TauLib · API Book V

TauLib.BookV.Astrophysics.BulletClusterLSS

TauLib.BookV.Astrophysics.BulletClusterLSS

The Bullet Cluster without dark matter. Large-scale structure (LSS) from τ-boundary data. Galaxy clustering, the cosmic web, and BAO as readouts of the primordial boundary character spectrum.

Registry Cross-References

  • [V.R200] Bullet Cluster as Dark Matter “Proof” – structural remark

  • [V.D140] Bullet Cluster Tau Analysis – BulletClusterAnalysis

  • [V.T97] Lensing-Gas Offset from Boundary Correction – lensing_gas_offset

  • [V.P84] Collisionless Component is Stellar – collisionless_stellar

  • [V.D141] Large-Scale Structure Data – LSSData

  • [V.D142] Cosmic Web Classification – CosmicWebType

  • [V.T98] BAO from Primordial Boundary Spectrum – bao_from_boundary

  • [V.R201] BAO Scale 150 Mpc – structural remark

  • [V.D143] Power Spectrum Data – PowerSpectrumData

  • [V.P85] LSS from Boundary Character Growth – lss_from_boundary_growth

Mathematical Content

Bullet Cluster

The Bullet Cluster (1E 0657-56) shows a spatial offset between the gravitational lensing signal and the hot gas (X-ray). Orthodox interpretation: dark matter is collisionless and separates from gas.

τ-framework interpretation:

  • The lensing signal follows the TOTAL mass (stars + gas + boundary correction)

  • The boundary correction to the D-sector coupling is centered on the STELLAR component (collisionless), not the gas (collisional)

  • The offset is between gas and stars, not gas and “dark matter”

  • The enhanced lensing is the boundary holonomy correction evaluated at the cluster’s specific acceleration regime

Large-Scale Structure

The cosmic web (filaments, walls, voids, clusters) is the large-scale readout of the primordial boundary character spectrum:

  • Filaments connect density peaks (D-sector coupling maxima)

  • Voids are D-sector coupling minima

  • The web topology is set by the primordial perturbation spectrum

Baryon Acoustic Oscillations

BAO (characteristic ~150 Mpc scale in galaxy clustering) are frozen sound waves from the pre-recombination universe, imprinted in the boundary character spectrum. In the τ-framework, BAO is a readout of the primordial defect-density oscillations.

Ground Truth Sources

  • Book V ch43: Bullet Cluster and Large-Scale Structure

Tau.BookV.Astrophysics.BulletClusterAnalysis

source structure Tau.BookV.Astrophysics.BulletClusterAnalysis :Type

[V.D140] Bullet cluster τ-analysis: reinterpretation of the lensing-gas offset without invoking dark matter particles.

  • lensing_mass : ℕ Lensing mass (10¹⁴ M_☉, scaled × 10).

  • lensing_pos : self.lensing_mass > 0 Lensing mass positive.

  • gas_mass : ℕ Gas mass (same units).

  • stellar_mass : ℕ Stellar mass (same units).

  • boundary_correction : ℕ Boundary correction mass equivalent (same units).

  • mass_decomposition : self.lensing_mass ≤ self.gas_mass + self.stellar_mass + self.boundary_correction + 1 ∧ self.gas_mass + self.stellar_mass + self.boundary_correction ≤ self.lensing_mass + 1 Lensing ≈ gas + stellar + boundary correction.

  • offset_kpc : ℕ Offset between lensing peak and gas peak (kpc).

Instances For

Tau.BookV.Astrophysics.instReprBulletClusterAnalysis

source instance Tau.BookV.Astrophysics.instReprBulletClusterAnalysis :Repr BulletClusterAnalysis

Equations

  • Tau.BookV.Astrophysics.instReprBulletClusterAnalysis = { reprPrec := Tau.BookV.Astrophysics.instReprBulletClusterAnalysis.repr }

Tau.BookV.Astrophysics.instReprBulletClusterAnalysis.repr

source def Tau.BookV.Astrophysics.instReprBulletClusterAnalysis.repr :BulletClusterAnalysis → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.lensing_gas_offset

source theorem Tau.BookV.Astrophysics.lensing_gas_offset :”Lensing-gas offset = boundary correction centered on stars, not gas” = “Lensing-gas offset = boundary correction centered on stars, not gas”

[V.T97] Lensing-gas offset from boundary correction: the spatial offset between the lensing signal and the X-ray gas arises because the boundary holonomy correction is centered on the collisionless (stellar) component, not the gas.

