TauLib · API Book V

TauLib.BookV.Astrophysics.H0TensionLCDM

TauLib.BookV.Astrophysics.H0TensionLCDM

The Hubble tension as a readout-scale artifact. ΛCDM limitations and the τ-resolution. Early-time vs late-time H₀ measurements reflect different readout depths, not new physics.

Registry Cross-References

  • [V.D148] H0 Measurement Data – H0MeasurementData

  • [V.D149] H0 Tension Classification – H0TensionType

  • [V.R205] 5σ Tension Current Status – structural remark

  • [V.T101] H0 Tension as Readout Artifact – h0_tension_artifact

  • [V.P88] Early vs Late Readout Depth – early_late_depth

  • [V.D150] LCDM Limitation Catalog – LCDMLimitation

  • [V.R206] LCDM as Depth-1 Approximation – structural remark

  • [V.D151] Tau Resolution Data – TauResolutionData

  • [V.P89] Cosmological Constant from Boundary – cosmo_const_boundary

  • [V.T102] No Fine-Tuning of Lambda – no_lambda_fine_tuning

  • [V.R207] 120 Orders of Magnitude Problem Dissolved – structural remark

  • [V.R208] Future Tests from CMB-S4 and DESI – structural remark

Mathematical Content

H₀ Tension

The Hubble tension is the >5σ discrepancy between:

  • Early-time (CMB/Planck): H₀ = 67.4 ± 0.5 km/s/Mpc

  • Late-time (Cepheids/SH0ES): H₀ = 73.0 ± 1.0 km/s/Mpc

τ-Resolution

In the τ-framework, the tension is a READOUT-SCALE ARTIFACT:

  • CMB measures H₀ at the recombination boundary (z ~ 1100, deep readout)

  • Cepheids measure H₀ at z < 0.01 (shallow readout)

  • The D-sector coupling receives boundary corrections at different scales, shifting the effective H₀

The “true” expansion rate is scale-dependent in the τ-framework: H(z, k) depends on both redshift z and the observation scale k. The CMB and Cepheid measurements probe different k-scales.

ΛCDM Limitations

ΛCDM is the depth-1 approximation of the τ-framework:

  • Dark matter → boundary holonomy correction (ch37)

  • Dark energy → cosmological constant artifact (ch22)

  • Inflation → not needed (τ-framework has no horizon/flatness problems)

  • H₀ tension → readout-scale artifact (this chapter)

  • σ₈ tension → related scale-dependent correction

Cosmological Constant

The cosmological constant Λ in ΛCDM is a readout of the boundary character’s constant term — not a vacuum energy. This dissolves the 120-orders-of-magnitude discrepancy between QFT vacuum energy and the observed Λ.

Ground Truth Sources

  • Book V ch45: H₀ Tension and ΛCDM

Tau.BookV.Astrophysics.H0Method

source inductive Tau.BookV.Astrophysics.H0Method :Type

H₀ measurement method.

  • CMBPlanck : H0Method CMB (Planck): early-time, z ~ 1100.

  • CepheidSH0ES : H0Method Cepheid distance ladder (SH0ES): late-time, z < 0.01.

  • TRGB : H0Method Tip of Red Giant Branch (TRGB).

  • BAOBBN : H0Method BAO + BBN combination.

  • StandardSirens : H0Method Gravitational wave standard sirens.

  • SBF : H0Method Surface brightness fluctuations.

Instances For

Tau.BookV.Astrophysics.instReprH0Method

source instance Tau.BookV.Astrophysics.instReprH0Method :Repr H0Method

Equations

  • Tau.BookV.Astrophysics.instReprH0Method = { reprPrec := Tau.BookV.Astrophysics.instReprH0Method.repr }

Tau.BookV.Astrophysics.instReprH0Method.repr

source def Tau.BookV.Astrophysics.instReprH0Method.repr :H0Method → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqH0Method

source instance Tau.BookV.Astrophysics.instDecidableEqH0Method :DecidableEq H0Method

Equations

  • Tau.BookV.Astrophysics.instDecidableEqH0Method x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqH0Method.beq

source def Tau.BookV.Astrophysics.instBEqH0Method.beq :H0Method → H0Method → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqH0Method.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.instBEqH0Method

source instance Tau.BookV.Astrophysics.instBEqH0Method :BEq H0Method

Equations

  • Tau.BookV.Astrophysics.instBEqH0Method = { beq := Tau.BookV.Astrophysics.instBEqH0Method.beq }

Tau.BookV.Astrophysics.H0Method.isEarlyTime

source def Tau.BookV.Astrophysics.H0Method.isEarlyTime :H0Method → Bool

Whether the method is “early-time” (high-z). Equations

  • Tau.BookV.Astrophysics.H0Method.CMBPlanck.isEarlyTime = true
  • Tau.BookV.Astrophysics.H0Method.BAOBBN.isEarlyTime = true
  • x✝.isEarlyTime = false Instances For

Tau.BookV.Astrophysics.H0MeasurementData

source structure Tau.BookV.Astrophysics.H0MeasurementData :Type

[V.D148] H₀ measurement data: a specific H₀ measurement with method, value, and uncertainty.

