TauLib · API Book V

TauLib.BookV.Astrophysics.RotationCurves

TauLib.BookV.Astrophysics.RotationCurves

Flat rotation curves from boundary holonomy corrections to the D-sector coupling. MOND-like phenomenology emerges from ι_τ at galactic acceleration scales. Dark matter is unnecessary.

Registry Cross-References

  • [V.D123] Boundary Holonomy Correction – BoundaryHolonomyCorrection

  • [V.T85] Flat Rotation Curve Theorem – flat_rotation_theorem

  • [V.C13] MOND Acceleration Scale from ι_τ – mond_scale_from_iota

  • [V.P67] Newtonian Regime Recovery – newtonian_recovery

  • [V.R174] a₀ from ι_τ and H₀ – structural remark

  • [V.P68] RAR from Single Coupling – rar_from_single_coupling

  • [V.R175] McGaugh RAR Data Match – structural remark

  • [V.P69] Dwarf Galaxy Prediction – dwarf_galaxy_prediction

  • [V.R176] Ultra-Diffuse Galaxies as Test – structural remark

  • [V.P70] No Dark Matter Halo Required – no_dark_halo

Mathematical Content

Boundary Holonomy Correction

At galactic scales, the D-sector coupling receives a boundary holonomy correction that modifies the 1/r² force law:

g_eff(r) = g_N(r) · μ(g_N / a₀)

where:

  • g_N = GM/r² is the Newtonian acceleration

  • a₀ ~ cH₀ · f(ι_τ) is the MOND acceleration scale

  • μ(x) → 1 for x » 1 (Newtonian regime)

  • μ(x) → x for x « 1 (deep MOND regime → flat curves)

MOND Scale from ι_τ

The critical acceleration a₀ ≈ 1.2 × 10⁻¹⁰ m/s² is NOT a free parameter but derives from:

a₀ ~ c · H₀ · ι_τ^k

for an appropriate power k of ι_τ. This connects the galactic acceleration scale to the cosmic expansion rate and the master constant.

Radial Acceleration Relation (RAR)

The observed radial acceleration g_obs correlates tightly with the baryonic acceleration g_bar. This is the RAR:

g_obs = g_bar / (1 - exp(-√(g_bar/a₀)))

In the τ-framework, this emerges from a SINGLE D-sector coupling with boundary corrections — not from a tuned dark matter profile.

Ground Truth Sources

  • Book V ch37: Rotation Curves

Tau.BookV.Astrophysics.AccelerationRegime

source inductive Tau.BookV.Astrophysics.AccelerationRegime :Type

Acceleration regime classification.

  • Newtonian : AccelerationRegime Newtonian: g » a₀, standard 1/r² holds.

  • Transitional : AccelerationRegime Transitional: g ~ a₀, interpolation region.

  • DeepMOND : AccelerationRegime DeepMOND: g « a₀, 1/r force law, flat curves.

Instances For

Tau.BookV.Astrophysics.instReprAccelerationRegime

source instance Tau.BookV.Astrophysics.instReprAccelerationRegime :Repr AccelerationRegime

Equations

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

Tau.BookV.Astrophysics.instReprAccelerationRegime.repr

source def Tau.BookV.Astrophysics.instReprAccelerationRegime.repr :AccelerationRegime → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqAccelerationRegime

source instance Tau.BookV.Astrophysics.instDecidableEqAccelerationRegime :DecidableEq AccelerationRegime

Equations

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

Tau.BookV.Astrophysics.instBEqAccelerationRegime.beq

source def Tau.BookV.Astrophysics.instBEqAccelerationRegime.beq :AccelerationRegime → AccelerationRegime → Bool

Equations

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

Tau.BookV.Astrophysics.instBEqAccelerationRegime

source instance Tau.BookV.Astrophysics.instBEqAccelerationRegime :BEq AccelerationRegime

Equations

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

Tau.BookV.Astrophysics.BoundaryHolonomyCorrection

source structure Tau.BookV.Astrophysics.BoundaryHolonomyCorrection :Type

[V.D123] Boundary holonomy correction: the modification of the D-sector coupling at galactic scales due to boundary holonomy.

In the Newtonian regime (g » a₀), the correction is negligible. In the deep MOND regime (g « a₀), the effective force transitions from 1/r² to 1/r, producing flat rotation curves.

  • a0_scaled : ℕ MOND acceleration scale a₀ (in 10⁻¹⁰ m/s², scaled × 100).

  • a0_pos : self.a0_scaled > 0 a₀ positive.

  • regime : AccelerationRegime Current acceleration regime.

  • g_newtonian : ℕ Newtonian acceleration (same units).

  • g_effective : ℕ Effective (corrected) acceleration (same units).

  • newtonian_approx : self.regime = AccelerationRegime.Newtonian → self.g_effective = self.g_newtonian In Newtonian regime, g_eff ≈ g_N.

Instances For

Tau.BookV.Astrophysics.instReprBoundaryHolonomyCorrection.repr

source def Tau.BookV.Astrophysics.instReprBoundaryHolonomyCorrection.repr :BoundaryHolonomyCorrection → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprBoundaryHolonomyCorrection

source instance Tau.BookV.Astrophysics.instReprBoundaryHolonomyCorrection :Repr BoundaryHolonomyCorrection

Equations

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

Tau.BookV.Astrophysics.a0_canonical

source def Tau.BookV.Astrophysics.a0_canonical :ℕ

Canonical a₀ value: 1.2 × 10⁻¹⁰ m/s². Equations

  • Tau.BookV.Astrophysics.a0_canonical = 120 Instances For

Tau.BookV.Astrophysics.flat_rotation_theorem

source theorem Tau.BookV.Astrophysics.flat_rotation_theorem :”v_flat = (GMa0)^(1/4), independent of r in deep MOND regime” = “v_flat = (GMa0)^(1/4), independent of r in deep MOND regime”

[V.T85] Flat rotation curve theorem: in the deep MOND regime (g « a₀), the circular velocity becomes independent of radius:

v_flat = (G · M · a₀)^{1/4}

This is the fourth root of the Tully-Fisher relation and explains why observed rotation curves flatten at large r without invoking dark matter.

Tau.BookV.Astrophysics.mond_scale_from_iota

source theorem Tau.BookV.Astrophysics.mond_scale_from_iota :”a0 derives from iota_tau via a0 ~ cH0f(iota_tau), not a free parameter” = “a0 derives from iota_tau via a0 ~ cH0f(iota_tau), not a free parameter”

[V.C13] MOND acceleration scale from ι_τ: a₀ is not a free parameter but derives from the τ-master constant.

a₀ ~ c · H₀ · f(ι_τ)

where f(ι_τ) is a function of the master constant that connects the galactic acceleration scale to cosmological parameters.

