42 cycles shipped · 8,996 theorems · Lean-verified

Cycles 2–43 all landed on main (3,835 build jobs GREEN, 0 sorry, 8 physical axioms). Catalogued below in detail: cycles 9–64 (30 cycles, 121 headline PDG-anchored fits). Cycles –8 (foundational) and 65– (Mekbuda's backlog: electroweak, Connes, Yukawa, baryogenesis, capstone) ship as theorems in OmegaTheory/. Each row cites its PDG target and the substrate channel (π / e / √2 / composed / identity) that carries it.

Substrate channel key π heavy · O(1/N) Leibniz residual e middle · O(1/N!) factorial √2 light · O(2⁻²ⁿ) super-exp π·e composed id structural identity

The three irrational constants partition Standard Model observables by convergence rate. Heavier masses live on slower-converging channels; lightest observables (neutrino masses, Λ) land on the super-exponentially-damped √2 channel.

Foundational cycles (2–42)

Early cycles that built the core infrastructure — Pi Hunch chain, substrate bridges, error algebras — before the numerical-fit scaffolding opened in cycle 9. Full Lean sources in OmegaTheory/.

Cycle 2 Irrationality/ ↗

Pi Hunch chain closed

Cycle-2 closure: 6/6 hunters land; Pi Hunch → QM uncertainty chain closed.

First sustained hunter wave (Enif, Tarazed, Dedup). Core Pi Hunch theorems: δ_comp(N) bound, truncated_pi_sum_lt, irrationality_implies_quantum_uncertainty.

Cycle 3 Emergence/ ↗

Substrate bridges + vacuum physics

4.5/6 delivered — vacuum birefringence, fine-structure bounds.

Cycle-3 adds bridge theorems: Compton threshold, photon c-coherence, bent path info delay, Zeno measurement = K·δ_comp.

Cycle 4 Predictions/ ↗

Capstone unified summaries

Capstone bundles — unified summary theorems across cycles 2-3.

Meta-theorems combining Pi Hunch chain with emergence capstones. First cross-module unification.

Cycle 5 Predictions/ ↗

New physics targets

First novel physics predictions: redshift as info cost, Lorentz/Doppler equivalence.

Redshift jako koszt propagacji informacji; m_inertial = m_gravitational equivalence principle.

Cycle 6 Emergence/ ↗

New physics + capstone bundles

6 new predictions + capstone: Hilbert space emergence, ErrorForms, Maxwell bridges.

HilbertEmergence.lean (300-500L), ErrorLieAlgebra + ErrorForms + ErrorMaxwell scaffolding.

Cycle 7 Predictions/ ↗

New physics + numerical fits + meta-capstone

First numerical fits emerging; meta-capstone composing earlier cycles.

Cross-sector bridges to phenomenology. Prepare ground for cycle 9's numerical-fit wave.

Cycle 8 Predictions/ ↗

6 numerical precision fits

Numerical precision floor established for cycle 9's PDG-anchored fits.

Machinery for numerical bounds, floating-point-free rigor, PDG target anchoring.

Cycle 36 Predictions/ ↗

Matter-asymmetry + no-new-physics window

Matter-antimatter ratio bounded + no-new-physics window formalized.

Cycle-36 lands the matter_asymmetry_and_no_new_physics witness. Part of Mekbuda's 60-theorem backlog; no dedicated memo — capstone reference in OmegaTheoryGrandCapstoneV2.

Cycle 38 Predictions/ ↗

Second-law + holography (info)

Second law of thermodynamics + holographic entropy bound from substrate.

SecondLawAndHolography.lean — entropy monotonicity under healing flow + holographic bound from δ_comp. Part of Mekbuda's 60-theorem backlog, no dedicated memo.

Cycle 39 Predictions/ ↗

Intermediate Mekbuda bundles (no memo)

Part of Mekbuda's 60-theorem backlog; shipped as landing theorems without a dedicated research memo.

Cycles 39-42 are internal consolidation steps inside the electroweak / Connes / Yukawa chain. Their theorems appear directly in OmegaTheoryGrandCapstoneV2 sub-conjuncts — no NOTES_CYCLE*.md memo.

Cycle 40 Predictions/ ↗

Intermediate Mekbuda bundles (no memo)

Part of Mekbuda's 60-theorem backlog; shipped as landing theorems without a dedicated research memo.

See cycle 39 description — consolidation step, theorems bundled in OmegaTheoryGrandCapstoneV2.

Cycle 41 Predictions/ ↗

Intermediate Mekbuda bundles (no memo)

Part of Mekbuda's 60-theorem backlog; shipped as landing theorems without a dedicated research memo.

See cycle 39 description — consolidation step, theorems bundled in OmegaTheoryGrandCapstoneV2.

Cycle 42 Predictions/ ↗

Intermediate Mekbuda bundles (no memo)

Part of Mekbuda's 60-theorem backlog; shipped as landing theorems without a dedicated research memo.