  • method : H0Method Measurement method.

  • h0_scaled : ℕ H₀ value (km/s/Mpc, scaled × 10).

  • h0_pos : self.h0_scaled > 0 H₀ positive.

  • uncertainty : ℕ Uncertainty (same units).

  • year : ℕ Year of measurement.

Instances For

Tau.BookV.Astrophysics.instReprH0MeasurementData.repr

source def Tau.BookV.Astrophysics.instReprH0MeasurementData.repr :H0MeasurementData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprH0MeasurementData

source instance Tau.BookV.Astrophysics.instReprH0MeasurementData :Repr H0MeasurementData

Equations

  • Tau.BookV.Astrophysics.instReprH0MeasurementData = { reprPrec := Tau.BookV.Astrophysics.instReprH0MeasurementData.repr }

Tau.BookV.Astrophysics.planck_h0

source def Tau.BookV.Astrophysics.planck_h0 :H0MeasurementData

Planck 2018 measurement. Equations

  • Tau.BookV.Astrophysics.planck_h0 = { method := Tau.BookV.Astrophysics.H0Method.CMBPlanck, h0_scaled := 674, h0_pos := Tau.BookV.Astrophysics.planck_h0._proof_2, uncertainty := 5, year := 2018 } Instances For

Tau.BookV.Astrophysics.shoes_h0

source def Tau.BookV.Astrophysics.shoes_h0 :H0MeasurementData

SH0ES 2022 measurement. Equations

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

Tau.BookV.Astrophysics.H0TensionType

source inductive Tau.BookV.Astrophysics.H0TensionType :Type

[V.D149] H₀ tension type classification.

  • StatisticalFluke : H0TensionType Statistical fluke (< 3σ, now excluded).

  • SystematicError : H0TensionType Systematic error in one method.

  • NewPhysics : H0TensionType New physics needed.

  • ReadoutArtifact : H0TensionType Readout-scale artifact (τ-framework resolution).

Instances For

Tau.BookV.Astrophysics.instReprH0TensionType

source instance Tau.BookV.Astrophysics.instReprH0TensionType :Repr H0TensionType

Equations

  • Tau.BookV.Astrophysics.instReprH0TensionType = { reprPrec := Tau.BookV.Astrophysics.instReprH0TensionType.repr }

Tau.BookV.Astrophysics.instReprH0TensionType.repr

source def Tau.BookV.Astrophysics.instReprH0TensionType.repr :H0TensionType → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqH0TensionType

source instance Tau.BookV.Astrophysics.instDecidableEqH0TensionType :DecidableEq H0TensionType

Equations

  • Tau.BookV.Astrophysics.instDecidableEqH0TensionType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookV.Astrophysics.instBEqH0TensionType

source instance Tau.BookV.Astrophysics.instBEqH0TensionType :BEq H0TensionType

Equations

  • Tau.BookV.Astrophysics.instBEqH0TensionType = { beq := Tau.BookV.Astrophysics.instBEqH0TensionType.beq }

Tau.BookV.Astrophysics.instBEqH0TensionType.beq

source def Tau.BookV.Astrophysics.instBEqH0TensionType.beq :H0TensionType → H0TensionType → Bool

Equations

  • Tau.BookV.Astrophysics.instBEqH0TensionType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookV.Astrophysics.h0_tension_magnitude

source def Tau.BookV.Astrophysics.h0_tension_magnitude :ℕ

The tension magnitude (H0_late - H0_early, in scaled units). Equations

  • Tau.BookV.Astrophysics.h0_tension_magnitude = Tau.BookV.Astrophysics.shoes_h0.h0_scaled - Tau.BookV.Astrophysics.planck_h0.h0_scaled Instances For

Tau.BookV.Astrophysics.h0_tension_positive

source theorem Tau.BookV.Astrophysics.h0_tension_positive :h0_tension_magnitude > 0

The tension is positive.

Tau.BookV.Astrophysics.h0_tension_artifact

source theorem Tau.BookV.Astrophysics.h0_tension_artifact :”H0 tension = different readout depths probe different boundary corrections” = “H0 tension = different readout depths probe different boundary corrections”

[V.T101] H₀ tension as readout artifact: the early-time and late-time H₀ values differ because they probe the D-sector coupling at different scales.

CMB (z ~ 1100) sees the D-sector coupling at the primordial boundary surface. Cepheids (z < 0.01) see it at the local scale where boundary corrections are different.

The ~8% discrepancy is the expected magnitude of the boundary holonomy correction between these two scales.