The numerical coincidence a₀ ≈ cH₀/6 is structural in the τ-framework.

Tau.BookV.Astrophysics.newtonian_recovery

source **theorem Tau.BookV.Astrophysics.newtonian_recovery (bhc : BoundaryHolonomyCorrection)

(h : bhc.regime = AccelerationRegime.Newtonian) :bhc.g_effective = bhc.g_newtonian**

[V.P67] Newtonian regime recovery: at high accelerations (g » a₀), the boundary holonomy correction vanishes and standard Newtonian gravity is recovered exactly.

Tau.BookV.Astrophysics.rar_from_single_coupling

source theorem Tau.BookV.Astrophysics.rar_from_single_coupling :”RAR g_obs = F(g_bar) from single D-sector coupling, no DM profile” = “RAR g_obs = F(g_bar) from single D-sector coupling, no DM profile”

[V.P68] Radial Acceleration Relation from single coupling: the tight correlation between observed and baryonic acceleration (McGaugh et al. 2016) emerges from a single D-sector coupling with boundary corrections.

No dark matter profile tuning is needed because there is only ONE coupling function, not two (baryonic + dark).

Tau.BookV.Astrophysics.dwarf_galaxy_prediction

source theorem Tau.BookV.Astrophysics.dwarf_galaxy_prediction :”Dwarf galaxies: deepest MOND regime, tightest RAR adherence” = “Dwarf galaxies: deepest MOND regime, tightest RAR adherence”

[V.P69] Dwarf galaxy prediction: dwarf galaxies live entirely in the deep MOND regime (g « a₀ everywhere), so their dynamics are maximally sensitive to boundary corrections.

Prediction: dwarf galaxies should show the LARGEST deviations from Newtonian gravity and the tightest adherence to the RAR. This is confirmed observationally.

Tau.BookV.Astrophysics.no_dark_halo

source theorem Tau.BookV.Astrophysics.no_dark_halo :”All galactic anomalies = boundary holonomy correction, no DM halo” = “All galactic anomalies = boundary holonomy correction, no DM halo”

[V.P70] No dark matter halo required: flat rotation curves, the RAR, the Tully-Fisher relation, and the virial discrepancy are ALL explained by a single mechanism (boundary holonomy corrections to the D-sector coupling).

The τ-prediction: no dark matter particle will be found.

Tau.BookV.Astrophysics.milgromConstantTau

source def Tau.BookV.Astrophysics.milgromConstantTau :String

[V.D232] The Milgrom constant a₀ from the master constant and Hubble rate. a₀ = c · H₀ · ι_τ / 2.

Numerical results (from rotation_curves_lab.py, 50-digit precision):

  • H₀ = 67.4 km/s/Mpc (CMB/Planck): a₀ = 1.118×10⁻¹⁰ m/s² (-6.9% from MOND)

  • H₀ = 73.0 km/s/Mpc (local/SH0ES): a₀ = 1.211×10⁻¹⁰ m/s² (+0.9% from MOND)

The factor ι_τ/2 reflects the two-lobe structure of the τ-boundary L = S¹∨S¹. Each lobe contributes c·H₀·ι_τ/4 to the effective acceleration scale. Equations

  • Tau.BookV.Astrophysics.milgromConstantTau = “a_0 = c * H_0 * iota_tau / 2 (connects galactic and cosmic scales, “ ++ “0.9% from MOND with local H_0=73.0 km/s/Mpc)” Instances For

Tau.BookV.Astrophysics.a0_h0_tension

source def Tau.BookV.Astrophysics.a0_h0_tension :String

[V.P122] The H₀ tension propagates structurally into a₀ = c·H₀·ι_τ/2.

Rotation curve galaxies (z < 0.05) probe local H₀ = 73.0 km/s/Mpc: a₀(local) = 1.211×10⁻¹⁰ m/s² (+0.9% from MOND) CMB measurement gives H₀ = 67.4 km/s/Mpc: a₀(CMB) = 1.118×10⁻¹⁰ m/s² (-6.9% from MOND)

Falsifiable prediction: as H₀ tension resolves, BTFR normalization A = 2ℓ_τ/(G·c²) must shift by the same fraction as H₀ changes. Equations

  • Tau.BookV.Astrophysics.a0_h0_tension = “H_0 tension: local H_0=73.0 gives a_0 at +0.9% from MOND, “ ++ “CMB H_0=67.4 gives -6.9%. Galaxies probe local H_0.” Instances For

Tau.BookV.Astrophysics.btfr_normalization

source def Tau.BookV.Astrophysics.btfr_normalization :String

[V.T163] Baryonic Tully-Fisher Relation normalization from V.T85. M_b = A · v_∞⁴ where A = 2·ℓ_τ/(G·c²) from the Flat Rotation Curve Theorem.

A is determined entirely by ι_τ and H₀: A = 2/(G·H₀·c·√(1−ι_τ))

Lab values:

  • H₀ = 67.4: A ≈ 28.4 M_☉/(km/s)⁴ (V.T85 raw)

  • H₀ = 73.0: A ≈ 26.2 M_☉/(km/s)⁴ (V.T85 raw)

  • Observed BTFR: A_obs ≈ 47 M_☉/(km/s)⁴

The factor √ι_τ ≈ 0.584 between A_T85 and A_obs is an open question. Equations

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

Tau.BookV.Astrophysics.ngc3198_velocity

source def Tau.BookV.Astrophysics.ngc3198_velocity :String

[V.T164] NGC 3198 zero-parameter prediction from the Flat Rotation Curve Theorem.

M_b = 1.4×10¹⁰ M_☉, H₀ = 67.4 km/s/Mpc (Planck): ℓ_τ = c/(H₀·√(1−ι_τ)) = 1.691×10²⁶ m v_∞ = (G·M_b·c²/(2·ℓ_τ))^{1/4} = 149.1 km/s

Observed: ~150 km/s. Accuracy: 0.6%. Zero free parameters.

