Physics Superposition3 From Jcost

Superposition coherence

Quantum superposition is the ability of a particle to be in several states at once, such as passing through two slits simultaneously. The fragile part is preserving that dual existence intact. Any interaction with the surroundings, a stray photon, a jostling air molecule, can act like a measurement and force the particle to pick one path. The time over which this collapse happens is the decoherence time.

In 2026, physicists in the Recognition Science framework proposed a way to estimate that timescale from a single number. They defined a cost, a penalty for how hard it is for reality to preserve a discrete record of events, and applied it to the ratio of two masses. The cost function, written J(x) = (x + 1/x)/2 - 1, is zero when the masses are equal and grows as they diverge. The framework's library of formal theorems proves this cost is always nonnegative for positive inputs and that a certain threshold value, phi - 3/2, is positive. These are general facts about the cost function itself, not specific claims about superposition.

The superposition estimate emerges when the cost is combined with the framework's constants. The proposed coherence time is T_coherence = hbar * phi / (J(phi) * k_B * T), where hbar is the reduced Planck constant, phi is the golden ratio, k_B is Boltzmann's constant, and T is temperature. At room temperature (300 K), this gives about 600 femtoseconds, or 6e-13 seconds. That lands in the picosecond-to-femtosecond range where molecular decoherence is observed.

What the module actually proves is narrower than the estimate. The machine-checked library establishes three facts about the cost function: it vanishes when the two masses are equal, it is nonnegative for positive masses, and the threshold phi - 3/2 is positive. The leap from these facts to a physical decoherence time requires identifying what the masses m and e represent in a real quantum system. That identification is a research note, not a theorem. The framework's own documentation says the module proves nothing specific to superposition because the cost is defined without reference to one.

The honest takeaway: the framework offers a new way to think about decoherence, one where the cost of recognition sets the timescale. But the specific 600 femtosecond number is a proposal awaiting a definition of m and e in physical terms. What is proved is the mathematical backbone of the cost function, which any such definition would have to respect.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty Superpos3Cert := ⟨cert⟩

What this page does not claim

The 600 femtosecond decoherence time is not proved, only proposed. The framework does not derive quantum mechanics from the cost function. The module does not identify what the masses m and e represent physically.

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/Physics/Superposition3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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