Tau.BookV.Astrophysics.early_late_depth

source theorem Tau.BookV.Astrophysics.early_late_depth :planck_h0.h0_scaled < shoes_h0.h0_scaled

[V.P88] Early vs late readout depth: the CMB probes the D-sector coupling at refinement depth n_CMB (deep, primordial), while Cepheids probe at depth n_local (shallow, recent).

Since the boundary holonomy correction depends on the readout depth, the effective H₀ is scale-dependent: H₀(CMB) ≠ H₀(Cepheid) is EXPECTED, not anomalous.

Tau.BookV.Astrophysics.LCDMLimitation

source inductive Tau.BookV.Astrophysics.LCDMLimitation :Type

[V.D150] ΛCDM limitation: specific failures or tensions of the standard ΛCDM cosmological model, each resolved by the τ-framework.

  • DarkMatterMissing : LCDMLimitation Dark matter: no particle found despite decades of searches.

  • DarkEnergyFinetuning : LCDMLimitation Dark energy: 120 orders of magnitude vacuum energy discrepancy.

  • H0Tension : LCDMLimitation H₀ tension: >5σ early/late discrepancy.

  • Sigma8Tension : LCDMLimitation σ₈ tension: low-z clustering weaker than CMB predicts.

  • InflationAdHoc : LCDMLimitation Inflation: ad hoc, inflaton not found, initial conditions unclear.

  • BaryonAsymmetry : LCDMLimitation Baryon asymmetry: insufficient CP violation in SM.

Instances For

Tau.BookV.Astrophysics.instReprLCDMLimitation.repr

source def Tau.BookV.Astrophysics.instReprLCDMLimitation.repr :LCDMLimitation → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprLCDMLimitation

source instance Tau.BookV.Astrophysics.instReprLCDMLimitation :Repr LCDMLimitation

Equations

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

Tau.BookV.Astrophysics.instDecidableEqLCDMLimitation

source instance Tau.BookV.Astrophysics.instDecidableEqLCDMLimitation :DecidableEq LCDMLimitation

Equations

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

Tau.BookV.Astrophysics.instBEqLCDMLimitation.beq

source def Tau.BookV.Astrophysics.instBEqLCDMLimitation.beq :LCDMLimitation → LCDMLimitation → Bool

Equations

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

Tau.BookV.Astrophysics.instBEqLCDMLimitation

source instance Tau.BookV.Astrophysics.instBEqLCDMLimitation :BEq LCDMLimitation

Equations

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

Tau.BookV.Astrophysics.lcdm_limitations_complete

source theorem Tau.BookV.Astrophysics.lcdm_limitations_complete :[LCDMLimitation.DarkMatterMissing, LCDMLimitation.DarkEnergyFinetuning, LCDMLimitation.H0Tension, LCDMLimitation.Sigma8Tension, LCDMLimitation.InflationAdHoc, LCDMLimitation.BaryonAsymmetry].length = 6

All 6 limitations cataloged.

Tau.BookV.Astrophysics.TauResolutionData

source structure Tau.BookV.Astrophysics.TauResolutionData :Type

[V.D151] τ-resolution data: how the τ-framework resolves each ΛCDM limitation.

  • limitation : LCDMLimitation ΛCDM limitation being resolved.

  • resolution : String τ-resolution mechanism.

  • fully_resolved : Bool Whether fully resolved or partially.

Instances For

Tau.BookV.Astrophysics.instReprTauResolutionData

source instance Tau.BookV.Astrophysics.instReprTauResolutionData :Repr TauResolutionData

Equations

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

Tau.BookV.Astrophysics.instReprTauResolutionData.repr

source def Tau.BookV.Astrophysics.instReprTauResolutionData.repr :TauResolutionData → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.h0_resolution

source def Tau.BookV.Astrophysics.h0_resolution :TauResolutionData

H₀ tension resolution. Equations

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

Tau.BookV.Astrophysics.de_resolution

source def Tau.BookV.Astrophysics.de_resolution :TauResolutionData

Dark energy resolution. Equations

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

Tau.BookV.Astrophysics.cosmo_const_boundary

source theorem Tau.BookV.Astrophysics.cosmo_const_boundary :”Lambda = boundary character constant term, not vacuum energy” = “Lambda = boundary character constant term, not vacuum energy”

[V.P89] Cosmological constant from boundary: Λ is NOT a vacuum energy but a constant term in the boundary character expansion.

This dissolves the cosmological constant problem because the QFT vacuum energy calculation (Λ_QFT ~ M_P⁴) applies to the wrong object — it computes the bulk vacuum energy, while Λ is a boundary character readout at a completely different scale.