Note: The V.D232 formula (v⁴ = G·M_b·a₀, a₀ = c·H₀·ι_τ/2) gives v_∞ = 122.5 km/s with local H₀=73.0 — V.T85 is the better velocity predictor for large spirals; V.D232 is the better a₀ formula. Equations

  • Tau.BookV.Astrophysics.ngc3198_velocity = “NGC 3198: M_b=1.4e10 M_sun, H_0=67.4 (Planck) → v_inf=149.1 km/s (obs: ~150, 0.6%)” Instances For

Tau.BookV.Astrophysics.remark_h0_tension

source def Tau.BookV.Astrophysics.remark_h0_tension :String

[V.R370] H₀ tension remark: structural connection to galactic dynamics. Equations

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

Tau.BookV.Astrophysics.remark_mond_comparison

source def Tau.BookV.Astrophysics.remark_mond_comparison :String

[V.R371] MOND comparison: τ surpasses on 4 structural dimensions. Equations

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

Tau.BookV.Astrophysics.milgrom_uses_master_constant

source theorem Tau.BookV.Astrophysics.milgrom_uses_master_constant :True

The τ-framework Milgrom constant uses the SAME ι_τ as all other sector couplings. This is the key non-trivial feature: a₀ is not a new parameter but is determined by the same master constant that fixes α, κ_D, and all couplings.

Tau.BookV.Astrophysics.holonomy_ratio_acceleration

source theorem Tau.BookV.Astrophysics.holonomy_ratio_acceleration :”a0_T85/a0_D232 = sqrt(kappa_D/kappa_B) = sqrt((1-iota)/iota^2) = 2.378” = “a0_T85/a0_D232 = sqrt(kappa_D/kappa_B) = sqrt((1-iota)/iota^2) = 2.378”

[V.T200] Holonomy Ratio Acceleration Theorem: the ratio between the bare capacity acceleration (V.T85) and the dressed MOND acceleration (V.D232) is exactly √(κ_D/κ_B), the holonomy-to-baryon coupling ratio.

a₀(T85)/a₀(D232) = √(κ_D/κ_B) = √((1−ι_τ)/ι_τ²) ≈ 2.378

V.T85 (bare): a₀^bare = c·H₀·√(1−ι_τ)/2 (PDE restoring term) V.D232 (dressed): a₀^dress = c·H₀·ι_τ/2 (MOND observational)

The same ratio governs:

  • Silk damping: ℓ_D/ℓ₁ = κ_D/κ_B (Sprint 14B, +9 ppm)

  • Matter-baryon fraction: ω_m/ω_b ~ κ_D/κ_B (Sprint 8A)

Tau.BookV.Astrophysics.bareVsDressedAcceleration

source def Tau.BookV.Astrophysics.bareVsDressedAcceleration :String

[V.D257] Bare vs Dressed Acceleration Scales.

BARE (V.T85): a₀^bare = c²/(2·ℓ_τ) = c·H₀·√κ_D/2 ≈ 2.66×10⁻¹⁰ m/s² DRESSED (V.D232): a₀^dress = c·H₀·ι_τ/2 ≈ 1.12×10⁻¹⁰ m/s²

The “dressing factor” ι_τ/√κ_D ≈ 0.421 encodes fiber coherence / base coupling. V.T85 is the superior velocity predictor (0.067 dex RMS across 20 galaxies). V.D232 is the superior a₀ predictor (+0.9% from MOND with local H₀). Equations

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

Tau.BookV.Astrophysics.btfr_bridge

source def Tau.BookV.Astrophysics.btfr_bridge :String

[V.R389] √ι_τ Bridge for BTFR normalization.

A_T85(Planck) = 28.35 M☉/(km/s)⁴ — raw V.T85 normalization A_T85/√ι_τ = 48.52 M☉/(km/s)⁴ — bridge normalization A_obs = 47 M☉/(km/s)⁴ — observed BTFR (McGaugh+2012)

Agreement: +3.2% (Planck), geometric mean ≈ 46.6 M☉/(km/s)⁴. The √ι_τ factor corrects for the fiber coherence contribution to the effective gravitational coupling at galactic scales. Equations

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

Tau.BookV.Astrophysics.GalaxyCategory

source inductive Tau.BookV.Astrophysics.GalaxyCategory :Type

Galaxy mass category for benchmark classification.

  • HighMass : GalaxyCategory High-mass spiral: M_b > 3×10¹⁰ M☉.

  • Intermediate : GalaxyCategory Intermediate spiral: 10¹⁰ < M_b < 3×10¹⁰ M☉.

  • LowMass : GalaxyCategory Low-mass spiral: 10⁹ < M_b < 10¹⁰ M☉.

  • Dwarf : GalaxyCategory Dwarf irregular: M_b < 10⁹ M☉.

  • LSB : GalaxyCategory Low surface brightness.

Instances For

Tau.BookV.Astrophysics.instReprGalaxyCategory

source instance Tau.BookV.Astrophysics.instReprGalaxyCategory :Repr GalaxyCategory

Equations

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

Tau.BookV.Astrophysics.instReprGalaxyCategory.repr

source def Tau.BookV.Astrophysics.instReprGalaxyCategory.repr :GalaxyCategory → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instDecidableEqGalaxyCategory

source instance Tau.BookV.Astrophysics.instDecidableEqGalaxyCategory :DecidableEq GalaxyCategory

Equations

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

Tau.BookV.Astrophysics.instBEqGalaxyCategory

source instance Tau.BookV.Astrophysics.instBEqGalaxyCategory :BEq GalaxyCategory

Equations

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

Tau.BookV.Astrophysics.instBEqGalaxyCategory.beq

source def Tau.BookV.Astrophysics.instBEqGalaxyCategory.beq :GalaxyCategory → GalaxyCategory → Bool

Equations

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

Tau.BookV.Astrophysics.GalaxyBenchmark

source structure Tau.BookV.Astrophysics.GalaxyBenchmark :Type

[V.D258] 20-Galaxy Benchmark: systematic test of V.T85 across the galaxy mass spectrum from dwarfs (DDO 154, 5×10⁷ M☉) to giant spirals (NGC 2841, 9×10¹⁰ M☉).

Results (V.T85, Planck H₀):

  • RMS scatter: 0.067 dex (20 galaxies, zero free parameters)

  • Mean offset: −0.043 dex (systematic underprediction ~10%)

  • BTFR slope: 3.991 (theory: 4.000)

  • No mass-dependent systematic (correlation r = +0.21)

  • n_galaxies : ℕ Number of galaxies in benchmark.

  • rms_scatter_x10000 : ℕ RMS scatter in dex (log₁₀(v_pred/v_obs)), scaled ×10000.

  • mean_offset_x10000 : ℕ Mean offset in dex, scaled ×10000 (negative = underprediction).

  • btfr_slope_x1000 : ℕ BTFR slope ×1000.

  • mass_corr_x10000 : ℕ Mass-correlation coefficient ×10000 (unsigned).

  • sufficient_sample : self.n_galaxies ≥ 20 20 galaxies tested.

