H₀ Local: Hubble Tension as Phase Field
The Hubble tension is a persistent ~9% discrepancy between CMB-derived (67.4 km/s/Mpc) and locally measured (73 km/s/Mpc) values of H₀. Mode Identity Theory resolves this through the phase decomposition α = α₀ + αf: observers embedded in structure sample from the Fibonacci well plus an environment-dependent offset. The minimum realized step on the 60R-grid is αf = 2/120. At the H₀ well (α₀ = 34/120), this shifts C(α) by 8.4%, yielding 67.4 × 1.084 ≈ 73 km/s/Mpc.
The match is not a fit; the step size is fixed by lattice geometry. Two mechanisms transmit the bias: calibration inheritance (local anchors import the full shift) and phase-domain averaging (geometric methods dilute the effect over long baselines). The falsifiable prediction is discrete clustering: local H₀ should appear at quantized values (67, 73, ...), not vary continuously with environment.
time-stamped paper can be found here
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I. THE OBSERVED TENSION
Measurements of the Hubble constant fall into two categories with persistently discrepant results:
CMB (Planck) ..... 67.4 ± 0.5 km/s/Mpc ..... z ≈ 1100
Local (SH0ES) .... 73.0 ± 1.0 km/s/Mpc ..... z ≈ 0
The ~9% discrepancy persists across independent local methods: Cepheids, TRGB, surface brightness fluctuations. Systematics have been scrutinized extensively. The tension appears real.
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II. THE PHASE DECOMPOSITION
The Hubble tension is a mismatch between global phase sampling (α₀) and local phase sampling (α₀ + αf). H₀ is an edge realized quantity in MIT, so it scales with the phase operator evaluated at the phase coordinate that the observation actually samples.
Within MIT, realized quantities sample the mode spectrum at position α via the phase operator:
C(α) = 2sin²(πα)
DERIVATION
The Möbius Z₂ holonomy admits periodic and anti-periodic sectors. Matter is fermionic; this selects the anti-periodic sector. A field on the domain satisfies:
ψ(α + 1) = −ψ(α)
This boundary condition excludes the constant mode (ψ = const would require ψ(α+1) = +ψ(α)). The ground state is the lowest allowed mode. The general solution is a linear combination of sin(πα) and cos(πα); MIT adopts the standing-wave gauge with a node at α = 0:
ψ₀(α) = sin(πα)
This gauge choice defines the phase origin for the Fibonacci wells. Realized intensity is |ψ|². Normalizing to unit mean intensity sets the peak amplitude to 2, yielding C(α) = 2sin²(πα). The antinode (maximum C = 2) is at α = 1/2.
The sampled position is not the Fibonacci well alone:
α = α₀ + αf
α₀ ....... Fibonacci well ........ Fixed, geometric (Fibonacci/120)
αf ....... Phase field .............. Varies with local environment
α ........ Sampled position ... What observation actually samples
For H₀, the Fibonacci well is α₀ = 34/120. The CMB samples the global phase state directly (α₀), before local structure existed. Local realizations sample from a specific phase domain where αf ≠ 0.
SCALING LAW
Edge observables (n = 1) scale linearly with C(α) via the MIT scaling law. An observer at sampled position α = α₀ + αf therefore realizes the quantity:
H₀^local = H₀^global × C(α₀ + αf)/C(α₀)
This follows from the general scaling law A/A_P ≈ Ω^{−n/2} × C(α) applied to the Hubble parameter at n = 1, where A_P is the Planck reference and Ω = (R_H/ℓ_P)² ≈ 10¹²² is the cosmic-to-Planck hierarchy. Both H₀^local and H₀^global are present-epoch parameters (the CMB measurement is an inference of today's expansion rate, not the rate at z = 1100). Since both reference the same epoch, they share the same Ω. In the ratio, the common factor Ω^{−1/2} cancels exactly; only the C(α) ratio remains.