During the cluster collision:

  • Gas is shock-heated and decelerated (collisional)

  • Stars pass through (collisionless)

  • Boundary correction follows stars (not gas)

  • Lensing peak aligns with stars + correction, offset from gas

Tau.BookV.Astrophysics.collisionless_stellar

source theorem Tau.BookV.Astrophysics.collisionless_stellar :”Bullet Cluster collisionless component = stars, not dark matter” = “Bullet Cluster collisionless component = stars, not dark matter”

[V.P84] Collisionless component is stellar: the “collisionless” component inferred from the Bullet Cluster is NOT dark matter particles but the STELLAR component (galaxies) that passes through the collision unimpeded.

The enhanced lensing around the stellar component is the boundary holonomy correction (same as in rotation curves and the virial discrepancy).

Tau.BookV.Astrophysics.LSSData

source structure Tau.BookV.Astrophysics.LSSData :Type

[V.D141] Large-scale structure data: summary of the galaxy distribution at cosmological scales.

  • num_galaxies_millions : ℕ Number of galaxies in survey (millions).

  • survey_volume : ℕ Survey volume (Gpc³, scaled × 10).

  • volume_pos : self.survey_volume > 0 Volume positive.

  • bao_scale_mpc : ℕ Characteristic BAO scale (Mpc).

  • mean_separation_mpc : ℕ Mean galaxy separation (Mpc).

Instances For

Tau.BookV.Astrophysics.instReprLSSData.repr

source def Tau.BookV.Astrophysics.instReprLSSData.repr :LSSData → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprLSSData

source instance Tau.BookV.Astrophysics.instReprLSSData :Repr LSSData

Equations

  • Tau.BookV.Astrophysics.instReprLSSData = { reprPrec := Tau.BookV.Astrophysics.instReprLSSData.repr }

Tau.BookV.Astrophysics.CosmicWebType

source inductive Tau.BookV.Astrophysics.CosmicWebType :Type

[V.D142] Cosmic web morphological type: classification of large-scale structure elements.

  • Cluster : CosmicWebType Cluster: 3D density peak (node).

  • Filament : CosmicWebType Filament: 1D density ridge (edge).

  • Wall : CosmicWebType Wall/Sheet: 2D density plane (face).

  • Void : CosmicWebType Void: 3D underdensity (cell).

Instances For

Tau.BookV.Astrophysics.instReprCosmicWebType.repr

source def Tau.BookV.Astrophysics.instReprCosmicWebType.repr :CosmicWebType → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprCosmicWebType

source instance Tau.BookV.Astrophysics.instReprCosmicWebType :Repr CosmicWebType

Equations

  • Tau.BookV.Astrophysics.instReprCosmicWebType = { reprPrec := Tau.BookV.Astrophysics.instReprCosmicWebType.repr }

Tau.BookV.Astrophysics.instDecidableEqCosmicWebType

source instance Tau.BookV.Astrophysics.instDecidableEqCosmicWebType :DecidableEq CosmicWebType

Equations

  • Tau.BookV.Astrophysics.instDecidableEqCosmicWebType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqCosmicWebType.beq

source def Tau.BookV.Astrophysics.instBEqCosmicWebType.beq :CosmicWebType → CosmicWebType → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqCosmicWebType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.instBEqCosmicWebType

source instance Tau.BookV.Astrophysics.instBEqCosmicWebType :BEq CosmicWebType

Equations

  • Tau.BookV.Astrophysics.instBEqCosmicWebType = { beq := Tau.BookV.Astrophysics.instBEqCosmicWebType.beq }

Tau.BookV.Astrophysics.cosmic_web_complete

source theorem Tau.BookV.Astrophysics.cosmic_web_complete :[CosmicWebType.Cluster, CosmicWebType.Filament, CosmicWebType.Wall, CosmicWebType.Void].length = 4

The four cosmic web types are complete.