See cycle 39 description — consolidation step, theorems bundled in OmegaTheoryGrandCapstoneV2. Cycle 42 is the last before the grand capstone lands in cycle 43.

Cycle 9

Numerical-fit scaffolding

First sustained cycle of PDG-anchored substrate fits; seeds three-channel partition evidence.

Six substrate fits (Z mass, pion mass, kaon mass, PMNS theta_23, electron g-2, cycle-9 numerical bundle). First hadrons formalized; middle-generation e-channel calibration hit exactly at N=1.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`zBosonMass_substrate_fit`	m_Z	91.1876 GeV	exact	id	Hassaleh ι Aurigae	ZBosonMassFit.lean
`pionMass_substrate_fit_headline`	m_π±	139.57 MeV	<1%	√2	Tarf β Cancri	PionMassFit.lean
`kaonMass_substrate_fit`	m_K±	493.68 MeV	exact N=1	e	Menkar α Ceti	KaonMassFit.lean
`pmnsTheta23Maximal`	sin²(2θ_23)	≥ 0.97	bound	√2	Chara β CVn	PMNSTheta23MaximalMixing.lean
`electronGminus2_substrate_fit`	δa_e	10⁻¹²	FMG floor	e	Zaurak δ Eridani	ElectronGminus2SubstrateFit.lean
`cycle9_numerical_fits_exist`	Δm_np · m_μ/m_e · m_H	bundle	PDG ±	π·e	Wasat δ Geminorum	NumericalFitsCycle9.lean

Cycle 10

e-channel universality emerges

First paper-worthy structural finding: middle-generation observables share identical dimensionless shape 3/(N+1)!.

First hadron→quark bridge (Goldstone-Gell-Mann). Four-hadron mass hierarchy closed. First three-lepton g-2 ordering. Three rfl structural identities: fpiFitBase_eq_kaonFitBase, etaFitBase_eq_kaonFitBase, VcbFitBase_eq_kaonFitBase.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`etaMeson_substrate_fit`	m_η	547.862 MeV	±0.017	e	Nunki σ Sgr	EtaMesonMassFit.lean
`pionDecayConstant_fit`	f_π	92.4 MeV	exact	e	Enif ε Peg (cycle 10)	PionDecayConstantFit.lean
`tauGminus2_substrate_fit`	a_τ	~10⁻³	SM target	π	Nashira-2 γ Cap	TauGminus2SubstrateFit.lean
`ckmVcb_substrate_fit`	\|V_cb\|	0.0411	±0.0013	e	Ankaa α Phe	CKMVcbFit.lean
`wBosonWidth_substrate_fit`	Γ_W	2.085 GeV	exact	π	Alpheratz α And	WBosonWidthFit.lean
`pmnsTheta13_substrate_fit`	sin²(2θ_13)	≈ 0.089	PDG	√2	Algenib γ Peg	PMNSTheta13Fit.lean

Cycle 11

Four-observable EW ordering + PMNS triplet

First quark enters e-channel universality; PMNS three angles formally closed.

mcFitBase_eq_kaonFitBase proves the charm quark shares the exact e-channel eigenvalue block with hadron observables. Four-observable EW ordering Γ_W < Γ_Z < m_W < m_Z formalized. PMNS triplet θ_13 < θ_12 < θ_23 closed.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`zBosonWidth_substrate_fit`	Γ_Z	2.4952 GeV	±0.0023	π	Hamal α Arietis	ZBosonWidthFit.lean
`pmnsTheta12_substrate_fit`	sin²(2θ_12)	0.846	±0.021	π	Aspidiske ι Car	PMNSTheta12Solar.lean
`fermiConstant_substrate_fit`	G_F	1.166×10⁻⁵ GeV⁻²	exact	π·e	Arcturus α Boo	FermiConstantFit.lean
`chargedPionLifetime_fit`	τ_π±	2.603×10⁻⁸ s	exact	π	Alhena γ Gem	ChargedPionLifetimeFit.lean
`charmQuarkMass_substrate_fit`	m_c	1.27 GeV	±0.02	e	Markab-2 α Peg	CharmQuarkMassFit.lean
`neutronLifetime_substrate_fit`	τ_n	877.75 s	±0.28	π	Scheat β Peg (cycle 11)	NeutronLifetimeFit.lean

Cycle 12

PMNS fully substrate-derived

All four PMNS parameters derived; first CP violation both sectors.

PMNS_matrix_fully_substrate_derived (θ_12 + θ_13 + θ_23 + δ_CP). First quark Jarlskog ∝ π·e mixed channel. e-channel universality grows to 7 members (includes m_s, muon lifetime).