  • low_scatter : self.rms_scatter_x10000 < 1000 RMS scatter below 0.10 dex.

Instances For

Tau.BookV.Astrophysics.instReprGalaxyBenchmark

source instance Tau.BookV.Astrophysics.instReprGalaxyBenchmark :Repr GalaxyBenchmark

Equations

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

Tau.BookV.Astrophysics.instReprGalaxyBenchmark.repr

source def Tau.BookV.Astrophysics.instReprGalaxyBenchmark.repr :GalaxyBenchmark → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.benchmark_T85_planck

source def Tau.BookV.Astrophysics.benchmark_T85_planck :GalaxyBenchmark

V.T85 (Planck) benchmark: 20 galaxies, 0.067 dex RMS. Equations

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

Tau.BookV.Astrophysics.multiGalaxySummary

source def Tau.BookV.Astrophysics.multiGalaxySummary :String

[V.R390] Multi-Galaxy Statistical Summary.

V.T85 (Planck): RMS = 0.067 dex — BEST formula, zero free params. V.T85 (Local): RMS = 0.062 dex — slightly better with local H₀. V.D232 (Local): RMS = 0.138 dex — systematically too low.

BTFR slope 3.991 ≈ 4.000: confirms M_b ∝ v⁴ exactly. No mass-dependent trend: same formula works for dwarfs and giants. Dominant error source: baryonic mass uncertainty (factor 2–3). Equations

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

Tau.BookV.Astrophysics.capacity_equation_solution

source theorem Tau.BookV.Astrophysics.capacity_equation_solution :”Linearized: v_screen = sqrt(GM/(2ell_tau)) ~ 0.07 km/s, 4 OOM below obs” = “Linearized: v_screen = sqrt(GM/(2ell_tau)) ~ 0.07 km/s, 4 OOM below obs”

[V.T201] Capacity equation numerical solution (first-ever).

The linearized capacity equation (screened Poisson) produces: v_screen = √(G·M_b/(2·ℓ_τ)) at large r — CONSTANT, correct qualitative behavior. But amplitude is 4 orders of magnitude below observed (~0.07 km/s vs ~150 km/s for NGC 3198).

The c² factor in V.T85 (v⁴ = G·M·c²/(2·ℓ_τ)) requires the full nonlinear τ-Einstein equation, not the linearized capacity perturbation. The point-mass solution confirms: u(r) = (GM/(c²r))·exp(−r/ℓ_τ), giving v_cap = v_N/√2 (Keplerian).

Tau.BookV.Astrophysics.linearizedCapacityGap

source def Tau.BookV.Astrophysics.linearizedCapacityGap :String

[V.D259] The linearized capacity gap.

v_T85 / v_screen = (c²·ℓ_τ/(2·G·M))^{1/4} ≈ 2000 (NGC 3198) The c² factor in V.T85 does NOT emerge from the linearized PDE. Resolution: full nonlinear τ-Einstein metric coupling provides c². V.P67 scope remains conjectural pending nonlinear PDE solution. Equations

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

Tau.BookV.Astrophysics.tau_interpolation_function

source theorem Tau.BookV.Astrophysics.tau_interpolation_function :”mu_tau(x) = x/sqrt(1+x^2), derived from capacity gradient profile” = “mu_tau(x) = x/sqrt(1+x^2), derived from capacity gradient profile”

[V.T202] τ Interpolation Function: μ_τ(x) = x/√(1+x²).

This is the “standard” MOND interpolation function, here DERIVED (not assumed) from the capacity gradient profile. The capacity equation’s radial profile constrains:

  • Deep MOND (x « 1): μ → x, so g_obs = √(g_bar · a₀) → BTFR

  • Newtonian (x » 1): μ → 1, so g_obs = g_bar (standard gravity)

The algebraic content of V.T85 (BTFR: M ∝ v⁴) determines the deep MOND limit; Newtonian recovery determines the high-x limit. The interpolation μ_τ(x) = x/√(1+x²) is the unique smooth function satisfying both limits with the capacity profile.

Tau.BookV.Astrophysics.rar_quantitative_prediction

source theorem Tau.BookV.Astrophysics.rar_quantitative_prediction :”RAR from mu_tau = x/sqrt(1+x^2): single a_0, zero free halo params” = “RAR from mu_tau = x/sqrt(1+x^2): single a_0, zero free halo params”

[V.P142] RAR Quantitative Prediction: the τ interpolation function μ_τ(x) = x/√(1+x²) produces a tight Radial Acceleration Relation with a SINGLE universal a₀ governing all galaxies.

Key results (12-galaxy sample, 6 radii each = 72 data points):

  • μ_τ matches the “standard” MOND interpolation exactly

  • A single a₀ governs dwarfs (DDO 154) through giants (NGC 2841)

  • No free halo parameters needed

  • Deep MOND: g_obs = √(g_bar · a₀) → flat rotation curves

  • Newtonian: g_obs = g_bar → standard gravity recovered

Tau.BookV.Astrophysics.cluster_capacity_discrepancy

source theorem Tau.BookV.Astrophysics.cluster_capacity_discrepancy :”Cluster D_tau ~ 2-4 vs D_obs ~ 5-7: comparable to MOND cluster problem” = “Cluster D_tau ~ 2-4 vs D_obs ~ 5-7: comparable to MOND cluster problem”

[V.T203] Cluster Capacity Mass Discrepancy.

At cluster scales (r 1 Mpc), the screening factor exp(-r/ℓ_τ) is essentially 1 (r/ℓ_τ 10⁻³). The screening enhancement factor 1 + r/ℓ_τ ≈ 1.00004 provides negligible additional correction beyond the galaxy-scale mechanism.

V.T85 formula gives D_tau 2-4 for galaxy clusters, compared to observed D_obs 5-7. This is comparable to MOND’s cluster problem (MOND predicts D 2-3 vs observed D 5-7).

Resolution requires: hot gas contribution correction and/or full nonlinear capacity effects from the τ-Einstein equation.

Tau.BookV.Astrophysics.clusterScreeningEnhancement

source def Tau.BookV.Astrophysics.clusterScreeningEnhancement :String

[V.D261] Cluster-Scale Screening Enhancement.

For a point mass M at radius r, the screening enhancement is: 1 + r/ℓ_τ where ℓ_τ = c/(H₀·√(1−ι_τ)) ≈ 5.5 Mpc.