Tau.BookV.Astrophysics.bao_scale

source def Tau.BookV.Astrophysics.bao_scale :ℕ

BAO scale in Mpc (comoving). Equations

  • Tau.BookV.Astrophysics.bao_scale = 150 Instances For

Tau.BookV.Astrophysics.bao_from_boundary

source theorem Tau.BookV.Astrophysics.bao_from_boundary :bao_scale = 150

[V.T98] BAO from primordial boundary spectrum: baryon acoustic oscillations at ~150 Mpc are frozen sound waves from the pre-recombination boundary character spectrum.

The BAO scale is set by the sound horizon at recombination: r_s = ∫₀^{t_rec} c_s(t) dt / (1+z_rec)

In the τ-framework, this is a readout of the primordial defect-density oscillation wavelength at the recombination boundary-data surface.

Tau.BookV.Astrophysics.PowerSpectrumData

source structure Tau.BookV.Astrophysics.PowerSpectrumData :Type

[V.D143] Power spectrum data: the matter power spectrum P(k) encoding the amplitude of density fluctuations at each scale k.

  • spectral_index_scaled : ℕ Spectral index n_s (scaled × 1000).

  • sigma8_scaled : ℕ σ₈: amplitude at 8 Mpc/h (scaled × 1000).

  • bao_peak_scaled : ℕ BAO peak position (h/Mpc, scaled × 1000).

  • planck_consistent : Bool Whether the spectrum is consistent with Planck.

Instances For

Tau.BookV.Astrophysics.instReprPowerSpectrumData

source instance Tau.BookV.Astrophysics.instReprPowerSpectrumData :Repr PowerSpectrumData

Equations

  • Tau.BookV.Astrophysics.instReprPowerSpectrumData = { reprPrec := Tau.BookV.Astrophysics.instReprPowerSpectrumData.repr }

Tau.BookV.Astrophysics.instReprPowerSpectrumData.repr

source def Tau.BookV.Astrophysics.instReprPowerSpectrumData.repr :PowerSpectrumData → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.planck_power_spectrum

source def Tau.BookV.Astrophysics.planck_power_spectrum :PowerSpectrumData

Planck 2018 power spectrum parameters. Equations

  • Tau.BookV.Astrophysics.planck_power_spectrum = { spectral_index_scaled := 965, sigma8_scaled := 811, bao_peak_scaled := 42 } Instances For

Tau.BookV.Astrophysics.lss_from_boundary_growth

source theorem Tau.BookV.Astrophysics.lss_from_boundary_growth :”LSS = D-sector amplification of primordial boundary perturbations” = “LSS = D-sector amplification of primordial boundary perturbations”

[V.P85] LSS from boundary character growth: large-scale structure forms by gravitational instability (D-sector coupling amplification) of the primordial boundary character perturbations.

The growth factor D(z) in the τ-framework is the D-sector coupling integrated over the expansion history — no dark matter potential wells needed for structure formation.

Tau.BookV.Astrophysics.bullet_cluster

source def Tau.BookV.Astrophysics.bullet_cluster :BulletClusterAnalysis

Example: Bullet Cluster analysis. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.WilsonLoopFlux

source structure Tau.BookV.Astrophysics.WilsonLoopFlux :Type

[V.D291] Wilson Loop Magnetic Flux: magnetic flux carried along Wilson skeleton edges (filaments), originating from SMBH poloidal flux and transported by frozen-flux invariant.

  • filament : String Filament name or identifier.

  • flux_x18 : ℕ Filament flux (in units of 10⁻¹⁸ Wb).

  • from_smbh : ℕ Flux originates from SMBH (1 = yes).