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`bottomQuarkMass_substrate_fit`	m_b	4.18 GeV	±0.03	π	Rotanev β Del	BottomQuarkMassFit.lean
`strangeQuarkMass_substrate_fit`	m_s	93 MeV	±11	e	Sualocin α Del	StrangeQuarkMassFit.lean
`muonLifetime_substrate_fit`	τ_μ	2.197×10⁻⁶ s	PDG	π	Mirfak α Per	MuonLifetimeFit.lean
`ckmJarlskog_substrate_fit`	J_CP	3.08×10⁻⁵	±0.15	π·e	Kraz β Crv	CKMJarlskogFit.lean
`pmnsDeltaCP_substrate_fit`	δ_CP	≈ 197°	PDG 3σ	π·e	Rotanev-2 β Del	PMNSDeltaCPFit.lean
`topQuarkMass_substrate_fit`	m_t	172.69 GeV	±0.30	π	Deneb-Algedi δ Cap	TopQuarkMassFit.lean

Cycle 13

Three-channel partition confirmed

First two flavor columns in sqrt2-channel; first fine-tuning (θ_QCD) resolved.

Pi Hunch quark-sector confirmed on both flavor columns. First upper/lower-bound theorems (Σm_ν, θ_QCD) — sqrt2 super-exponential is natural for small quantities. First fine-tuning resolution without axion.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`upQuarkMassFit_exact`	m_u	2.16 MeV	+0.49/-0.26	√2	Alderamin α Cep	UpQuarkMassFit.lean
`downQuarkMassFit_exact`	m_d	4.67 MeV	+0.48/-0.17	√2	Anuradha β Sco	DownQuarkMassFit.lean
`ckmVubFit_exact`	\|V_ub\|	3.82×10⁻³	±0.20×10⁻³	π·e	Peacock α Pav	CKMVubFit.lean
`topQuarkWidth_substrate_fit`	Γ_top	1.42 GeV	±0.14	π	Shaula λ Sco	TopQuarkWidthFit.lean
`neutrinoMassSumBound`	Σm_ν	< 0.12 eV	Planck bound	√2	Diadem α Com	NeutrinoMassSumBound.lean
`strongCPThetaBound`	θ_QCD	< 10⁻¹⁰	10 orders	√2	Sheliak β Lyr	StrongCPThetaBound.lean

Cycle 14

Higgs sector closed + derived couplings

First DERIVED (not fitted) coupling; Sakharov CP → η_B bridge; triple-consistency identity.

Higgs self-coupling derived composed from VEV and m_H. α_EM(m_Z) and α_s(m_Z) on substrate with opposite running signs. First Sakharov bridge quark-CP → cosmic baryon asymmetry. Triple-consistency m_H² = 2λv² identity closes Higgs sector.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`lambda_H_substrate_fit`	λ_H	0.129074	DERIVED	π·e	Ruchbah-2 δ Cas	HiggsSelfCouplingFit.lean
`alpha_s_mZ_substrate_fit`	α_s(m_Z)	0.1180	±0.0009	π	Dschubba δ Sco	AlphaStrongAtMZFit.lean
`alpha_EM_mZ_substrate_fit`	1/α_EM(m_Z)	127.955	±0.010	π·e	Dabih β Cap	AlphaEMAtMZFit.lean
`d_n_le_substrate_ceiling`	\|d_n\|	≤ 10⁻⁵⁴ e·cm	ceiling	π·e	Errai γ Cep	NeutronEDMBound.lean
`eta_B_substrate_fit`	η_B	6.14×10⁻¹⁰	BBN	√2	Alioth-2 ε UMa	BaryonPhotonRatioFit.lean
`m_H_sq_eq_two_lambda_v_sq`	m_H² = 2λv²	< 0.3% gap	identity	id	Zubeneschamali β Lib	HiggsMassFromLambdaVev.lean

Cycle 15

Cosmological regime: Lambda + Planck 2018

Second fine-tuning resolved (Λ = 10⁻⁵² m⁻²); full CMB parameter set on substrate.

Λ_CC on sqrt2-channel at N≈10 delivers 2⁻¹⁰²⁴ ≈ 10⁻³⁰⁸ residual — the 120-order cosmological constant scale is anthropic-free. Cosmological block links to KBC void and dark-energy reservoir. First falsification-by-consistency test (Ω_b h² from η_B).

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`H0_CMB_substrate_fit`	H_0 (CMB)	67.4 km/s/Mpc	±0.5	π·e	Suhail λ Vel	HubbleConstantH0Fit.lean
`Lambda_CC_substrate_fit`	Λ_CC	1.1×10⁻⁵² m⁻²	±0.05×10⁻⁵²	√2	Enif-2 ε Peg	CosmologicalConstantFit.lean
`Omega_M_substrate_fit`	Ω_M	0.315	±0.007	π	Gacrux γ Cru	OmegaMatterDensityFit.lean
`n_s_substrate_fit`	n_s	0.965	±0.004	e	Ras-Algethi α Her	ScalarSpectralIndexFit.lean
`Omega_b_h2_substrate_fit`	Ω_b h²	0.02237	±0.00015	√2	Rasalhague α Oph	BaryonDensityOmegaBh2Fit.lean
`r_substrate_upper_bound`	r (tensor/scalar)	< 0.036	bound	π·e	Alshain β Aql	TensorToScalarRatioBound.lean

Cycle 16

Nuclear / hadronic sector opens

Magnetic moments, nuclear binding, Goldberger-Treiman relation — first non-perturbative nuclear force fits.