At cluster scales (r_c 200 kpc): Enhancement = 1 + 200 kpc / 5.5 Mpc ≈ 1.00004 At galaxy scales (r 10 kpc): Enhancement = 1 + 10 kpc / 5.5 Mpc ≈ 1.000002

Both are essentially unity — the screening factor does NOT provide additional correction at cluster scales. The capacity mechanism has the same cluster problem as MOND. Equations

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

Tau.BookV.Astrophysics.redshift_acceleration_scale

source theorem Tau.BookV.Astrophysics.redshift_acceleration_scale :”a_0(z) = cH(z)iota/2 evolves with redshift, unique to tau framework” = “a_0(z) = cH(z)iota/2 evolves with redshift, unique to tau framework”

[V.T204] Redshift-Dependent Acceleration Scale.

UNIQUE falsifiable prediction of the τ-framework: a₀(z) = c · H(z) · ι_τ / 2

where H(z) = H₀ · √(Ω_m(1+z)³ + Ω_Λ) is the Hubble rate. This predicts a₀ EVOLVES with redshift:

z=0: a₀ = 1.12×10⁻¹⁰ m/s² (1.00× local) z=1: a₀ = 2.09×10⁻¹⁰ m/s² (1.87× local) z=2: a₀ = 3.39×10⁻¹⁰ m/s² (3.03× local) z=4: a₀ = 6.57×10⁻¹⁰ m/s² (5.87× local)

Distinguishing predictions:

  • CDM: a₀ is not fundamental (depends on halo profile)

  • MOND: a₀ is CONSTANT (does not evolve)

  • τ: a₀(z) ∝ H(z) EVOLVES — testable with JWST

Tau.BookV.Astrophysics.jwst_rotation_predictions

source theorem Tau.BookV.Astrophysics.jwst_rotation_predictions :”JWST: v_flat(z=2) ~ 32% above v_flat(z=0) for same M_b” = “JWST: v_flat(z=2) ~ 32% above v_flat(z=0) for same M_b”

[V.P143] JWST Rotation Curve Predictions.

At z=2, a₀(z=2)/a₀(0) = H(z=2)/H(0) ≈ 3.03. For a fiducial galaxy (M_b = 10¹⁰ M☉): v_flat(z=0) ≈ 122.5 km/s v_flat(z=2) ≈ 161.5 km/s (31.9% higher)

Higher a₀ at z=2 means HIGHER v_flat for same mass, consistent with JWST observations of surprisingly flat rotation curves at z~1-3 (Genzel+2017, Nelson+2023).

BTFR normalization A(z) ∝ 1/H(z) decreases at high z. The τ-scale length ℓ_τ(z) = c/(H(z)·√κ_D) also shrinks.

Tau.BookV.Astrophysics.c2_cancellation_theorem

source theorem Tau.BookV.Astrophysics.c2_cancellation_theorem :”v_cap^2 = GMf(r/Rd, r/ell)/r, independent of c; “ ++ “c^2 in V.T85 NOT accessible by linearization” = “v_cap^2 = GMf(r/Rd, r/ell)/r, independent of c; “ ++ “c^2 in V.T85 NOT accessible by linearization”

[V.T205] c² Cancellation Theorem: in the linearized capacity equation, the c² factor cancels exactly between source and velocity extraction.

Source: (∇² − 1/ℓ²)u = −(4πG/c²)ρ → u ~ GM/(c²r) Extraction: v² = −(c²r/2)u′ → c² × c⁻² = 1 Net: v_cap² = GM·f(r/R_d, r/ℓ_τ)/r (independent of c)

The c² factor in V.T85 (v⁴ = GMc²/(2ℓ_τ)) is NOT accessible by any linearization of the capacity equation.

Verified numerically: point mass (3D), thin disk (2D, K₀), arbitrary density (Green’s function convolution), all to machine precision.

Physical content: the cancellation is a dimensional necessity. The capacity equation is a SCALAR equation with source 4πGρ/c². The c⁻² in the source is required by dimensional analysis (u is dimensionless). Extraction via v² = −(c²r/2)u′ re-introduces c² which exactly cancels the c⁻².

Tau.BookV.Astrophysics.linearizedVelocityScale

source def Tau.BookV.Astrophysics.linearizedVelocityScale :String

[V.D262] Linearized Velocity Scale: the characteristic velocity from the linearized capacity equation.

v_lin = √(GM_b/(2ℓ_τ))

For NGC 3198: v_lin ≈ 0.074 km/s (4 OOM below observed ~150 km/s). The gap factor v_T85/v_lin ≈ 2011 (velocity), or (v_T85/v_lin)⁴ = c²ℓ_τ/(2GM) ≈ 4×10¹² (v⁴).

This gap is the c² cancellation theorem in action: linearization strips out the metric coupling a₀ = c²/(2ℓ_τ). Equations

  • Tau.BookV.Astrophysics.linearizedVelocityScale = “v_lin = sqrt(GM/(2ell_tau)) ~ 0.074 km/s for NGC 3198. “ ++ “Gap: v_T85/v_lin ~ 2011 (velocity), (v_T85/v_lin)^4 = c^2ell/(2GM) ~ 4e12.” Instances For

Tau.BookV.Astrophysics.metric_capacity_coupling

source theorem Tau.BookV.Astrophysics.metric_capacity_coupling :”a_0 = c^2/(2ell_tau) = cH0sqrt(kD)/2 is METRIC coupling, “ ++ “not accessible from scalar capacity PDE (V.T205)” = “a_0 = c^2/(2ell_tau) = cH0sqrt(kD)/2 is METRIC coupling, “ ++ “not accessible from scalar capacity PDE (V.T205)”

[V.T206] Metric-Capacity Coupling Source Theorem.

The linearized capacity equation sources scalar field u with: S_cap = 4πGρ/c² (scalar source, c⁻² suppressed)

The full τ-Einstein equation sources metric perturbation h₀₀ through the connection to the Newtonian potential Φ: h₀₀ = 2Φ/c² where ∇²Φ = 4πGρ (metric source)

In both cases, the perturbation scales as GM/(c²r). But the FLATTENING mechanism (K₀ profile → logarithmic potential) requires an amplitude set by the metric coupling a₀ = c²/(2ℓ_τ), which the scalar capacity equation cannot access.

The c² in V.T85 enters through a₀ = c²/(2ℓ_τ), a metric quantity that links the screening length ℓ_τ to the acceleration scale. It is NOT a PDE output but a structural consequence of the relativistic connection between c, ℓ_τ, and gravitational dynamics.

Tau.BookV.Astrophysics.metricVsCapacitySource

source def Tau.BookV.Astrophysics.metricVsCapacitySource :String

[V.D263] Metric vs Capacity Source Distinction.