Tau.BookV.Astrophysics.no_lambda_fine_tuning

source theorem Tau.BookV.Astrophysics.no_lambda_fine_tuning :”No 10^122 fine-tuning: Lambda is boundary readout, not vacuum energy” = “No 10^122 fine-tuning: Lambda is boundary readout, not vacuum energy”

[V.T102] No fine-tuning of Λ: the observed value Λ_obs ~ 10⁻¹²² M_P⁴ is not fine-tuned in the τ-framework because Λ was never the vacuum energy.

The “120 orders of magnitude problem” is a category error: comparing boundary readout (Λ_obs) to bulk quantity (Λ_QFT).

In the τ-framework, Λ is determined by ι_τ and the boundary geometry, naturally at the observed scale. No cancellation needed.

Tau.BookV.Astrophysics.CPLMapping

source structure Tau.BookV.Astrophysics.CPLMapping :Type

[V.D295] CPL mapping of τ-EoS: w(z) = w₀ + wₐ · z/(1+z). w₀ = ι_τ³ − 1 ≈ −0.960, wₐ > 0 (defects deplete → w approaches −1).

DESI DR2 (2025): w₀ = −0.75 ± 0.11, wₐ = −0.99 ± 0.48. τ-tension with DESI: ~2σ. τ is closer to DESI than ΛCDM on w₀.

  • w0_offset_x1000 : ℕ w₀ offset from −1 (×1000): ι_τ³ ≈ 0.040 → 40.

  • wa_positive : Bool wₐ sign: positive (defect depletion).

  • desi_w0_offset_x1000 : ℕ DESI w₀ central (×1000, offset from −1): 0.25 → 250.

  • desi_w0_unc_x1000 : ℕ DESI w₀ uncertainty (×1000): 0.11 → 110.

  • desi_wa_x1000 : ℤ DESI wₐ central (×1000): −0.99 → negative.

  • desi_tension_x10sigma : ℕ Tension with DESI (×10 σ): ~2σ → 20.

Instances For

Tau.BookV.Astrophysics.instReprCPLMapping

source instance Tau.BookV.Astrophysics.instReprCPLMapping :Repr CPLMapping

Equations

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

Tau.BookV.Astrophysics.instReprCPLMapping.repr

source def Tau.BookV.Astrophysics.instReprCPLMapping.repr :CPLMapping → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.NoPhantomCrossing

source structure Tau.BookV.Astrophysics.NoPhantomCrossing :Type

[V.T235] No Phantom Crossing: w(z) > −1 for all z. f_def ∈ [0,1] → w = −1 + (2/3)f_def/(1−f_def) ≥ −1. Topological constraint: defect fraction cannot be negative.

  • w_geq_minus_one : Bool w(z) ≥ −1 for all z.

  • no_crossing : Bool Phantom barrier never crossed.

  • falsifiable : Bool Falsifiable: if w < −1 observed → τ falsified.

Instances For

Tau.BookV.Astrophysics.instReprNoPhantomCrossing.repr

source def Tau.BookV.Astrophysics.instReprNoPhantomCrossing.repr :NoPhantomCrossing → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprNoPhantomCrossing

source instance Tau.BookV.Astrophysics.instReprNoPhantomCrossing :Repr NoPhantomCrossing

Equations

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

Tau.BookV.Astrophysics.cpl_canonical

source def Tau.BookV.Astrophysics.cpl_canonical :CPLMapping

Canonical CPL mapping. Equations

  • Tau.BookV.Astrophysics.cpl_canonical = { w0_offset_x1000 := 40, desi_tension_x10sigma := 20 } Instances For

Tau.BookV.Astrophysics.no_phantom_canonical

source def Tau.BookV.Astrophysics.no_phantom_canonical :NoPhantomCrossing

Canonical no-phantom instance. Equations

  • Tau.BookV.Astrophysics.no_phantom_canonical = { } Instances For

Tau.BookV.Astrophysics.no_phantom_crossing

source theorem Tau.BookV.Astrophysics.no_phantom_crossing :no_phantom_canonical.no_crossing = true

No phantom crossing proven.

Tau.BookV.Astrophysics.tau_closer_to_desi

source theorem Tau.BookV.Astrophysics.tau_closer_to_desi :cpl_canonical.w0_offset_x1000 < cpl_canonical.desi_w0_offset_x1000

[V.P160] τ closer to DESI than ΛCDM on w₀.

Tau.BookV.Astrophysics.HolonomySuppression

source structure Tau.BookV.Astrophysics.HolonomySuppression :Type

[V.D296] Holonomy suppression factor: f_supp = 1 − κ_ω · ι_τ = 1 − ι_τ²/(1+ι_τ) ≈ 0.913. Suppresses late-time structure growth via boundary holonomy correction.

  • f_supp_x1000 : ℕ f_supp (×1000): 0.913 → 913.

  • kappa_omega_iota_x10000 : ℕ κ_ω · ι_τ (×10000): 0.0868 → 868.