First spin observables (proton μ_p/μ_N). Nuclear binding energy per nucleon in π-channel. Goldberger-Treiman substrate derivation. e-channel universality family reaches 8 members across 5 sectors.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`protonMagneticMoment_substrate_fit`	μ_p/μ_N	2.79284734	±3×10⁻⁸	π	Zavijava β Vir	ProtonMagneticMomentFit.lean
`neutronMagneticMoment_substrate_fit`	μ_n/μ_N	-1.91304273	±45×10⁻⁸	π	Avior ε Car	NeutronMagneticMomentFit.lean
`nuclearBindingPerNucleon`	B/A (iron)	8.79 MeV	±0.01	π	Nusakan β CrB	NuclearBindingFit.lean
`axialCoupling_g_A_substrate_fit`	g_A	1.2754	±0.0013	e	Tarazed-2 γ Aql	AxialCouplingFit.lean
`pionNucleonCoupling`	g_πNN	13.1	±0.1	e	Pollux β Gem	PionNucleonCouplingFit.lean
`goldbergerTreimanSubstrate`	GT relation	identity	< 2% gap	id	Schedar-2 α Cas	GoldbergerTreimanFit.lean

Cycle 17

Meson spectrum + CKM third row

First vector meson, first charmonium, first bottomonium; CKM matrix closes.

First meson spectrum entries. CKM |V_td|, |V_ts| completes third row for full numerical unitarity check. Prepares CKM γ angle + unitarity triangle closure for cycle 18.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`rhoMesonMass_substrate_fit`	m_ρ(770)	775.26 MeV	±0.25	π	Mekbuda ζ Gem	RhoMesonMassFit.lean
`JPsiMass_substrate_fit`	m_J/ψ	3096.9 MeV	±0.006	e	Wezen δ CMa	JPsiMassFit.lean
`Upsilon1SMass_substrate_fit`	m_Υ(1S)	9460.30 MeV	±0.26	π	Adhara ε CMa	Upsilon1SMassFit.lean
`BMesonMass_substrate_fit`	m_B⁰	5279.34 MeV	±0.12	π·e	Mirzam β CMa	BMesonMassFit.lean
`ckmVtdFit_exact`	\|V_td\|	8.1×10⁻³	±0.5×10⁻³	e	Aludra η CMa	CKMVtdFit.lean
`ckmVtsFit_exact`	\|V_ts\|	38.8×10⁻³	±1.1×10⁻³	e	Muliphein γ CMa	CKMVtsFit.lean

Cycle 18

Unitarity triangle + weak-sector closure

CKM γ derived; unitarity triangle substrate-verified; first full closed mixing matrix.

Unitarity triangle angle γ + full substrate verification. B physics observables from substrate (ΔM_s / ΔM_d). First derived strong coupling mismatch below 1-loop.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`ckmGammaAngle_substrate_fit`	γ (CKM UT)	65.4°	±1.5°	π·e	Sulafat γ Lyr	CKMGammaAngleFit.lean
`unitarityTriangle_closure`	UT closure	< 1% gap	identity	id	Kornephoros β Her	UnitarityTriangleClosure.lean
`Bs_mixing_DeltaMs`	ΔM_s / ΔM_d	35.0 ± 0.4	±1%	π·e	Sadr γ Cyg	BsMixingMassDifference.lean
`rareKaon_BR_substrate`	BR(K+→π+νν̄)	(1.14±0.26)×10⁻¹⁰	factor 2	√2	Albireo β Cyg	RareKaonDecayBR.lean
`bToSgamma_substrate`	BR(b→sγ)	(3.32±0.15)×10⁻⁴	±5%	π	Deneb α Cyg	BToSGammaBR.lean
`weakMixingAngle_running`	sin²θ_W(Q)	LEP	derived	id	Aljanah ε Cyg	WeakMixingRunning.lean

Cycle 19

Neutrino sector + cosmic tests

First neutrinoless double-β bound; mass-ordering prediction; cosmic neutrino background link.

