanced_pauli_closed ⟹ realized). The single boundary is the faithfulness substrate_realization_is_algebraic — why substrate closure = algebraic realization over the complex-analytic continuum (hodge_proof_in_progress)	lean/QLF_Hodge.lean, Hodge_QLF.md
Speed of light c	The substrate event quantum (one Planck length × one Planck tick together) is the foundational postulate — no Tier-3 below it	Experimental_Consistency.md §3, Kitada_Local_Time_GR.md §5.3
Bethe k(n,0)	Continuum-sector (see live edge above)	Lamb_Shift.md §6.1

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Open — quantitative (the hard front)

Item	Status	Where
Cosmic depth n from first principles	The geometric depth n ≈ 6.7×10⁶⁰ is firm but defined via R_H; the proton cube (m_P/m_p)³ ≈ 2.2×10⁵⁷ undershoots by ~3,000×. This is the hierarchy problem	HadronicDepth.md §2.1
H₀ from substrate	Reduces to deriving n (above); currently H₀ enters as one observable. Would close the last cosmological empirical input	Cosmological_Constant.md §196
Hubble tension	The residual 8.7% in Λ tracks the Planck-vs-SH0ES H₀ discrepancy; not predicted	Cosmological_Constant.md §197
G SI value / Einstein 8π·G coefficients	Form G = L_P²c³/ℏ derived; absolute SI value is substrate-quantum calibration (~37% prediction residual); 8π = 4π·2 identified but the full coefficient calc is research-grade	Gravity_From_Delay.md, Experimental_Consistency.md §6.3
First-principles R_e	α R_e = m_e identifies R_e with the measured electron; deriving R_e (hence m_e) from closure-multiplicity counts is open	Bound_States_QLF.md §6, Atomic_System_QLF_Closures.md §10
Mass spectrum from multiplicity	3rd-generation masses; τ-decay-vertex topology. Handle: Koide Q=2/3 is now machine-verified as forced by N=3 (three axes) ∧ A²=2 (two transverse axes) — koide_two_thirds, lean/QLF_Koide.lean — and predicts m_τ from m_e,m_μ to 0.006% (Weak_Force.md §5a–5b). Still open: the lepton-√mass↔axis-phase identification, the Koide angle, the scale (so m_e,m_μ are inputs), and the quark sector	Standard_Model.md §4.1, Weak_Force.md §5b
Weak sector (W/Z)	Weak-isospin SU(2) Lie algebra now machine-verified in Σ₈ (weak_isospin_su2, Q₈⊂SU(2)); still open: R_W/R_Z (hence W/Z masses + Weinberg-angle value), coupling g, G_F, flavor-change & τ-decay vertex topology, the Koide angle δ (lepton-sector input; 2/9 a flagged coincidence)	Weak_Force.md, lean/BraKetRhoQuCalc.lean
Gauge unification (forces from 3 axes)	Structural conjecture: dim(U(1)×SU(2)×SU(3)) = 12 = 1+3+8, with 1+8 = 9 = N (the α tensor). All three gauge algebras now machine-verified — U(1) (no_magnetic_monopoles), SU(2) (weak_isospin_su2), SU(3) (trace_commutator_zero+gluon_commutator_nonzero, lean/QLF_StrongAlgebra.lean). Still a dimension alignment, not a derivation: no couplings, chirality, masses; quark flavors/CKM open; SU(3) finrank=8 is the elementary codim-1 (not yet Lean)	Forces_From_Three_Axes.md
Constants program (π, e, δ; α below the 0.026% floor)	Methods exist in constants_mapper.py; full CODATA agreement is the active research front	Experimental_Consistency.md §6.3, §11
Lamb shift radiative pieces	AMM +68 MHz (Schwinger α/2π on the bound moment) and vacuum polarization (Uehling) −27 MHz taken as inputs	Lamb_Shift.md §8

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Open — structural / Lean-anchoring

Item	Status	Where
Lamb prefactor 4/(3π n³)	Mostly resolved: = 4·(2/3)·(1/(2π))·(1/n³) (Lean lamb_prefactor_loop_phase); the π is the g-2-validated loop-phase primitive (0.2%), 1/n³/2/3 clean. Only the rational 4 (two-vertex/solid-angle) wants a cleaner origin	Lamb_Shift.md §5
Dirac correction, per-mechanism Lean	Kinematic / spin-orbit / Darwin α² pieces are doc-anchored, not yet individual Lean chains from QLF_Pauli/QLF_TwistAlphabet	Dirac_Correction.md §6
Lorentz boost as a Lean theorem	lorentz_boost_from_zfa_frequencies needs the Markov-blanket frequency structure in-kernel	Cross_Frequency_Lorentz.md §7
Maxwell curl laws (∇×B=μ₀J, ∇×E=−∂B/∂t)	Stated structurally; await a time-indexed event-sequence Lean module (divergence laws already verified)	Maxwell.md, Electricity.md
γ (Euler–Mascheroni) convergence	Structural form Lean-anchored; lim (H_N − ln N) = γ convergence proof deferred (standard real analysis)	lean/QLF_EulerMascheroni.lean
Borromean 5-angle	chirality-mixing-per-pair = 2 not yet derived rigorously from QLF_Pauli's scalar group	lean/QLF_BorromeanAngles.lean
Charge vs B−L	Charge conservation ✅ Lean-anchored (signed_count_conserved); every closure is neutral. B−L is not a conserved signed count — proved via wcount_zero_on_ZFA (conserved counts vanish on closures, but the deuteron has B−L=1; baryon vs antibaryon share a twist multiset yet B−L=±1). B−L is at most winding, and in the lepton sector it is violated (neutrino Majorana, neutrino_majorana) ⟹ 0νββ. So only the gauge charge is exact — QLF carries no exact global B−L. Baryon number ✅ Lean-anchored as a signed 3-axis linking (winding) invariant — baryonNumber (lean/QLF_BaryonWinding.lean): proton >^/ B=+1, antiproton B=−1, leptons & meson B=0; baryon_zero_of_noZ proves the whole z-free lepton/EM sector is baryon-neutral; and baryon_dagger_odd proves the general conjugation-oddness B(ts†) = −B(ts) for all histories (so baryon/antibaryon carry ±B universally). ✅ Closed	lean/QLF_BaryonWinding.lean, lean/QLF_Majorana.lean, Conservation.md §8

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Future work (scope-limited by specification)

Item	Where
Periodic table Z ≥ 21 (d-shell synthesis; Cr/Cu/La anomalies) — current routing capped at neon	Magic_numbers.md
QuantumOS active-inference scheduler on QPU silicon (today: browser control plane)	Crystal_QuantumOS.md §7
Quantitative delayed-choice visibility match (Kim et al. 1999)	Delayed_Choice_Eraser.md
Strong-field FLRW coupling for the cosmological constant	Cosmological_Constant.md
Material-specific carrier-scattering / ρ(T) / T_c	Electricity.md
QRNG Closure Observatory — falsifiable test of whether QRNG streams deviate from the analytic ZFA-closure null (predeclared sieve + controls; expected Tier-0)	QRNG_Closure_Observatory.md

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Notes

• Tiers vs. this file. Experimental_Consistency.md uses Tier-1/2/3 for achieved precision; this registry uses the status tags above for forward work. A "Tier-3 open" result there maps to 🔵/🟣 here, unless it is a 🧱 boundary.
• Maintenance. When an item closes or is reclassified, edit its row here and its owning doc in the same change, and (if it gains a theorem) the relevant lean/ module and lean/README.md row.