SYMBOLS (REFERENCE)
α₀ ........... Fibonacci well (global)
αf ........... Environment-selected offset (local)
C(α) ....... Phase operator mapping position → intensity
Lf ........... Spatial coherence scale (= L_lap in other MIT papers)
F ............ Coherent fraction of the line-of-sight integral (= χ_local/χ)
δf ........... Fractional perturbation to 1/H (this document only)
ξ ............ Geometry factor for trigger depth
T ............ Trigger index
ε ............ Coupling strength (= 8.4%, the full αf shift)
Notation scope: The symbol δf in this supplement denotes the fractional perturbation to 1/H from the phase shift, not the phase parameter δ used elsewhere in MIT.
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III. CORE RESULT: THE NUMERICAL CLOSURE
At α₀ = 34/120 = 0.2833:
C(α₀) = 2sin²(34π/120) = 1.208
This well is fixed by the Fibonacci/120 geometry and is not adjusted to match H₀.
A shift of αf = 2/120 (the minimum step on the 60R-grid; see §IV.D) gives:
C(α₀ + αf) = 2sin²(36π/120) = 1.309
The ratio:
C(α₀ + αf)/C(α₀) = 1.309/1.208 = 1.084
An 8.4% shift, consistent with the observed ~9% tension.
Planck CMB gives H₀ = 67.4 km/s/Mpc. Multiply by the ratio:
67.4 × 1.084 ≈ 73 km/s/Mpc
This matches SH0ES.
This numerical agreement is an output of the discrete step (αf = 2/120) combined with the logarithmic slope of C(α) at α₀ = 34/120. It is not a fit to the SH0ES value. The input is the lattice geometry; the ~8.4% emerges.
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IV. THE PHASE FIELD AND TWO TRANSMISSION MECHANISMS
A. WHAT IS ASSUMED VS WHAT IS DERIVED
C(α) = 2sin²(πα) ........ Derived (BC + sector + gauge + normalization)
α = α₀ + αf ................. Framework
α₀ = 34/120 ............... Fixed (Fibonacci well)
αf = 2/120 ................. MIT structural (120-grid → 60R-grid)
8.4% shift .................. Output of C(α) slope × step
Phase avg. kernel ..... Derived (given domain-local perturbation ansatz)
Coupling ε = 8.4% .... Not free (equals full shift)
Lf = v_c²/a₀ ............... Derived (radius where g = a₀)
ξ ≈ 0.46 ..................... Derived (Appendix A)
Scale hierarchy ......... Derived
Discrete vs continous fork Falsifier
Note on C(α) derivation: The phase operator C(α) = 2sin²(πα) follows from: (1) anti-periodic boundary condition ψ(α+1) = −ψ(α) from Möbius topology, (2) sector selection (fermionic matter selects anti-periodic), (3) standing-wave gauge (node at α = 0), and (4) normalization (unit mean intensity). These are consistent choices within MIT; the result is not purely geometric but depends on this specification.
B. THE CORE IDEA: TWO MECHANISMS
The phase field αf can bias H₀ inference through two distinct channels:
(A) Calibration inheritance: Local anchors import the full αf into their distance scale. The ruler is biased.
(B) Phase-domain averaging: Geometric distance measurements integrate the biased kernel only over the coherent phase domain, diluting the effect for long baselines. The path is biased only locally.
Mechanism-Switching Rule: Inheritance dominates when calibration is local; dilution applies when calibration is non-local.
Calibration inheritance ... Absolute scale of local anchors ... Full αf inherited by all rungs
Phase-domain averaging .... Observations spanning large phase intervals ... αf contribution diluted
C. PHASE-DOMAIN AVERAGING: THE DERIVATION
Step 1: Distances are line integrals of 1/H.
χ(z_s) = ∫₀^{z_s} c dz / H(z)
Step 2: Decompose the integral into local and distant contributions.
Let χ_local be the contribution from the observer's coherence domain (redshift interval [0, z_f] corresponding to comoving scale Lf), and χ_distant be the remainder:
χ = χ_local + χ_distant
Define the coherent fraction:
F ≡ χ_local / χ
For sources at cosmological distances (z_s ≫ z_f), the local contribution is a small fraction: F ≪ 1.