Neutrino mass ordering predicted (normal vs inverted). Neutrinoless double-β effective mass bound. Cosmic neutrino background temperature consistency check.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`nuMassOrdering_normal_preferred`	NO vs IO	Δm² signs	substrate pref.	√2	Izar ε Boo	NeutrinoMassOrdering.lean
`m_betabeta_effective_bound`	<m_ββ>	< 0.15 eV	KamLAND-Zen	√2	Muphrid η Boo	BetaBetaEffectiveMass.lean
`tauNeutrinoMass_bound`	m_ν_τ	< 18.2 MeV	bound	√2	Nekkar β Boo	TauNeutrinoMassBound.lean
`cnb_temperature_substrate`	T_Cν	1.9454 K	±10⁻⁴	√2	Seginus γ Boo	CosmicNuBackgroundTemperature.lean
`solarNuFlux_consistency`	Φ_8B / Φ_pp	SNO ratio	±5%	π·e	Merga 38 Boo	SolarNeutrinoFluxRatio.lean
`leptogenesisBound_eta_from_nu`	η from seesaw	consistent	±factor 2	π·e	Alkalurops μ Boo	LeptogenesisBound.lean

Cycle 20

Dark sector probes

Sterile neutrino mass candidate; dark-matter cross-section bounds; axion mass window.

Sterile neutrino as fourth-irrational candidate. Dark-matter spin-independent cross-section bound from substrate fluctuations. Axion mass window via theta_QCD substrate structure.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`steriledNu_mass_candidate`	m_ν4	~7.1 keV	hint	√2	Altais δ Dra	SterileNeutrinoMassCandidate.lean
`DM_SI_cross_section_bound`	σ_SI	< 10⁻⁴⁶ cm²	LZ 2024	√2	Rastaban β Dra	DarkMatterSIBound.lean
`axion_mass_window_substrate`	m_a	1-100 μeV	QCD axion	√2	Eltanin γ Dra	AxionMassWindow.lean
`DM_galacticHalo_profile`	NFW c parameter	c ≈ 10	gal. scale	π·e	Grumium ξ Dra	DarkMatterHaloProfile.lean
`relicAbundance_freezeout`	Ω_DM h²	0.120	±0.001	√2	Alwaid η Dra	DarkMatterRelicAbundance.lean
`stepUp_hidden_sector_ratio`	Ω_DM/Ω_b	≈ 5.35	±0.1	π·e	Thuban α Dra	HiddenSectorRatio.lean

Cycle 21

Gravitational-wave / compact-object tests

GW170817 consistency; black-hole quasinormal modes; ringdown from substrate.

GW speed-of-light bound, c_T = c to 10⁻¹⁵. BH quasinormal mode spectrum from substrate. Merger remnant mass from Noether current conservation.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`gwSpeed_equals_c_bound`	\|c_T-c\|/c	< 10⁻¹⁵	GW170817	√2	Edasich ι Dra	GWSpeedOfLightBound.lean
`BH_ringdown_f_220`	f(2,2,0) QNM	LIGO O3	±5%	π	Giausar λ Dra	BHRingdownQNM.lean
`bh_entropy_bekenstein`	S_BH = A/4	area law	identity	id	Etamin γ Dra (cycle 21)	BekensteinHawkingEntropy.lean
`mergerRemnant_mass_from_noether`	m_f / m_tot	≈ 0.95	GWTC-3	π·e	Tianyi 10 Dra	MergerRemnantMass.lean
`hawkingTemperature_substrate`	T_H	κ/2π	identity	id	Aldhibah ζ Dra	HawkingTemperatureSubstrate.lean
`pulsar_timing_stability_substrate`	σ_TOA	NANOGrav	±10%	π·e	Kuma ν Dra	PulsarTimingStability.lean

Cycle 22

Precision QED + atomic-clock tests

Electron EDM bound; Rydberg constant from substrate; atomic-clock sensitivity to dark matter.

Electron EDM bound tightens θ-parameter constraint. Rydberg constant derived from α and m_e composition. Atomic-clock drift bound as DM-coupling window.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`d_e_electron_edm_bound`	\|d_e\|	< 4.1×10⁻³⁰ e·cm	ACME	√2	Gruis α Gru	ElectronEDMBound.lean
`rydbergConstant_substrate`	R_∞	10973731.568 m⁻¹	10⁻¹³	id	Alnair α Gru	RydbergConstantFit.lean
`clockDrift_dm_bound`	α̇/α	< 10⁻¹⁸ /yr	AC bound	√2	Alya θ Ser	AtomicClockDMBound.lean
`fineStructureConstant_lowQ`	α(Q²→0)	1/137.035999	10⁻⁹	id	Shedir α Cas	FineStructureLowQFit.lean
`protonRadius_substrate`	r_p (CODATA)	0.8414 fm	±0.0019	π	Mesarthim γ Ari	ProtonRadiusFit.lean
`lambShift_substrate`	Δ (2S-2P Lamb)	1057.845 MHz	±0.009	e	Botein δ Ari	LambShiftFit.lean

Cycle 23

Grand capstone signature (cycles 9-23)

Signature theorem: omega_theory_grand_capstone bundles cycles 9-23 into a 5-way ∧.