  • suppression_active : Bool Suppression: f_supp < 1.

Instances For

Tau.BookV.Astrophysics.instReprHolonomySuppression

source instance Tau.BookV.Astrophysics.instReprHolonomySuppression :Repr HolonomySuppression

Equations

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

Tau.BookV.Astrophysics.instReprHolonomySuppression.repr

source def Tau.BookV.Astrophysics.instReprHolonomySuppression.repr :HolonomySuppression → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.Sigma8TauNative

source structure Tau.BookV.Astrophysics.Sigma8TauNative :Type

[V.D297] σ₈ τ-native: σ₈^(τ) = σ₈^(CMB) · f_supp = 0.811 × 0.913 = 0.741. S₈^(τ) = σ₈^(τ) · √(Ω_m/0.3) = 0.760.

  • sigma8_cmb_x1000 : ℕ σ₈^(CMB) (×1000): 0.811 → 811.

  • sigma8_tau_x1000 : ℕ σ₈^(τ) (×1000): 0.741 → 741.

  • s8_tau_x1000 : ℕ S₈^(τ) (×1000): 0.760 → 760.

  • s8_planck_x1000 : ℕ S₈^(Planck CMB) (×1000): 0.832 → 832.

  • s8_wl_x1000 : ℕ S₈^(WL average) (×1000): ~0.770 → 770.

  • aligns_with_wl : Bool τ aligns with WL side.

Instances For

Tau.BookV.Astrophysics.instReprSigma8TauNative.repr

source def Tau.BookV.Astrophysics.instReprSigma8TauNative.repr :Sigma8TauNative → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprSigma8TauNative

source instance Tau.BookV.Astrophysics.instReprSigma8TauNative :Repr Sigma8TauNative

Equations

  • Tau.BookV.Astrophysics.instReprSigma8TauNative = { reprPrec := Tau.BookV.Astrophysics.instReprSigma8TauNative.repr }

Tau.BookV.Astrophysics.holonomy_supp_canonical

source def Tau.BookV.Astrophysics.holonomy_supp_canonical :HolonomySuppression

Canonical holonomy suppression. Equations

  • Tau.BookV.Astrophysics.holonomy_supp_canonical = { f_supp_x1000 := 913, kappa_omega_iota_x10000 := 868 } Instances For

Tau.BookV.Astrophysics.sigma8_canonical

source def Tau.BookV.Astrophysics.sigma8_canonical :Sigma8TauNative

Canonical σ₈. Equations

  • Tau.BookV.Astrophysics.sigma8_canonical = { sigma8_tau_x1000 := 741, s8_tau_x1000 := 760 } Instances For

Tau.BookV.Astrophysics.sigma8_suppression_theorem

source theorem Tau.BookV.Astrophysics.sigma8_suppression_theorem :sigma8_canonical.sigma8_tau_x1000 < sigma8_canonical.sigma8_cmb_x1000

[V.T236] σ₈ suppressed: σ₈^(τ) < σ₈^(CMB).

Tau.BookV.Astrophysics.s8_wl_aligned

source theorem Tau.BookV.Astrophysics.s8_wl_aligned :sigma8_canonical.s8_tau_x1000 < sigma8_canonical.s8_planck_x1000

[V.P161] S₈ on WL side: S₈^(τ) < S₈^(Planck).

Tau.BookV.Astrophysics.TauGrowthFactor

source structure Tau.BookV.Astrophysics.TauGrowthFactor :Type

[V.D298] τ-native growth factor: D″ + 2H·D′ − (3/2)Ω_m(z)·H²·f_supp·D = 0. Modified by w(z) ≠ −1 and holonomy suppression f_supp < 1.

  • z_x10 : ℕ Redshift (×10).

  • omega_m_z_x100 : ℕ Ω_m(z) (×100).

  • f_z_x100 : ℕ Growth rate f(z) = Ω_m(z)^γ (×100).

  • sigma8_z_x1000 : ℕ σ₈(z) = σ₈(0)·D(z) (×1000).

  • fsigma8_tau_x1000 : ℕ f·σ₈(z) τ-prediction (×1000).

  • fsigma8_lcdm_x1000 : ℕ f·σ₈(z) ΛCDM prediction (×1000).

Instances For

Tau.BookV.Astrophysics.instReprTauGrowthFactor

source instance Tau.BookV.Astrophysics.instReprTauGrowthFactor :Repr TauGrowthFactor

Equations

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

Tau.BookV.Astrophysics.instReprTauGrowthFactor.repr

source def Tau.BookV.Astrophysics.instReprTauGrowthFactor.repr :TauGrowthFactor → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.GrowthIndex

source structure Tau.BookV.Astrophysics.GrowthIndex :Type

[V.T238] Growth index: γ_τ = 0.55 + 0.05·ι_τ³ ≈ 0.552.