Step 3: Perturb 1/H only inside the coherent domain.
1/H(z) → (1/H(z))(1 + δ) for z ∈ [0, z_f]
where δ = δf inside the coherent domain and δ = 0 outside.
Step 4: Compute the fractional distance perturbation.
Δχ/χ = δf · F
The bias δf applies only to the fraction F of the integral.
Evaluation of F:
For nearby sources (z_s small), χ_local/χ → 1, so F → 1.
For distant sources, F → χ_local/χ_total. Taking Lf ≈ 13 kpc and χ_total ~ Gpc scales, F ~ 10⁻⁵ for cosmological baselines.
The key point is that F is determined by geometry, not fitted.
Mapping to H₀: The phase shift increases H by factor 1.084. The exact perturbation to 1/H is δf = 1/(1+ε) − 1 = −ε/(1+ε) ≈ −0.077 for ε = 0.084. To first order, δf ≃ −ε = −0.084. We use the first-order form throughout:
Δd_L/d_L ≈ −0.084 · F
The coupling strength equals the full αf shift (8.4%) in the linearized regime.
Applicability: This dilution formula applies to any line-of-sight distance integral:
— For time-delay lenses: d_Δt ∝ ∫(1+z)/H(z) dz → same dilution factor
— For standard sirens: d_L(z) = (1+z)χ(z) → same kernel structure
— For BAO+BBN: the ruler r_d is early-time, so only late-time integration is diluted (minimal effect)
Limiting behavior:
— Local limit (z_s ~ z_f): F → 1, full bias applies
— Long-baseline limit (z_s ≫ z_f): F → 0, bias dilutes
D. WHY THE MINIMUM STEP IS 2/120
The minimum observable step αf = 2/120 is inherited from MIT’s 120‑grid structure. The binary icosahedral group 2I (order 120) defines the full phase lattice on S3. Observation, however, samples only the 60 realized positions of the icosahedral group I, the 60R‑grid. Because the observer accesses every second point of the 120‑lattice, the smallest realized shift is 2/120 rather than 1/120.
Phase domain ...... |2I| = 120 ...... Full structure
Realization grid ...... |I| = 60R ...... Geometric measurement
E. WHICH MECHANISM APPLIES WHERE
Local ladder (Cepheid/SN, TRGB) ... Local anchors .... Calibration inheritance (full αf)
Megamasers ............. Local geometry ............ Calibration inheritance (full αf)
Time-delay lenses .... Geometric, late-time ...... Phase-domain averaging (partial αf)
Standard sirens ........ Geometric, late-time ...... Phase-domain averaging (partial αf)
BAO + BBN ................ Early-Universe ruler (r_d) Phase-domain averaging (minimal αf)
CMB .......................... Early-Universe physics .... Neither (samples α₀ directly)
The key distinction is not the redshift of the realized object, but where the absolute calibration is set.
Why CMB is distinct: CMB parameter inference is dominated by early-time physics (z ≈ 1100, before Local Group-scale environment exists). The CMB does not "average over" αf; it observes a phase epoch where αf had not yet developed. This is why CMB returns α₀ directly.
F. COHERENCE STRUCTURE
The phase field decomposes into two components:
αf = αf^env + δαf^micro
αf^env .......... Lf = v_c²/a₀ (~13 kpc for MW) .. Coherent; shifts H₀
δαf^micro .... Small, time-varying .................... Noise; averages out
The Hubble tension is attributed to αf^env, not micro-variability. The coherence scale Lf is derived in §VII.B.
Micro-variability bound: δαf^micro must be ≲ 10⁻⁴, or it would appear as seasonal or site-to-site scatter in precision ladder calibrations. No such scatter is observed. This is a stability requirement, not a derived bound.
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V. PREDICTIONS
The sharpest test is the discrete-vs-continuous fork (§VII.C).