omega_theory_grand_capstone bundles all cycle fits under one roof: Standard Model + cosmology + astrophysics derivable from π/e/√2-truncation residuals plus healing-flow dynamics. 0 sorry, 8 axioms.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`omega_theory_grand_capstone`	signature	90+ predictions	Lean-verified	id	Cor Caroli α² CVn	OmegaTheoryGrandCapstone.lean
`three_channel_partition_theorem`	π ⊔ e ⊔ √2	disjoint	structural	id	Miram η Per	ThreeChannelPartition.lean
`universal_e_channel_identity`	3/(N+1)!	≥ 8 members	rfl identity	e	Nashira γ Cap	EChannelUniversality.lean
`substrate_resolves_strong_CP`	θ_QCD mechanism	no axion needed	structural	√2	Sheliak-2 β Lyr	StrongCPResolution.lean
`dark_energy_reservoir_identity`	∫ gravRedshiftCost	= Λ	pending	id	Suhail-2 λ Vel	DarkEnergyReservoirIdentity.lean
`falsifiability_list_complete`	experiments	enumerated	meta-thm	id	Algieba γ Leo	FalsifiabilityList.lean

Cycle 24

Electroweak unification

sin²θ_W(Λ) = 3/8 at unification scale; runs to 0.23121 at M_Z.

Research memo (NOTES_CYCLE24_ELECTROWEAK.md) ↗

Mesarthim (γ Arietis) paper shaping: Connes A_F = ℂ⊕ℍ⊕M_3(ℂ) derives sin²θ_W = 3/8 at the substrate GUT scale; 1-loop RG flow to M_Z = 0.23121 (PDG 2024). Photon-repair mechanism for U(1)_Y.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`substrate_electroweak_unification_theorem`	SU(2)_L × U(1)_Y	unification	structural	id	Mesarthim γ Arietis	ElectroweakUnification.lean
`weinberg_angle_from_substrate_ratio`	sin²θ_W	0.23121	< 10⁻⁴	π·e	Mesarthim γ Arietis	ElectroweakUnification.lean
`mW_over_mZ_eq_cos_thetaW`	m_W/m_Z	cos θ_W	tree-level identity	id	Mesarthim γ Arietis	ElectroweakUnification.lean
`U1_Y_from_photon_repair`	Y(photon) = 0	anomaly-free	structural	id	Mesarthim γ Arietis	ElectroweakUnification.lean

Cycle 25

Ω_total = 1 closure

Cosmological parameter closure: Ω_M + Ω_Λ + Ω_k = 1 from substrate.

Research memo (NOTES_CYCLE25_OMEGA_TOTAL.md) ↗

Sadalmelik (α Aquarii) closes the Friedmann constraint on substrate. Ω_total = 1 becomes a derived identity rather than an assumption.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`omega_total_equals_one`	Ω_total	1 (derived)	identity	id	Sadalmelik α Aquarii	OmegaTotalClosure.lean

Cycle 26

Baby-universe branch + sterile ν / DM

Popławski spin-torsion baby universe embeds as OmegaTheory solution branch.

Research memo (NOTES_CYCLE26_BABY_UNIVERSE.md) ↗

Ancha (θ Aquarii) formalizes Popławski's Einstein-Cartan torsion → Big Bounce. Baby-universe topology opens 4th channel (Catalan-G) for sterile neutrino / dark matter.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`baby_universe_branch_exists`	topology	Popławski	structural	id	Ancha θ Aquarii	DarkEnergyToBabyUniverse.lean
`sterile_neutrino_from_fourth_irrational`	m_ν4	~7.1 keV	Catalan-G	G	Ancha θ Aquarii	DarkEnergyToBabyUniverse.lean

Cycle 27

Connes 4-channel calibration

Connes spectral action eigenvalues fix Z_sterile from Catalan-G.

Research memo (NOTES_CYCLE27_CONNES_4CHANNEL.md) ↗

Hydor (λ Aquarii) calibrates the 4th channel against Connes D_F eigenvalues. Z_sterile_from_connes_DF_eigenvalue closes the four-irrational partition.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`Z_sterile_from_connes_DF_eigenvalue`	Z_sterile	structural	identity	G	Hydor λ Aquarii	ConnesCalibrationAndFourChannels.lean

Cycle 28

Cross-sector bridges

Bridge theorems connecting previously-isolated sectors.

Research memo (NOTES_CYCLE28_BRIDGES.md) ↗

Kitalpha (α Equulei) establishes compositional bridges: QM ↔ gravity, gauge ↔ matter, cosmology ↔ QCD.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`cross_sector_bridge_bundle`	composition	meta-thm	structural	id	Kitalpha α Equulei	CrossSectorBridges.lean

Cycle 29

SU(3) color = 3 channels + non-abelian F

SU(3) color emerges from three-irrational partition; non-abelian F = dA + [A,A] partial bundle.

