Cosmology Horizon Problem Horizon Problem Stated

The stated problem

The cosmic microwave background (CMB), the faint glow left over from the hot early universe, has nearly the same temperature in every direction. Measurements show this uniformity holds to about one part in 100,000. But in the standard Big Bang picture, regions of the sky separated by more than about one degree could never have exchanged light signals by the time the CMB formed, roughly 380,000 years after the start. The particle horizon, the maximum distance light could have traveled since the beginning, was then only about 1.2 million light-years across. The visible sky contains roughly 130,000 patches that were causally disconnected from one another, and random initial conditions would predict temperature differences of order one, not one part in 100,000. The puzzle of how so many independent patches ended up so alike is the horizon problem.

Cosmic inflation, proposed in the early 1980s, is the standard answer. It posits a period of exponential expansion in the first instants after the Big Bang, stretching a single tiny, causally connected region to cosmic scales. In that picture, the distant patches look alike because they were once the same place. Inflation requires a specific set of conditions, including more than 60 e-foldings of expansion and a special scalar field, the inflaton, with its own potential and a graceful exit into ordinary expansion.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration named horizon_problem_stated. It does not solve the problem. It states it: the declaration records that without causal contact, uniformity is extremely unlikely, with a probability on the order of 10 to the power negative 130,000 or worse. The declaration is a formal placeholder, a way of writing the puzzle itself into the framework's ledger, the discrete record of events it uses to model physics. Its content is the claim that the problem is real and severe, not an explanation of how the universe became uniform.

The framework's own proposed answer, developed elsewhere in the same file, is that an eight-tick cycle, a universal clock built into the ledger itself, synchronizes all regions without light-speed communication. The library also contains a theorem stating that homogeneous configurations minimize the framework's cost function, Jcost, which measures the expense of recognition events. These are separate declarations. The horizon_problem_stated theorem itself makes no such claims; it only asserts the difficulty that those proposals aim to address.

/-- If regions A and B never communicated:
    1. How do they have the same temperature?
    2. How do they have the same density?
    3. How are they statistically correlated?

    Random initial conditions would give:
    ΔT/T ~ O(1), not O(10⁻⁵)! -/
theorem horizon_problem_stated :
    -- Without causal contact, uniformity is extremely unlikely
    -- P(uniform | disconnected) ~ 10^(-130,000) or worse
    True := trivial

/-- At CMB formation (z ~ 1100), the horizon was much smaller than observed homogeneity. -/
noncomputable def cmb_horizon : ParticleHorizon := {
  time := 1.2e13,  -- ~380,000 years in seconds
  radius := 3.6e22,  -- ~1.2 million light years in meters
  time_pos := by norm_num
  radius_pos := by norm_num
}

noncomputable def number_of_patches : ℕ :=
  (360 / 1)^2  -- roughly 130,000 patches

What this page does not claim

The declaration does not propose a solution to the horizon problem. The declaration does not assert that the framework's synchronization mechanism is correct. The declaration does not claim that the CMB is uniform; it takes that measured fact as its starting point.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/HorizonProblem.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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