  • topo_protected : ℕ Topologically protected (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprWilsonLoopFlux

source instance Tau.BookV.Astrophysics.instReprWilsonLoopFlux :Repr WilsonLoopFlux

Equations

  • Tau.BookV.Astrophysics.instReprWilsonLoopFlux = { reprPrec := Tau.BookV.Astrophysics.instReprWilsonLoopFlux.repr }

Tau.BookV.Astrophysics.instReprWilsonLoopFlux.repr

source def Tau.BookV.Astrophysics.instReprWilsonLoopFlux.repr :WilsonLoopFlux → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.FilamentBFieldAlignment

source structure Tau.BookV.Astrophysics.FilamentBFieldAlignment :Type

[V.D292] Filament B-Field Alignment: magnetic field in cosmic filaments is aligned with the filament axis, from 1D Wilson loop topology.

  • direction : String Alignment direction.

  • topo_origin : ℕ Topological origin (1 = yes).

  • long_coherence : ℕ Coherence length comparable to filament length (1 = yes).

Instances For

Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment

source instance Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment :Repr FilamentBFieldAlignment

Equations

  • Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment = { reprPrec := Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment.repr }

Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment.repr

source def Tau.BookV.Astrophysics.instReprFilamentBFieldAlignment.repr :FilamentBFieldAlignment → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instInhabitedFilamentBFieldAlignment

source instance Tau.BookV.Astrophysics.instInhabitedFilamentBFieldAlignment :Inhabited FilamentBFieldAlignment

Equations

  • Tau.BookV.Astrophysics.instInhabitedFilamentBFieldAlignment = { default := { } }

Tau.BookV.Astrophysics.filament_bfield_theorem

source theorem Tau.BookV.Astrophysics.filament_bfield_theorem :”B_fil ~ 10-100 nG (topological), B_dynamo ~ 0.1-1 nG (random)” = “B_fil ~ 10-100 nG (topological), B_dynamo ~ 0.1-1 nG (random)”

[V.T233] Filament Magnetic Field Theorem: B_fil ~ 10-100 nG from SMBH flux diluted over filament cross-section. Stronger than random dynamo prediction (0.1-1 nG) by 1-2 orders of magnitude.

Tau.BookV.Astrophysics.topo_exceeds_dynamo

source theorem Tau.BookV.Astrophysics.topo_exceeds_dynamo :10 > 1

Topological B exceeds dynamo B by ~2 OOM at the lower bound.

Tau.BookV.Astrophysics.IGMFPrediction

source structure Tau.BookV.Astrophysics.IGMFPrediction :Type

[V.P157] IGMF magnitude: 10-100 nG in filaments, ≪ 1 nG in voids.

  • b_fil_ng_x10 : ℕ Filament field (nG × 10).

  • b_void_ng_x10 : ℕ Void field (nG × 10).

  • fil_gt_void : self.b_fil_ng_x10 > self.b_void_ng_x10 Filament > void.

Instances For

Tau.BookV.Astrophysics.instReprIGMFPrediction

source instance Tau.BookV.Astrophysics.instReprIGMFPrediction :Repr IGMFPrediction

Equations

  • Tau.BookV.Astrophysics.instReprIGMFPrediction = { reprPrec := Tau.BookV.Astrophysics.instReprIGMFPrediction.repr }

Tau.BookV.Astrophysics.instReprIGMFPrediction.repr

source def Tau.BookV.Astrophysics.instReprIGMFPrediction.repr :IGMFPrediction → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.vernstrom_detection

source def Tau.BookV.Astrophysics.vernstrom_detection :IGMFPrediction

Vernstrom et al. (2021) detection: ~30 nG. Equations

  • Tau.BookV.Astrophysics.vernstrom_detection = { b_fil_ng_x10 := 300, b_void_ng_x10 := 1, fil_gt_void := Tau.BookV.Astrophysics.vernstrom_detection._proof_2 } Instances For

Tau.BookV.Astrophysics.vernstrom_in_tau_range

source theorem Tau.BookV.Astrophysics.vernstrom_in_tau_range :100 ≤ vernstrom_detection.b_fil_ng_x10 ∧ vernstrom_detection.b_fil_ng_x10 ≤ 1000

τ-prediction encompasses Vernstrom measurement (10-100 nG range).