A. PRIMARY: CALIBRATION-CLASS ORDERING
This is a classification prediction: sort methods by calibration anchor, not by the redshift of the realized object.
Local ladder ................. ~73 km/s/Mpc ........ Full αf inheritance
Late-time geometric ...... Intermediate .......... Partial αf (averaging)
Early-Universe ruler ..... ~67 km/s/Mpc ........ Minimal αf
Test: H₀ values should stratify by calibration source, not by target redshift.
B. SECONDARY: PHASE-INTERVAL DEPENDENCE
For geometric methods with non-local calibration, the distance bias maps to an H₀ bias. At fixed redshift and background cosmology, luminosity distance scales as d_L ∝ 1/H₀. (This proportionality is exact for d_L = cz/H₀ at low z; at higher z, it holds to leading order when other cosmological parameters are fixed.) If the measured distance is biased low by factor (1 − ε·F), the inferred H₀ is biased high. To first order in ε:
ΔH₀/H₀ = −Δd_L/d_L ≈ +ε · F
Therefore (to first order):
H₀^inferred(z) ≈ H₀^global(1 + ε · F(z))
where F(z) = χ_local/χ(z) is the coherent fraction at source redshift z.
At low z (small χ), the αf-biased segment dominates (F → 1) → higher inferred H₀. At high z (large χ), the bias dilutes (F → 0) → convergence toward global value.
Test: Time-delay lenses and standard sirens at different redshifts should show declining H₀ with increasing z (if calibration is non-local).
C. RAR SCATTER AS αf VARIATION
The Radial Acceleration Relation shows ~0.13 dex intrinsic scatter. If αf varies across galactic environments, the observed acceleration a_obs = g_bar × f(α) would inherit this variation.
Prediction: RAR residuals should correlate with environment metrics (local density, potential depth).
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VI. CANDIDATE MECHANISM: GRAVITATIONAL REFRACTIVE INDEX
A. FROM DENSITY TO PHASE
General relativity predicts that gravitational potentials affect the coordinate speed of light (Shapiro delay). This is equivalent to an effective refractive index n > 1 in gravity wells:
n_eff = 1 − 2Φ/c² ≈ 1 + 2|Φ|/c²
where Φ < 0 is the Newtonian potential. Throughout this paper, g denotes the magnitude of the gravitational acceleration (g = |dΦ/dr| ≥ 0), and Φ_rel denotes the gauge-invariant relative potential depth (Φ_rel ≥ 0); see Appendix A for precise definitions.
If realizations sample phase on S¹, then phase accumulation rate scales with optical path length. Local structure creates a coherent phase lag for all embedded observers. The trigger index T (defined precisely in §VII.A) is:
T = (2/c²Lf) ∫₀^{Lf} Φ_rel(l) dl
This is standard GR (Shapiro delay) reframed as phase accumulation on the mode spectrum. The trigger T varies continuously with potential depth; the phase field αf responds discretely (see §VII.A).
B. WHY THE EFFECT IS COHERENT
The Local Group potential is smooth on Mpc scales and embeds all local calibrators, so αf acts as a coherent shift, not noise.
C. THE AMPLITUDE PROBLEM: GR IS TOO SMALL
For the Local Group (M ~ 2.5 × 10¹² M☉, R ~ 1 Mpc):
|Φ|/c² ~ GM/(Rc²) ≈ 1.2 × 10⁻⁷
The required αf ≈ 0.017 is 10⁵ times larger than naive GR accumulation. This rules out continuous sourcing.
Resolution: The weak potential acts as a trigger selecting the lattice state, not as the amplitude source. Trigger ≠ amplitude source. The amplitude is fixed by lattice geometry (one step on the 60R-grid = 2/120 ≈ 0.017). This is analogous to phonon coupling in superconductivity: the coupling triggers the transition, but the gap is set by the condensate structure.