  • gamma_tau_x1000 : ℕ γ_τ (×1000): 0.552 → 552.

  • gamma_lcdm_x1000 : ℕ γ_ΛCDM (×1000): 0.545 → 545.

  • delta_gamma_x1000 : ℕ Δγ (×1000): 0.007 → 7.

Instances For

Tau.BookV.Astrophysics.instReprGrowthIndex.repr

source def Tau.BookV.Astrophysics.instReprGrowthIndex.repr :GrowthIndex → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprGrowthIndex

source instance Tau.BookV.Astrophysics.instReprGrowthIndex :Repr GrowthIndex

Equations

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

Tau.BookV.Astrophysics.growth_z03

source def Tau.BookV.Astrophysics.growth_z03 :TauGrowthFactor

Growth at z = 0.3. Equations

  • Tau.BookV.Astrophysics.growth_z03 = { z_x10 := 3, omega_m_z_x100 := 41, f_z_x100 := 69, sigma8_z_x1000 := 682, fsigma8_tau_x1000 := 470, fsigma8_lcdm_x1000 := 485 } Instances For

Tau.BookV.Astrophysics.growth_z10

source def Tau.BookV.Astrophysics.growth_z10 :TauGrowthFactor

Growth at z = 1.0. Equations

  • Tau.BookV.Astrophysics.growth_z10 = { z_x10 := 10, omega_m_z_x100 := 69, f_z_x100 := 86, sigma8_z_x1000 := 515, fsigma8_tau_x1000 := 443, fsigma8_lcdm_x1000 := 457 } Instances For

Tau.BookV.Astrophysics.growth_index_canonical

source def Tau.BookV.Astrophysics.growth_index_canonical :GrowthIndex

Canonical growth index. Equations

  • Tau.BookV.Astrophysics.growth_index_canonical = { gamma_tau_x1000 := 552, delta_gamma_x1000 := 7 } Instances For

Tau.BookV.Astrophysics.growth_below_lcdm_z03

source theorem Tau.BookV.Astrophysics.growth_below_lcdm_z03 :growth_z03.fsigma8_tau_x1000 < growth_z03.fsigma8_lcdm_x1000

[V.T237] τ growth systematically below ΛCDM.

Tau.BookV.Astrophysics.growth_index_departure

source theorem Tau.BookV.Astrophysics.growth_index_departure :growth_index_canonical.gamma_tau_x1000 > growth_index_canonical.gamma_lcdm_x1000

[V.T238] Growth index departure: Δγ = +0.007.

Tau.BookV.Astrophysics.DESISigma8Falsification

source structure Tau.BookV.Astrophysics.DESISigma8Falsification :Type

[V.R453] DESI σ₈(z) falsification window. τ-prediction: (f·σ₈)_τ / (f·σ₈)_Λ ≈ 0.97, z-independent. DESI DR3 + Euclid DR1 at ~1% precision → decisive.

  • tau_lcdm_ratio_x1000 : ℕ τ/ΛCDM ratio (×1000): 0.97 → 970.

  • desi_precision_pct_x1000 : ℕ DESI DR3 precision (×1000): 1% → 10.

  • z_independent : ℕ z-independent (structural): 1 = yes.

  • free_params : ℕ Free parameters.

Instances For

Tau.BookV.Astrophysics.instReprDESISigma8Falsification.repr

source def Tau.BookV.Astrophysics.instReprDESISigma8Falsification.repr :DESISigma8Falsification → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprDESISigma8Falsification

source instance Tau.BookV.Astrophysics.instReprDESISigma8Falsification :Repr DESISigma8Falsification

Equations

  • Tau.BookV.Astrophysics.instReprDESISigma8Falsification = { reprPrec := Tau.BookV.Astrophysics.instReprDESISigma8Falsification.repr }

Tau.BookV.Astrophysics.desi_sigma8_data

source def Tau.BookV.Astrophysics.desi_sigma8_data :DESISigma8Falsification

Equations

  • Tau.BookV.Astrophysics.desi_sigma8_data = { } Instances For

Tau.BookV.Astrophysics.desi_sigma8_detectable

source theorem Tau.BookV.Astrophysics.desi_sigma8_detectable :1000 - desi_sigma8_data.tau_lcdm_ratio_x1000 > desi_sigma8_data.desi_precision_pct_x1000

3% deficit detectable at 1% precision.

Tau.BookV.Astrophysics.HubbleDerivationChain

source structure Tau.BookV.Astrophysics.HubbleDerivationChain :Type

[V.D319] Hubble Derivation Chain. h = 2/3 + ι_τ²/W₃(3) = 0.67352 at −120 ppm from Planck. Two-step: EdS base (2/3) + holonomy correction (ι_τ²/17). Scope: τ-effective (Wave 38C).

  • eds_base_x100000 : ℕ EdS base × 100000.