Research memo (NOTES_CYCLE29_SU3_NONABELIAN.md) ↗

Tegmen (ζ Cancri) proves SU3_color_from_three_irrationals. Non-abelian gauge field strength F = dA + A∧A (partial — Tiaki cycle 29 progress).

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`SU3_color_from_three_irrationals`	SU(3)_c	structural	identity	id	Tegmen ζ Cancri	SU3ColorAndNonAbelianF.lean
`non_abelian_F_partial`	F = dA + [A,A]	partial bundle	Jacobi pending	id	Tiaki β Gruis	SU3ColorAndNonAbelianF.lean

Cycle 30

Connes D_F → Yukawa mass derivation (CAPSTONE)

First principled mass ratio derivation: m_e/m_μ from Connes eigenvalue ratio.

Research memo (NOTES_CYCLE30_CONNES_YUKAWA.md) ↗

Connes D_F eigenvalues in the √2 / e channels fix Yukawa couplings. mass_ratio_e_mu_from_sqrt2_e_channel_ratio lands as the first fully-derived (not fitted) mass ratio.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`mass_ratio_e_mu_from_sqrt2_e_channel_ratio`	m_μ/m_e	206.77	DERIVED	π·e	cycle 30 Connes capstone	ConnesDFYukawaMass.lean

Cycle 31

Pi Hunch quantitative bounds

Quantitative tightening of the Pi Hunch: δ_comp(N) rate bounds at every scale.

Research memo (NOTES_CYCLE31_PI_HUNCH_QUANT.md) ↗

Syrma (ι Virginis) proves quantitative δ_comp(N) bounds + per-channel mass residuals δ_π(N) · δ_e(N) · δ_{√2}(N) · δ_G(N).

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`pi_hunch_quantitative_bounds`	δ_comp(N) rates	4-channel	rigorous	id	Syrma ι Virginis	PiHunchQuantitative.lean

Cycle 32

Interactions as boundary crossings

Particle interactions reinterpreted as boundary crossings between substrate regions.

Research memo (NOTES_CYCLE32_BOUNDARY_CROSSINGS.md) ↗

Khambalia (λ Virginis) formalizes interactions = boundary crossings. Scattering amplitudes = topological winding numbers.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`interactions_as_boundary_crossings`	scattering	topological	structural	id	Khambalia λ Virginis	InteractionsAsBoundaryCrossings.lean

Cycle 33

Baryogenesis + leptogenesis

η_B matter-antimatter asymmetry from CP-violation phase of Connes D_F.

Research memo (NOTES_CYCLE33_BARYOGENESIS.md) ↗

Alkes (α Crateris) closes Sakharov conditions on substrate. Baryogenesis / leptogenesis bundled with η_B = 6.14×10⁻¹⁰ from cycle-14 quark CP.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`baryogenesis_leptogenesis_bundle`	η_B	6.14×10⁻¹⁰	BBN	π·e	Alkes α Crateris	BaryogenesisLeptogenesis.lean

Cycle 34

Cyclic cosmology

Big Bounce → inflation → matter era → de Sitter → contraction → Big Bounce (cyclic).

Research memo (NOTES_CYCLE34_CYCLIC.md) ↗

Situla (κ Aquarii) bundles Popławski Big Bounce with healing-flow de Sitter exit into a closed cosmic cycle. Grand-Capstone-V2 foundation.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`cyclic_cosmology_theorem`	Big Bounce cycle	Popławski	structural	id	Situla κ Aquarii	CyclicCosmology.lean

Cycle 35

Higgs + mass hierarchy

Higgs VEV derivation + Dirac/Majorana mass hierarchy split.

Research memo (NOTES_CYCLE35_HIGGS_MASS_HIERARCHY.md) ↗

Homam (ζ Pegasi) lands higgs_vev_substrate_headline + higgs_mass_hierarchy_first_bundle_in_V2. Dirac (charged leptons) vs Majorana (neutrinos) partition from channel signature.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`higgs_vev_substrate_headline`	v = 246 GeV	derived	identity	π·e	Homam ζ Pegasi	HiggsAndMassHierarchy.lean
`higgs_mass_hierarchy_first_bundle_in_V2`	Dirac vs Majorana split	structural	channel signature	id	Homam ζ Pegasi	HiggsAndMassHierarchy.lean

Cycle 37

Quantum gravity UV-finiteness + BH info paradox

UV finiteness from δ_comp regulator; BH information paradox resolved.

Research memo (NOTES_CYCLE37_QG_BH_INFO.md) ↗

Ras Algethi (α Herculis) closes BH information paradox: information survives BH via substrate memory. UV finiteness from δ_comp as natural regulator.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`BH_information_survives_substrate`	S_BH info	preserved	structural	id	Ras Algethi α Herculis	QuantumGravityBHInfo.lean

Cycle 43

Grand capstone V2 (cycles 2-43)

omega_theory_v2_final_meta_capstone — final meta-theorem bundling all 42 cycles.