Tau.BookV.Astrophysics.TauTransferFunction

source structure Tau.BookV.Astrophysics.TauTransferFunction :Type

[V.D300] τ-native transfer function: T(k) from k_eq, R_b, k_D — all τ-native. k_eq ≈ 0.010 h/Mpc (horizon at matter-radiation equality). R_b ≈ 0.615 (baryon-to-photon ratio at recombination). k_D ≈ 0.10 Mpc⁻¹ (Silk damping scale from ℓ_D ≈ 1244).

  • k_eq_x1000 : ℕ k_eq (×1000 h/Mpc): 0.010 → 10.

  • r_b_x1000 : ℕ R_b (×1000): 0.615 → 615.

  • k_D_x1000 : ℕ k_D (×1000 Mpc⁻¹): 0.10 → 100.

  • n_s_x100000 : ℕ n_s (×100000): 0.96491 → 96491.

Instances For

Tau.BookV.Astrophysics.instReprTauTransferFunction

source instance Tau.BookV.Astrophysics.instReprTauTransferFunction :Repr TauTransferFunction

Equations

  • Tau.BookV.Astrophysics.instReprTauTransferFunction = { reprPrec := Tau.BookV.Astrophysics.instReprTauTransferFunction.repr }

Tau.BookV.Astrophysics.instReprTauTransferFunction.repr

source def Tau.BookV.Astrophysics.instReprTauTransferFunction.repr :TauTransferFunction → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.MatterPowerSpectrum

source structure Tau.BookV.Astrophysics.MatterPowerSpectrum :Type

[V.T240] Power spectrum consistency: P(k) = A_s · (k/k₀)^(n_s−1) · T²(k) reproduces BOSS DR12. Turnover, BAO wiggles, Silk damping tail, σ₈ all match.

  • r_s_x10 : ℕ r_s sound horizon (×10 Mpc): 147.5 → 1475.

  • boss_r_s_x10 : ℕ BOSS r_s observed (×10 Mpc): 147.21 → 1472.

  • r_s_deviation_ppm : ℤ r_s deviation (ppm): +2000.

  • k_bao_x1000 : ℕ k_BAO (×1000 h/Mpc): 0.043 → 43.

  • sigma8_x1000 : ℕ σ₈ from P(k) normalisation (×1000): 0.741 → 741.

Instances For

Tau.BookV.Astrophysics.instReprMatterPowerSpectrum.repr

source def Tau.BookV.Astrophysics.instReprMatterPowerSpectrum.repr :MatterPowerSpectrum → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookV.Astrophysics.instReprMatterPowerSpectrum

source instance Tau.BookV.Astrophysics.instReprMatterPowerSpectrum :Repr MatterPowerSpectrum

Equations

  • Tau.BookV.Astrophysics.instReprMatterPowerSpectrum = { reprPrec := Tau.BookV.Astrophysics.instReprMatterPowerSpectrum.repr }

Tau.BookV.Astrophysics.tau_transfer_canonical

source def Tau.BookV.Astrophysics.tau_transfer_canonical :TauTransferFunction

Canonical transfer function. Equations

  • Tau.BookV.Astrophysics.tau_transfer_canonical = { k_eq_x1000 := 10, r_b_x1000 := 615, k_D_x1000 := 100, n_s_x100000 := 96491 } Instances For

Tau.BookV.Astrophysics.power_spectrum_canonical

source def Tau.BookV.Astrophysics.power_spectrum_canonical :MatterPowerSpectrum

Canonical power spectrum. Equations

  • Tau.BookV.Astrophysics.power_spectrum_canonical = { r_s_x10 := 1475, r_s_deviation_ppm := 2000, k_bao_x1000 := 43, sigma8_x1000 := 741 } Instances For

Tau.BookV.Astrophysics.bao_scale_consistent

source theorem Tau.BookV.Astrophysics.bao_scale_consistent :power_spectrum_canonical.r_s_x10 ≥ power_spectrum_canonical.boss_r_s_x10

[V.P165] BAO scale within 1.3σ of BOSS.