D. CONNECTION TO SCALE HIERARCHY
Continuous venue fields are subject to volume dilution: intensities scale as (√Ω)^{−n} where √Ω ≈ 10⁶¹. A percent-level local effect cannot arise from a diluted continuous mode. The observed αf ≈ 0.017 must therefore be a discrete state index: coordinates do not dilute; only intensities do.
The 120-grid inherited from |2I| on S³ provides this discretization. The observable grid has 60R positions (see §IV.D), making 2/120 the minimum visible step. The environment selects which lattice state is accessed; the lattice fixes how far the state shifts.
What remains open: The physical operator implementing the snap from T < Tc to T > Tc. Observational confirmation awaits data (§VII.C).
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VII. THE THRESHOLD PROGRAM
The quantized response requires a threshold: a condition that separates observers in the Fibonacci well (αf = 0) from those shifted to the first lattice step (αf = 2/120).
A. GAUGE-INVARIANT TRIGGER DEFINITION
Potentials carry an arbitrary additive constant; accelerations do not. The trigger must be defined in terms of potential differences.
Define the relative potential referenced to the outer edge of a coherence domain of scale Lf:
Φ_rel(l) ≡ Φ(l_out) − Φ(l) ≥ 0
The dimensionless trigger index is the mean relative depth:
T ≡ (2/c²Lf) ∫₀^{Lf} Φ_rel(l) dl
This is manifestly gauge-invariant.
The threshold condition is:
αf = (2/120) · Θ(T − Tc)
where Θ is the Heaviside step function and Tc is the critical trigger value.
B. DERIVED PARAMETERS
The a₀ transition as threshold: The quantized phase response activates when the observer's coherence domain contains a region with characteristic gravitational acceleration at or below the MOND transition scale a₀.
Coherence scale (derived): Define Lf as the galactocentric radius where the gravitational field enters the MOND regime: g(Lf) = a₀.
For flat rotation curves (g(r) = v_c²/r):
Lf = v_c²/a₀
For the MW (v_c ≈ 220 km/s, a₀ ≈ 1.2 × 10⁻¹⁰ m/s²):
Lf(MW) ≈ 13 kpc
(Appendix A uses Lf as the integration length for the ξ calculation; this identifies the characteristic radius with a segment length.)
Geometry factor (derived): The geometry factor ξ relates the mean potential depth to the characteristic scales:
ξ ≡ ⟨Φ_rel⟩ / (g(R) · Lf)
where g(R) = v_c²/R is the local gravitational acceleration. Explicit calculation from halo models (isothermal, NFW, Hernquist) gives:
ξ ≈ 0.46 ± 0.02
The full derivation is in Appendix A.
Trigger evaluation: From the definition T = 2⟨Φ_rel⟩/c² and the geometry factor:
T = 2ξ · g(R) · Lf / c²
For the MW (R ≈ 8 kpc, Lf ≈ 13 kpc, ξ ≈ 0.46):
T_MW ≈ 8 × 10⁻⁷
Threshold condition: The critical threshold Tc marks the minimum trigger depth for phase-field activation. From the coherence definition g(Lf) = a₀, we define the threshold scale for a given environment:
Tc ≡ 2ξ · a₀ · Lf / c² = 2ξv_c²/c²
Note: Tc depends on v_c and is therefore environment-specific. The threshold condition is T > Tc, which reduces to the dimensionless criterion Lf/R > 1 (see below). For the MW, Tc ≈ 5 × 10⁻⁷.
Numerical check: From the definitions, T = 2ξg(R)Lf/c² with g(R) = v_c²/R, and Tc = 2ξv_c²/c². The ratio:
T/Tc = g(R) · Lf / v_c² = (v_c²/R) · Lf / v_c² = Lf/R
For the MW: Lf/R ≈ 13/8 ≈ 1.6, so T_MW/Tc ≈ 1.6. The threshold condition T > Tc is equivalent to Lf > R: the observer must be inside the radius where g drops to a₀. This is a consistency identity confirming the coherence definition places typical disk galaxy observers above threshold.
The falsifiable content is different: The real test is whether the selected state is quantized and universal or continuous and environment-dependent.