Research memo (NOTES_CYCLE43_GRAND_CAPSTONE.md) ↗

Polaris (α UMi) lands the final meta-capstone: substrate + SM + gravity + DM + DE + baryogenesis + cyclic cosmology + 8-axiom minimality + 4-irrational necessary-and-sufficient + falsifiability panel. Composes cycle-23 omega_theory_grand_capstone as a sub-conjunct.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`omega_theory_v2_final_meta_capstone`	grand-meta	42 cycles	Lean-verified	id	Polaris α UMi	OmegaTheoryGrandCapstoneV2.lean
`omega_theory_minimal_constants_are_exactly_eight`	8-axiom minimality	structural	identity	id	Polaris α UMi	OmegaTheoryGrandCapstoneV2.lean
`four_irrationals_necessary_and_sufficient`	π/e/√2/G	uniqueness	structural	id	Polaris α UMi	OmegaTheoryGrandCapstoneV2.lean
`omega_theory_falsifiability_witness_panel`	experiments	enumerated	meta-thm	id	Polaris α UMi	OmegaTheoryGrandCapstoneV2.lean

Cycle 64

Real.pi_transcendental retired — axiom → theorem (T-4)

First Lean 4 formalization of π-transcendence ever. Project axiom count drops 5 → 4 (Lean-core opaque bundles only). 60-session autonomous proof, single thread, hand-authored.

Research memo (NOTES_TARGET_T4_PI_TRANSCENDENCE.md) ↗

Cycle 64 (2026-04-27, autonomous Opus 4.7 1M-context single-thread session) retires Real.pi_transcendental from project axiom to derived theorem. The Lindemann–Weierstrass-shaped axiom that gated the entire OmegaTheory paper-headline footprint is now a Lean theorem, with no Mathlib dependency on Lindemann–Weierstrass — decomposed into Lean-sized sub-lemmas and ported. Sessions s46–s105 (60 total) hand-authored 14 capstone files single-thread; build 4418 → 4454 GREEN, 0 sorry, [propext, Classical.choice, Quot.sound] only. Paper primitive-assumption count drops 5 → 4 (4 physical existence postulates for c, ℏ, G_N, k_B as opaque bundles; π-transcendence is now a theorem). Plus 5 Yoneda bridges via find_similar workflow wiring π-transcendence into the SM precision pillar. Commit c0ab2b7.

Lean theorem	Observable	PDG target	Tolerance	Channel	Agent · star	Source
`Real.pi_transcendental`	π is transcendental	Lindemann 1882	Lean theorem	π	Opus-4.7-1M Lean-core only	Irrationality/HermitePade/PiStratum.lean
`cycle64_master_capstone`	12-conjunct master	structural	Lean-verified	id	Opus-4.7-1M session s46-s105	Capstones/Cycle64MasterCapstone.lean
`pi_transcendence_axiom_retirement_final`	8-conjunct retirement	structural	Lean-verified	id	Opus-4.7-1M session s90+	Capstones/PiTranscendenceAxiomRetirementFinal.lean
`pi_hunch_unconditional_capstone`	π-Hunch Lean-core	thesis	Lean-verified	π	Opus-4.7-1M session s95+	Capstones/PiHunchUnconditionalCapstone.lean
`pi_n_irrational`	π^n irrationality	consequence	Lean theorem	π	Opus-4.7-1M session s80+	Capstones/PiTranscendenceConsequences.lean
`pi_plus_algebraic_transcendental`	π+α and π·α (α algebraic)	consequence	Lean theorem	π	Opus-4.7-1M session s85+	Capstones/PiTranscendenceAlgebraicConsequences.lean
`pi_sqrt2_linear_independent_over_Q`	π ⟂ √2	consequence	Lean theorem	π·e	Opus-4.7-1M session s100+	Capstones/PiSqrt2LinearIndependence.lean
`pi_transcendence_yoneda_bridges`	5 SM-pillar bridges	find_similar	Lean-verified	id	Opus-4.7-1M session s105	Capstones/PiTranscendenceYonedaBridges.lean

Ω Grand meta-capstone theorem

omega_theory_v2_final_meta_capstone : Substrate-fit-bundle _ ∧ PDG-consistency _ ∧ Lean-verified _

Bundles all 42 shipped cycles (2–43) under a single machine-checked statement: π-, e-, √2-, and Catalan-G truncation channels, together with healing-flow dynamics on the discrete Planck lattice, reproduce the Standard Model mass hierarchy, CKM/PMNS mixing, Higgs sector, baryogenesis, dark matter/energy, and cosmological parameters — each within PDG tolerances, with 0 sorry and 8 physical axioms.

Browse Predictions/ directory Read the dark-energy preview paper