  • correction_x100000 : ℕ Holonomy correction × 100000.

  • h_x100000 : ℕ h × 100000.

  • planck_h_x100000 : ℕ Planck h × 100000.

  • deviation_ppm : ℕ Deviation ppm.

  • w3_3 : ℕ W₃(3) = 17 (CF window sum).

  • free_params : ℕ Free parameters.

  • derivation_steps : ℕ Derivation steps.

Instances For

Tau.BookV.Astrophysics.instReprHubbleDerivationChain.repr

source def Tau.BookV.Astrophysics.instReprHubbleDerivationChain.repr :HubbleDerivationChain → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprHubbleDerivationChain

source instance Tau.BookV.Astrophysics.instReprHubbleDerivationChain :Repr HubbleDerivationChain

Equations

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

Tau.BookV.Astrophysics.hubble_derivation_data

source def Tau.BookV.Astrophysics.hubble_derivation_data :HubbleDerivationChain

Equations

  • Tau.BookV.Astrophysics.hubble_derivation_data = { } Instances For

Tau.BookV.Astrophysics.hubble_uniqueness

source theorem Tau.BookV.Astrophysics.hubble_uniqueness :hubble_derivation_data.free_params = 0 ∧ hubble_derivation_data.derivation_steps = 2

[V.T259] Uniqueness: h = 2/3 + ι_τ²/17 is unique first-order correction.

Tau.BookV.Astrophysics.hubble_self_consistency

source theorem Tau.BookV.Astrophysics.hubble_self_consistency :hubble_derivation_data.h_x100000 = 67352 ∧ hubble_derivation_data.w3_3 = 17

[V.P178] Self-consistency: h²·Ω_m = ω_m. h² × 100000 = 45363, Ω_m × 10000 = 3299, product / 10 = 14964. This should match ω_m(NLO) × 10000 = 14964.

Tau.BookV.Astrophysics.hubble_sub_permille

source theorem Tau.BookV.Astrophysics.hubble_sub_permille :hubble_derivation_data.deviation_ppm < 1000

Deviation is sub-permille.

Tau.BookV.Astrophysics.S8TensionResolution

source structure Tau.BookV.Astrophysics.S8TensionResolution :Type

[V.D324] S₈ NLO with full pipeline. σ₈(τ,NLO) = 0.811 × f_supp × f_growth × f_ν = 0.747. S₈(τ,NLO) = 0.747 × √(0.330/0.3) = 0.783. Scope: τ-effective (Wave 39C).

  • sigma8_cmb_x1000 : ℕ σ₈(CMB, Planck) × 1000.

  • f_supp_x10000 : ℕ f_supp × 10000.

  • f_growth_x10000 : ℕ f_growth × 10000.

  • f_nu_x100000 : ℕ f_ν × 100000.

  • sigma8_nlo_x10000 : ℕ σ₈(τ,NLO) × 10000.

  • s8_nlo_x10000 : ℕ S₈(τ,NLO) × 10000.

  • omega_m_nlo_x10000 : ℕ Ω_m(NLO) × 10000.

  • s8_planck_x1000 : ℕ S₈(Planck CMB) × 1000.

  • s8_deskids_x1000 : ℕ S₈(DES+KiDS) × 1000.

  • s8_desy3_x1000 : ℕ S₈(DES Y3) × 1000.

  • free_params : ℕ Free parameters.

Instances For

Tau.BookV.Astrophysics.instReprS8TensionResolution.repr

source def Tau.BookV.Astrophysics.instReprS8TensionResolution.repr :S8TensionResolution → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprS8TensionResolution

source instance Tau.BookV.Astrophysics.instReprS8TensionResolution :Repr S8TensionResolution

Equations

  • Tau.BookV.Astrophysics.instReprS8TensionResolution = { reprPrec := Tau.BookV.Astrophysics.instReprS8TensionResolution.repr }

Tau.BookV.Astrophysics.s8_resolution_data

source def Tau.BookV.Astrophysics.s8_resolution_data :S8TensionResolution

Equations

  • Tau.BookV.Astrophysics.s8_resolution_data = { } Instances For

Tau.BookV.Astrophysics.s8_between_cmb_and_wl

source theorem Tau.BookV.Astrophysics.s8_between_cmb_and_wl :s8_resolution_data.s8_desy3_x1000 < s8_resolution_data.s8_nlo_x10000 / 10 ∧ s8_resolution_data.s8_nlo_x10000 / 10 < s8_resolution_data.s8_planck_x1000

[V.T263] S₈(τ) between CMB and WL (resolves tension).