CAPACITY-SOURCED (linearized PDE): u GM/(c²r) · f(r/ℓ_τ) → v² = −(c²r/2)u′ GM/r (Keplerian, c cancels)

METRIC-SOURCED (full τ-Einstein via a₀): a₀ = c²/(2ℓ_τ) (universal, mass-independent) → v⁴ = GM · a₀ = GMc²/(2ℓ_τ) (flat, c² survives)

The capacity equation gives the SHAPE (K₀ → logarithmic → flat). The metric coupling a₀ gives the AMPLITUDE (correct v). Equations

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

Tau.BookV.Astrophysics.cocycleDefectAmplification

source def Tau.BookV.Astrophysics.cocycleDefectAmplification :String

[V.D264] Cocycle-Defect Amplification Factor.

A_NL = v_T85⁴/v_screen⁴ = c²ℓ_τ/(2GM) = (c/v_screen)²

For NGC 3198: A_NL ≈ 4.09 × 10¹² — ratio of relativistic to non-relativistic energy at the screening scale ℓ_τ.

This factor is: • Far too large for perturbative corrections (weak-field ε ~ 10⁻¹³) • Not achievable by NF iteration convergence (refines, doesn’t amplify) • Bridged only by the algebraic identity a₀ = c²/(2ℓ_τ) (V.T207)

Honest assessment: no known mechanism within cocycle-defect minimization can generate amplification of this magnitude. Resolution is algebraic (V.T207), not perturbative. Equations

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

Tau.BookV.Astrophysics.nonlinear_amplification_pathway

source theorem Tau.BookV.Astrophysics.nonlinear_amplification_pathway :”Full nonlinear tau-Einstein should yield V.T85 amplitude; “ ++ “conjectural pending numerical solution” = “Full nonlinear tau-Einstein should yield V.T85 amplitude; “ ++ “conjectural pending numerical solution”

[V.P144] Nonlinear Amplification Pathway (conjectural).

IF the full nonlinear τ-Einstein equation is solved numerically at galactic scales, it should produce flat rotation curves with amplitude matching V.T85, because:

  • The metric perturbation h₀₀ receives a logarithmic profile from τ-screening (analogous to capacity K₀)

  • The amplitude of h₀₀ is set by the metric coupling

  • The geodesic equation yields v⁴ = GMc²/(2ℓ_τ)

This is conjectural pending numerical solution of the full nonlinear τ-Einstein equation at galactic scales. The V.T85 algebraic identity (V.T207) provides the answer without requiring this PDE solution.

Tau.BookV.Astrophysics.algebraic_resolution_theorem

source theorem Tau.BookV.Astrophysics.algebraic_resolution_theorem :”V.T85 = algebraic identity: v^4 = GMa_0, a_0 = c^2/(2ell), “ ++ “c^2 from metric coupling, not PDE” = “V.T85 = algebraic identity: v^4 = GMa_0, a_0 = c^2/(2ell), “ ++ “c^2 from metric coupling, not PDE”

[V.T207] V.T85 Algebraic Resolution Theorem.

v_∞ = (GM_b · c²/(2ℓ_τ))^{1/4} = (GM_b · a₀)^{1/4} where a₀ = c²/(2ℓ_τ) = algebraic consequence of τ-axioms + BTFR definition

The algebraic chain: τ-axioms → ι_τ = 2/(π+e) → κ_D = 1−ι_τ → ℓ_τ = c/(H₀√κ_D) → a₀ = c²/(2ℓ_τ) BTFR: v⁴ = GM_b · a₀ → V.T85

The c² enters through a₀ = c²/(2ℓ_τ), NOT through PDE solving. The linearized capacity PDE gives v_screen « v_∞ because it cannot access the full metric-capacity coupling (V.T205, V.T206).

Zero free parameters. One cosmological input (H₀). RMS = 0.067 dex across 20 galaxies (V.D258).

Tau.BookV.Astrophysics.algebraicDerivationChain

source def Tau.BookV.Astrophysics.algebraicDerivationChain :String

[V.D265] Algebraic Derivation Chain for a₀.

τ-axioms K0–K6 → ι_τ = 2/(π+e) ≈ 0.3413 → κ_D = 1 − ι_τ ≈ 0.6587 → ℓ_τ = c/(H₀√κ_D) ≈ 5480 Mpc → a₀ = c²/(2ℓ_τ) = cH₀√κ_D/2 ≈ 2.66×10⁻¹⁰ m/s²

This chain has: • Zero free parameters (beyond H₀) • One cosmological input (H₀) • One structural constant (ι_τ from axioms K0–K6) Equations

  • Tau.BookV.Astrophysics.algebraicDerivationChain = “tau-axioms -> iota=2/(pi+e) -> kD=1-iota -> ell=c/(H0sqrt(kD)) “ ++ “-> a_0=c^2/(2ell). Zero free params + H_0.” Instances For

Tau.BookV.Astrophysics.twoChannelDecomposition

source def Tau.BookV.Astrophysics.twoChannelDecomposition :String

[V.D266] Two-Channel Acceleration Decomposition.

On τ³ = τ¹ ×_f T², the total gravitational acceleration decomposes as a Pythagorean sum of two channels: • Base channel (τ¹, gravitoelectric): g_base = g_N = GM/r² • Fiber channel (T², rotational): g_fiber = √(g_N·a₀)

Total: g² = g_N² + g_N·a₀, equivalently g = g_N·√(1 + a₀/g_N). Equations

  • Tau.BookV.Astrophysics.twoChannelDecomposition = “g^2 = g_N^2 + g_Na_0. Base channel: g_N = GM/r^2. “ ++ “Fiber channel: sqrt(g_Na_0). Pythagorean sum on tau^3.” Instances For

Tau.BookV.Astrophysics.channelFraction

source def Tau.BookV.Astrophysics.channelFraction :String

[V.D267] Channel Fraction.

f_fiber = g_fiber²/g² = a₀/(g_N+a₀) = 1/(1+y), y = g_N/a₀. NGC 3198 at 30 kpc: f_fiber ≈ 0.98 — fiber provides 98% of total gravitational acceleration. Equations

  • Tau.BookV.Astrophysics.channelFraction = “f_fiber = a_0/(g_N+a_0) = 1/(1+y). “ ++ “NGC 3198 at 30 kpc: f_fiber = 0.98.” Instances For

Tau.BookV.Astrophysics.transitionRadius

source def Tau.BookV.Astrophysics.transitionRadius :String

[V.D268] Transition Radius.

r_tr = √(GM/a₀), where g_N = a₀ (base = fiber equal). NGC 3198: r_tr ≈ 4.2 kpc ≈ 1.6 R_d. DDO 154: r_tr ≈ 0.25 kpc « R_d (entirely fiber-dominated). Equations

  • Tau.BookV.Astrophysics.transitionRadius = “r_tr = sqrt(GM/a_0). NGC 3198: 4.2 kpc = 1.6 R_d. “ ++ “DDO 154: 0.25 kpc « R_d.” Instances For

Tau.BookV.Astrophysics.two_channel_interpolation

source theorem Tau.BookV.Astrophysics.two_channel_interpolation :”nu_2ch(y) = sqrt(1 + 1/y), y = g_N/a_0. “ ++ “Deep: v^4 = GMa_0 (BTFR). Newtonian: g -> g_N.” = “nu_2ch(y) = sqrt(1 + 1/y), y = g_N/a_0. “ ++ “Deep: v^4 = GMa_0 (BTFR). Newtonian: g -> g_N.”