C. THE FORK TEST (FALSIFICATION)
Local H₀ values fill 67-73 continuously ...... Quantized αf is wrong
Local ladders cluster near 73 independent of environment ...... Supports quantized response
Geometric methods drift toward 67 with increasing z ...... Supports phase averaging
Void-embedded calibrators return H₀ ≈ 70 ± 1 ...... Intermediate state exists; 2/120 step not fundamental
Environment-binned H₀ shows smooth gradient ...... Phase field is continuously sourced, not triggered
What would change my mind: A continuous environment gradient in H₀ with no discrete clustering would falsify the quantized-step hypothesis, even if the mean CMB-vs-local difference remains ~9%.
Operationally: Histogram H₀ by environment bin and test for bimodality vs continuous trend.
The MIT prediction is that all disk galaxies with developed halos select the same quantized state. The H₀ distribution should be bimodal (CMB-anchored at ~67, locally-calibrated at ~73), not a continuous spread.
Near-term test (without moving Earth):
— Ladder calibrations built from hosts in void-like vs overdense regions
— Late-time geometric methods (lenses/sirens) binned by environment
— Check whether inferred H₀ shifts systematically with density metrics
The test is operational now, even if precision is limited.
D. EDGE CASES: SHARPEST PROBES
The threshold analysis assumes a flat rotation curve. Environments where this fails:
1. Dwarf galaxies with rising rotation curves: Lf may not be well-defined
2. Void galaxies with truncated halos: If v_c profile doesn't flatten, the derivation doesn't apply
3. Very early epochs: Before halos develop flat curves, threshold crossing isn't guaranteed
These edge cases are where the quantized picture could show structure beyond "all or nothing."
E. THRESHOLD HIERARCHY
The 120-grid factors as 2³ × 3 × 5; the 120 → 60 projection reduces this to 60 = 2² × 3 × 5. Different prime factors might create distinct threshold tiers:
Factor 2 ......... Binary (fundamental) ... Any structure vs vacuum
Factor 3 ......... Intermediate ................. Moderate overdensity
Factor 5 ......... Fine ............................... Strong overdensity
If the threshold is the binary tier (factor of 2), then Tc is simply "nonzero structure exists." If higher tiers are active, multiple discrete H₀ values might be observable. This structure requires an operator that couples to these factors to be physically grounded.
F. EPOCH CONSTRAINT
The tension "turns on" when typical observer environments exceed the threshold.
Prediction: At redshifts where typical galactic environments have T < Tc, local and global H₀ inferences should agree.
z > z_threshold ..... T < Tc ..... Expected H₀ tension: None (local = global)
z < z_threshold ..... T > Tc ..... Expected H₀ tension: Present (local > global)
Test: H₀ inference from intermediate-z geometric methods (z ~ 0.5-2), binned by environment. A sharp onset would confirm quantized response; gradual drift would favor continuous.
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VIII. SUMMARY
C(α) = 2sin²(πα) .......... Derived (BC + sector + gauge + normaliz.)
α = α₀ + αf .................... Framework (MIT)
αf = 2/120 min. step ... MIT structural (120-grid → 60R-grid)
8.4% shift .................... Output (not fit to SH0ES)
Two mechanisms ....... Clarified (inherit vs average)
Phase-avg. integral ..... Derived
Coupling ε = 8.4% ....... Not free
Scale hier. (√Ω dilt.) .... Derived (coordinates don't dilute)
Lf = v_c²/a₀ .................. Derived (radius where g = a₀)
ξ ≈ 0.46 ........................ Derived (Appendix A)
Tc = 2ξv_c²/c² ............. Defined (environment-specific threshold)
T/Tc = Lf/R .................. Consistency identity
Fork test ....................... Falsifiable
Cal.-class ordering ...... Primary prediction
Phase-interval test ...... Secondary prediction
Edge cases .................. Sharpest probes
Threshold hierarchy .... Exploratory
The Hubble tension, within MIT, is the phase field manifesting. The required αf = 2/120 is inherited from MIT's 120-grid structure, and the scale hierarchy rules out continuous sourcing.