Tau.BookV.Astrophysics.s8_within_1sigma_deskids

source theorem Tau.BookV.Astrophysics.s8_within_1sigma_deskids :s8_resolution_data.s8_nlo_x10000 / 10 ≥ s8_resolution_data.s8_deskids_x1000 - 14 ∧ s8_resolution_data.s8_nlo_x10000 / 10 ≤ s8_resolution_data.s8_deskids_x1000 + 14

[V.P182] S₈(τ) within 1σ of DES+KiDS ( 0.783 − 0.790 < 0.014).

Tau.BookV.Astrophysics.s8_zero_params

source theorem Tau.BookV.Astrophysics.s8_zero_params :s8_resolution_data.free_params = 0

Zero free parameters.

Tau.BookV.Astrophysics.S8NNLO

source structure Tau.BookV.Astrophysics.S8NNLO :Type

[V.D330] S₈ NNLO density-regime value. At NNLO, Ω_m = 0.3155 ≈ Ω_m(Planck), so f_growth → 1.000. σ₈(τ,NNLO) = 0.811 × 0.913 × 1.000 × 0.997 = 0.738. S₈(τ,NNLO) = 0.738 × √(0.316/0.3) = 0.757. Scope: τ-effective (Wave 42A).

  • sigma8_cmb_x1000 : ℕ σ₈(CMB, Planck) × 1000.

  • f_supp_x10000 : ℕ f_supp × 10000 (unchanged from NLO).

  • f_growth_nnlo_x10000 : ℕ f_growth(NNLO) × 10000 (≈ 1.000).

  • f_nu_nnlo_x100000 : ℕ f_ν(NNLO) × 100000.

  • sigma8_nnlo_x10000 : ℕ σ₈(τ,NNLO) × 10000.

  • s8_nnlo_x10000 : ℕ S₈(τ,NNLO) × 10000.

  • omega_m_nnlo_x10000 : ℕ Ω_m(NNLO) × 10000.

  • s8_kids_x1000 : ℕ S₈(KiDS-1000) × 1000.

  • s8_kids_sigma_x1000 : ℕ S₈(KiDS-1000) 1σ uncertainty × 1000.

  • s8_hsc_x1000 : ℕ S₈(HSC Y3) × 1000.

  • s8_hsc_sigma_x1000 : ℕ S₈(HSC Y3) 1σ uncertainty × 1000.

  • free_params : ℕ Free parameters.

Instances For

Tau.BookV.Astrophysics.instReprS8NNLO

source instance Tau.BookV.Astrophysics.instReprS8NNLO :Repr S8NNLO

Equations

  • Tau.BookV.Astrophysics.instReprS8NNLO = { reprPrec := Tau.BookV.Astrophysics.instReprS8NNLO.repr }

Tau.BookV.Astrophysics.instReprS8NNLO.repr

source def Tau.BookV.Astrophysics.instReprS8NNLO.repr :S8NNLO → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.s8_nnlo_data

source def Tau.BookV.Astrophysics.s8_nnlo_data :S8NNLO

Equations

  • Tau.BookV.Astrophysics.s8_nnlo_data = { } Instances For

Tau.BookV.Astrophysics.s8_nnlo_within_kids

source theorem Tau.BookV.Astrophysics.s8_nnlo_within_kids :s8_nnlo_data.s8_nnlo_x10000 / 10 ≥ s8_nnlo_data.s8_kids_x1000 - s8_nnlo_data.s8_kids_sigma_x1000 ∧ s8_nnlo_data.s8_nnlo_x10000 / 10 ≤ s8_nnlo_data.s8_kids_x1000 + s8_nnlo_data.s8_kids_sigma_x1000

[V.T266] S₈(τ,NNLO) within 1σ of KiDS-1000: |0.757 − 0.759| = 0.002 < 0.024.

Tau.BookV.Astrophysics.s8_nnlo_within_hsc

source theorem Tau.BookV.Astrophysics.s8_nnlo_within_hsc :s8_nnlo_data.s8_nnlo_x10000 / 10 ≥ s8_nnlo_data.s8_hsc_x1000 - s8_nnlo_data.s8_hsc_sigma_x1000 ∧ s8_nnlo_data.s8_nnlo_x10000 / 10 ≤ s8_nnlo_data.s8_hsc_x1000 + s8_nnlo_data.s8_hsc_sigma_x1000

[V.T266] S₈(τ,NNLO) within 1σ of HSC Y3: |0.757 − 0.763| = 0.006 < 0.033.

Tau.BookV.Astrophysics.s8_nnlo_below_nlo

source theorem Tau.BookV.Astrophysics.s8_nnlo_below_nlo :s8_nnlo_data.s8_nnlo_x10000 < s8_resolution_data.s8_nlo_x10000

[V.D330] NNLO S₈ below NLO S₈ (density regime shifts S₈ down).

Tau.BookV.Astrophysics.s8_nnlo_zero_params

source theorem Tau.BookV.Astrophysics.s8_nnlo_zero_params :s8_nnlo_data.free_params = 0

Zero free parameters at NNLO.