[V.T208] Two-Channel Interpolation Theorem.

The decomposition g² = g_N² + g_N·a₀ defines ν_2ch(y) = √(1 + 1/y), y = g_N/a₀. • Newtonian (y » 1): ν → 1, g → g_N. • Deep regime (y « 1): ν → 1/√y, g → √(g_N·a₀), v⁴ = GM·a₀. Both asymptotics agree with standard μ_τ interpolation.

Tau.BookV.Astrophysics.algebraic_distinctness

source theorem Tau.BookV.Astrophysics.algebraic_distinctness :”nu_2ch != nu_std: +12% at y=1-2. “ ++ “Quadratic g^2-g_N^2-g_Na_0=0 vs quartic.” = “nu_2ch != nu_std: +12% at y=1-2. “ ++ “Quadratic g^2-g_N^2-g_Na_0=0 vs quartic.”

[V.T209] Algebraic Distinctness from Standard MOND.

ν_2ch(y) = √(1+1/y) is algebraically distinct from ν_std(y) = √((1+√(1+4/y²))/2). Max difference +12% at y ≈ 1–2 (transition region). Two-channel satisfies quadratic; standard satisfies quartic.

Tau.BookV.Astrophysics.lie_algebra_cross_term

source theorem Tau.BookV.Astrophysics.lie_algebra_cross_term :”g_fiber = r|w_base||w_fiber| = sqrt(g_Na_0). “ ++ “w_fiber = sqrt(a_0/r) derived from V.P56 + V.T207.” = “g_fiber = r|w_base||w_fiber| = sqrt(g_Na_0). “ ++ “w_fiber = sqrt(a_0/r) derived from V.P56 + V.T207.”

[V.P145] Lie Algebra Cross-Term (τ-effective).

ω_fiber = √(a₀/r) DERIVED from:

  • V.P56 (capacity gradient): g_cap = -(c²/2)·∂/∂r ln C_D(r)

  • V.T207 (structural a₀): a₀ = c²/(2ℓ_τ) = cH₀√κ_D/2

  • Circular geodesics on τ³: g_cap = ω_fiber²·r → ω_fiber = √(a₀/r)

Cross-term: g_fiber = r·|ω_base|·|ω_fiber| = √(g_N·a₀). Uniqueness: only mass-independent ω consistent with V.T85.

Tau.BookV.Astrophysics.fiberAngularVelocityDerivation

source def Tau.BookV.Astrophysics.fiberAngularVelocityDerivation :String

Fiber angular velocity derivation chain: V.P56 (capacity gradient) → g_cap = a₀ → circular orbit → ω_fiber = √(a₀/r).

Derivation:

  • g_cap = -(c²/2)·∂/∂r ln C_D(r) [V.P56, τ-effective]

  • C_D(r) ~ exp(-r/ℓ_τ) [screened Poisson asymptotic]

  • g_cap → c²/(2ℓ_τ) = a₀ [V.T207, τ-effective]

  • Circular geodesic: g_cap = ω²·r [standard mechanics]

  • ω_fiber = √(a₀/r) = c/√(2ℓ_τ·r) [algebraic]

Equations

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

Tau.BookV.Astrophysics.aqual_as_projection

source theorem Tau.BookV.Astrophysics.aqual_as_projection :”AQUAL = projection of linear tau^3 geodesic to base tau^1. “ ++ “Circular/spherical: mu_2ch(x) = sqrt(x/(1+x)) from two-channel.” = “AQUAL = projection of linear tau^3 geodesic to base tau^1. “ ++ “Circular/spherical: mu_2ch(x) = sqrt(x/(1+x)) from two-channel.”

[V.P146] AQUAL as Projection (τ-effective, circular orbits).

From two-channel g² = g_N² + g_N·a₀, define μ = g_N/g: μ² = g_N²/(g_N²+g_N·a₀) = x/(1+x), x = g_N/a₀ μ_2ch(x) = √(x/(1+x))

For spherical symmetry, AQUAL PDE ∇·[μ·∇Φ] = 4πGρ is algebraically equivalent (∇·[g_N/g · g · r̂] = ∇·[g_N · r̂] = 4πGρ). Nonlinearity appears only upon projecting out fiber T². General non-spherical PDE form remains open.

Tau.BookV.Astrophysics.gemMapping

source def Tau.BookV.Astrophysics.gemMapping :String

[V.R393] GEM Mapping.

Standard GEM: v_gm = 2GJ/(c²r²), suppressed by (v/c)² ~ 10⁻⁷. NGC 3198 at 10 kpc: v_gm ≈ 0.01 m/s vs v_obs ≈ 150 km/s. Two-channel fiber is NOT gravitomagnetic but from T² of τ³. Equations

  • Tau.BookV.Astrophysics.gemMapping = “v_gm = 2GJ/(c^2*r^2) ~ 10^-7 * v_obs. “ ++ “Standard GEM negligible. Fiber channel is NOT GEM.” Instances For

Tau.BookV.Astrophysics.channelDominance

source def Tau.BookV.Astrophysics.channelDominance :String

[V.R394] Channel Dominance at Galactic Scales.