The falsifiable signature is discrete: local H₀ should cluster at quantized values (67, 73, ...), not vary continuously with environment.
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APPENDIX A: GEOMETRY FACTOR DERIVATION
A.1 SETUP
Place the observer coherence domain on a radial chord of length Lf, centered at galactocentric radius R:
r_in = R − Lf/2, r_out = R + Lf/2
Define the gauge-invariant relative potential:
Φ_rel(r) ≡ Φ(r_out) − Φ(r) ≥ 0
The mean depth across the chord is:
⟨Φ_rel⟩ ≡ (1/Lf) ∫_{r_in}^{r_out} (Φ(r_out) − Φ(r)) dr
A.2 DEFINITION OF ξ
The geometry factor is:
ξ ≡ ⟨Φ_rel⟩ / (g(R) · Lf)
where g(R) = v_c²/R is the characteristic local acceleration.
A.3 GENERAL IDENTITY
For a spherically symmetric potential with g(r) = |dΦ/dr| (the magnitude of the gravitational acceleration), the relative depth can be written:
Φ_rel(r) = ∫_r^{r_out} g(s) ds
(The potential is deeper at smaller r, so integrating g outward gives the depth difference.)
Therefore:
⟨Φ_rel⟩ = (1/Lf) ∫_{r_in}^{r_out} ∫_r^{r_out} g(s) ds dr = (1/Lf) ∫_{r_in}^{r_out} (s − r_in) g(s) ds
This yields:
ξ = (1/(g(R) Lf²)) ∫_{r_in}^{r_out} (r − r_in) g(r) dr
Physical interpretation: If g is constant across the segment, then ξ = 1/2 exactly. If g decreases outward (real halos), ξ shifts slightly below 1/2.
A.4 ISOTHERMAL SPHERE (ANALYTIC)
For a flat rotation curve with v_c = const:
g(r) = v_c²/r
The relative depth is Φ_rel(r) = ∫_r^{r_out} g(s) ds = v_c² ln(r_out/r), and the mean depth evaluates to:
⟨Φ_rel⟩ = (v_c²/Lf)(Lf − r_in ln(r_out/r_in))
Using g(R) = v_c²/R:
ξ_iso = (R/Lf²)(Lf − r_in ln(r_out/r_in))
For MW parameters (R = 8 kpc, Lf = 13 kpc):
ξ_iso ≈ 0.45
A.5 PROFILE COMPARISON
Evaluating the integral numerically for NFW and Hernquist profiles with MW-like parameters (v_c ≈ 220 km/s at R = 8 kpc, scale radii 15-25 kpc):
Isothermal ........ ξ = 0.448
NFW .................. ξ = 0.45-0.47
Hernquist .......... ξ = 0.44-0.47
Result: Across the standard halo family:
ξ ≈ 0.46 ± 0.02
The geometry factor is insensitive to profile choice in the Milky Way regime.
A.6 PHYSICAL INTERPRETATION
The threshold becomes a natural boundary between regimes:
Below Tc ...... No coherent a₀-scale region ..... Fibonacci well (αf = 0)
Above Tc ..... Contains a₀-scale region ........... Shifted (αf = 2/120)
The "MOND regime" and the "shifted H₀ regime" are the same phase transition.
Self-consistency: MIT derives a₀ from a different Fibonacci well (α₀ = 13/120 for galactic dynamics, vs 34/120 for H₀). The same phase structure that produces the MOND transition determines the threshold for phase-field activation.
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REFERENCES
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2. Planck Collaboration, "Planck 2018 results. VI. Cosmological parameters," Astron. Astrophys. 641, A6 (2020).
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4. S. S. McGaugh, F. Lelli, and J. M. Schombert, "Radial Acceleration Relation in Rotationally Supported Galaxies," Phys. Rev. Lett. 117, 201101 (2016).
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