Derivation 4: Spectral Tilt from Mode-Locking Structure#
Strip away “cost.” What remains is phase, frequency, coupling, and the rational numbers.
The problem with Derivation 02#
Derivation 02 modeled the CMB spectral tilt (n_s ≈ 0.965) as a synchronization cost gradient across scales, deriving P(k) ∝ 1/C(τ) and attributing the 3.5% deviation from scale-invariance to the slope of that gradient. It used P(k) ∝ 1/C(τ), where C is a synchronization “cost function.” A systematic scan of 7 cost function families (cost_function_scan.py) showed that every monotonically decreasing C(τ) produces positive running. This is a theorem, not a fitting problem: for any such C, the function ln(1/C(τ₀/k)) is convex in ln k.
The Planck central value for running is weakly negative (-0.005 ± 0.007). Standard slow-roll inflation also predicts small negative running.
The failure is ontological, not parametric. “Cost” carries optimization structure that obscures the mechanics. We don’t need it.
What a synchronizing system actually has#
A Kuramoto system has:
Phases θ_i(t) — the state of each oscillator
Natural frequencies ω_i — drawn from a distribution g(ω)
Coupling K — interaction strength
Order parameter r(t) — amplitude of the mean field
Collective frequency Ω(t) — frequency of the mean field
No “cost.” The dynamics are:
dθ_i/dt = ω_i + (K r / N) Σ sin(θ_j - θ_i)
The fluctuation spectrum#
Above the critical coupling K_c, the system synchronizes: r > 0. The oscillators split into a locked population (|ω_i - Ω| < Kr, phase-locked to the mean field) and a drifting population.
The fluctuations of the order parameter around its steady state have a spectrum. For the locked oscillators, the variance of phase fluctuations at natural frequency ω is:
⟨δθ²⟩(ω) ∝ g(ω) / (K r)²
This is a direct mechanical result: the fluctuation power at frequency ω is proportional to the density of oscillators at that frequency, divided by the restoring force (coupling × coherence) squared.
The mapping#
Identify:
wavenumber k ↔ natural frequency ω
P(k) ↔ ⟨δθ²⟩(ω) ∝ g(ω) / (K r(ω))²
The power spectrum of primordial perturbations is the fluctuation spectrum of the synchronized state. The spectral shape comes from g(ω) — the distribution of natural frequencies — modulated by r(ω), the scale-dependent order parameter.
This is simpler and more mechanical than Derivation 02. No cost function, no “power as inverse cost.” Just: how are the oscillators distributed in frequency, and how coherent is each sub-ensemble?
Why rational numbers#
In coupled oscillator systems, synchronization occurs preferentially at rational frequency ratios. This is mode-locking: two oscillators with frequencies in ratio p/q (integers) return to the same phase relationship after q cycles of one and p of the other. The phase relationship is periodic, so constructive interference persists. Irrational ratios never repeat — phase coherence drifts.
This produces Arnold tongues in the (K, ω₁/ω₂) plane: wedge-shaped regions of parameter space where oscillators lock to each rational ratio. As coupling increases, the tongues widen. The fraction of frequency space that is locked follows a devil’s staircase — a monotone function that is flat (constant, locked) on every rational, with measure-zero transitions between them.
The frequency distribution g(ω) is not smooth. It is structured by mode-locking: concentrated at rational multiples of the fundamental, depleted at irrationals. At sufficient coupling, g(ω) approaches a devil’s staircase.
The spectral tilt from g(ω)#
If g(ω) is a devil’s staircase (or an approximation to one), then:
n_s - 1 = d ln g / d ln ω + d ln(1/(Kr)²) / d ln ω
The tilt has two contributions:
The slope of g: how the density of locked oscillators varies with frequency. If more oscillators lock at lower frequencies (larger Arnold tongues for small-integer ratios), g decreases with ω, giving a red tilt.
The scale-dependent coherence: if r decreases at higher frequencies (smaller ensembles synchronize less completely), this adds to the red tilt.
Both contributions are red (n_s < 1), consistent with observation.
The running from the staircase#
The running dn_s/d ln k depends on the local curvature of ln g(ω) at the pivot scale. A devil’s staircase has:
Flat segments (locked plateaus): d ln g / d ln ω = 0, and the running is determined entirely by the r(ω) term.
Transition regions between plateaus: g changes rapidly, and the curvature can be positive or negative depending on which side of the transition you’re on.
Self-similar structure: the curvature at any scale depends on the local rational approximation structure.
Unlike a smooth cost function (which always gives positive running), the staircase structure can produce negative running if the pivot scale sits on the upper side of a transition — where g is concave (curving downward) as you approach the next locked plateau from below.
Physical interpretation: at the pivot scale, the system is transitioning from one mode-locked ratio to the next. The density of states is bending over (concave) as it approaches the new plateau. This concavity produces negative running.
The hierarchy of ratios#
Small-integer ratios have wider Arnold tongues (more robust locking):
1:1 > 2:1 > 3:2 > 4:3 > 5:4 > ...
The tongue width decreases roughly as 1/q for ratio p/q. This means:
The fundamental (1:1) dominates — most oscillators lock here
The octave (2:1) is next — the first harmonic
The fifth (3:2), fourth (4:3), etc. follow
The frequency distribution g(ω) has its largest concentration at the fundamental, with decreasing concentrations at each successive rational ratio. The envelope of these concentrations determines the large-scale shape of g, and therefore the spectral tilt.
The fine structure — the specific curvature between adjacent rationals — determines the running.
What this changes#
Quantity |
Old (Derivation 02) |
New |
|---|---|---|
P(k) |
1/C(τ) |
g(ω) / (Kr)² |
Tilt source |
Cost gradient |
Frequency distribution slope |
Running sign |
Always positive |
Depends on staircase position |
Free parameter |
x_* (pivot depth) |
Position on staircase |
Physical picture |
Cost minimization |
Mode-locking |
The “cost function” was a smooth approximation to the envelope of a devil’s staircase. The smoothing destroyed the curvature structure that determines the running sign.
Computational chain#
The following Python scripts implement and verify this derivation
(read in order — see INDEX.md for the full dependency graph):
Script |
Role |
|---|---|
Shows all monotonic cost functions give wrong running sign |
|
Circle map, Arnold tongues, devil’s staircase |
|
Zoom into 1/φ, identify it as the pivot |
|
Sample staircase on Stern-Brocot tree, not decimal grid |
|
Central result: exact φ² self-similarity at 1/φ |
|
The k↔Ω mapping: rate, amplitude, running |
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Non-Fibonacci rationals as overtone structure |
|
3D representations: Arnold surface, Poincaré disk, curvature |
|
The ψ-eigenmode and the alternating convergence |
Shared utilities: circle_map_utils.py