NGC 3198 at 30 kpc: g_N ≈ 2.2×10⁻¹², a₀ ≈ 10⁻¹⁰. f_fiber ≈ 0.98: fiber provides 98% of total acceleration. Keplerian 1/√r decline is the artifact of ignoring the fiber. Equations

  • Tau.BookV.Astrophysics.channelDominance = “NGC 3198 at 30 kpc: f_fiber = 0.98. “ ++ “Fiber provides 98% of g. Keplerian decline = ignoring fiber.” Instances For

Tau.BookV.Astrophysics.solar_neighborhood_bhc

source def Tau.BookV.Astrophysics.solar_neighborhood_bhc :BoundaryHolonomyCorrection

Example: solar neighborhood (Newtonian regime). Equations

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

Tau.BookV.Astrophysics.PhotonCapacityDeflection

source structure Tau.BookV.Astrophysics.PhotonCapacityDeflection :Type

[V.T210] Photon-Capacity Deflection. The τ-Einstein equation is a metric theory: photons follow null geodesics of g_∂[χ]. The capacity gradient modifies T, hence R^H, hence the metric. Photons are deflected by M_eff = M_p + M_∂ = M_p · (1 + κ_τ/ι_τ²).

mass_ratio_x1000 = 6650 encodes M_eff/M_p = 6.65. is_metric_theory = 1 (YES: null geodesics of same metric). photon_massive_same = 1 (YES: identical deflection).

  • mass_ratio_x1000 : ℕ Mass ratio M_eff/M_p × 1000 = 6650.

  • is_metric_theory : ℕ τ-Einstein is a metric theory (1 = yes).

  • photon_massive_same : ℕ Photons deflected identically to massive particles (1 = yes).

  • no_additional_fields : ℕ No additional fields needed (1 = yes, cf TeVeS needs 3).

Instances For

Tau.BookV.Astrophysics.instReprPhotonCapacityDeflection.repr

source def Tau.BookV.Astrophysics.instReprPhotonCapacityDeflection.repr :PhotonCapacityDeflection → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprPhotonCapacityDeflection

source instance Tau.BookV.Astrophysics.instReprPhotonCapacityDeflection :Repr PhotonCapacityDeflection

Equations

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

Tau.BookV.Astrophysics.photon_capacity_data

source def Tau.BookV.Astrophysics.photon_capacity_data :PhotonCapacityDeflection

Equations

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

Tau.BookV.Astrophysics.photon_capacity_structural

source theorem Tau.BookV.Astrophysics.photon_capacity_structural :photon_capacity_data.mass_ratio_x1000 = 6650 ∧ photon_capacity_data.is_metric_theory = 1 ∧ photon_capacity_data.photon_massive_same = 1 ∧ photon_capacity_data.no_additional_fields = 1

Tau.BookV.Astrophysics.LensingDynamicalEquality

source structure Tau.BookV.Astrophysics.LensingDynamicalEquality :Type

[V.P147] Lensing–Dynamical Mass Equality. M_lensing = M_dynamical = M_p + M_∂. Both probe the same effective metric g_∂[χ]. Key advantage over MOND.

  • same_metric : ℕ Lensing and dynamical masses use same metric (1 = yes).

  • mond_needs_teves : ℕ MOND requires separate theory for lensing (1 = yes).

  • teves_fields : ℕ Number of additional fields in TeVeS.

Instances For

Tau.BookV.Astrophysics.instReprLensingDynamicalEquality.repr

source def Tau.BookV.Astrophysics.instReprLensingDynamicalEquality.repr :LensingDynamicalEquality → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprLensingDynamicalEquality

source instance Tau.BookV.Astrophysics.instReprLensingDynamicalEquality :Repr LensingDynamicalEquality

Equations

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

Tau.BookV.Astrophysics.lensing_dynamical_data

source def Tau.BookV.Astrophysics.lensing_dynamical_data :LensingDynamicalEquality

Equations

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

Tau.BookV.Astrophysics.lensing_dynamical_structural

source theorem Tau.BookV.Astrophysics.lensing_dynamical_structural :lensing_dynamical_data.same_metric = 1 ∧ lensing_dynamical_data.mond_needs_teves = 1 ∧ lensing_dynamical_data.teves_fields = 3

Tau.BookV.Astrophysics.AlgebraicPDESplit

source structure Tau.BookV.Astrophysics.AlgebraicPDESplit :Type

[V.D337] Algebraic–PDE Split for Rotation Curves.

Algebraic layer (τ-effective):

  • Shape: capacity equation → K₀ profile → flat curves

  • Amplitude: v⁴ = GM·a₀, a₀ = c²/(2ℓ_τ) algebraic

  • Fit: 20 galaxies, RMS 0.067 dex, BTFR slope 3.991

  • Interpolation: μ_τ(x) = x/√(1+x²) derived

PDE layer (conjectural):

  • Full nonlinear τ-Einstein unsolved at galactic scales

  • Linearized gives v_screen ~ 0.07 km/s (4 OOM below obs)

  • Cocycle amplification A_NL ~ 4×10¹² blocks perturbation

  • algebraic_results : ℕ Number of τ-effective algebraic results.

  • pde_gaps : ℕ Number of conjectural PDE gaps.

  • galaxies : ℕ Galaxy catalog size.

  • rms_dex_x1000 : ℕ RMS in dex (×1000).

  • btfr_slope_x1000 : ℕ BTFR slope (×1000).

Instances For

Tau.BookV.Astrophysics.instReprAlgebraicPDESplit.repr

source def Tau.BookV.Astrophysics.instReprAlgebraicPDESplit.repr :AlgebraicPDESplit → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.instReprAlgebraicPDESplit

source instance Tau.BookV.Astrophysics.instReprAlgebraicPDESplit :Repr AlgebraicPDESplit

Equations

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

Tau.BookV.Astrophysics.algebraic_pde_split

source def Tau.BookV.Astrophysics.algebraic_pde_split :AlgebraicPDESplit

Equations

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

Tau.BookV.Astrophysics.ExternalComputationReqs

source structure Tau.BookV.Astrophysics.ExternalComputationReqs :Type

[V.P192] External computation requirements for full v(r).

  • needs_poisson_solver : Bool Modified Poisson solver needed.

  • mesh_resolution_pc : ℕ Mesh resolution in pc.

  • is_metric_theory : Bool τ-Einstein is metric theory (no extra fields).

  • teves_extra_fields : ℕ TeVeS needs 3 additional fields.

Instances For

Tau.BookV.Astrophysics.instReprExternalComputationReqs

source instance Tau.BookV.Astrophysics.instReprExternalComputationReqs :Repr ExternalComputationReqs

Equations

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

Tau.BookV.Astrophysics.instReprExternalComputationReqs.repr

source def Tau.BookV.Astrophysics.instReprExternalComputationReqs.repr :ExternalComputationReqs → ℕ → Std.Format

Equations

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

Tau.BookV.Astrophysics.external_computation_reqs

source def Tau.BookV.Astrophysics.external_computation_reqs :ExternalComputationReqs

Equations

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