Physics Aharonov Ananda3 From Jcost

The geometric phase and its cost

The Aharonov-Anandan phase is a generalization of the better-known Berry phase. When a quantum system evolves slowly and returns to its original physical state, it can acquire a phase factor that depends only on the path taken in the space of states, not on how fast the journey happened. The Aharonov-Anandan phase extends this idea to arbitrary cyclic evolutions, removing the slow-evolution requirement. It is computed as an integral over the path, and its value is a geometric property of that path.

In the Recognition Science framework, the cost function J(x) = (x + 1/x)/2 - 1 measures the price of recognizing one value in terms of another. The module ledger (a discrete record of recognition events) defines a domain cost as J applied to the ratio of two parameters, m and e. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when the two parameters are equal, it is never negative for positive inputs, and a canonical threshold value, phi minus 1.5, is always positive. These are pure statements about the cost function itself; they contain no physics specific to the Aharonov-Anandan phase.

The research note attached to the module states the intended physical connection: at the coupling strength where J equals phi, the Aharonov-Anandan phase equals J times the geometric Berry phase, numerically about 0.118 times 2 pi times an integer winding number. This is a hypothesis, not a proved result. The formal proofs in the module do not define m and e in terms of any physical quantities, so the connection remains a research direction rather than a theorem.

What the module does establish, in plain language, is a set of consistency conditions any candidate physical interpretation must satisfy. If m and e are ever given physical meaning, the cost of recognition must vanish when they coincide, must stay nonnegative for positive values, and must sit above a positive threshold determined by the golden ratio. These are the rails on which a future derivation would run.

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]

What this page does not claim

The module does not prove the Aharonov-Anandan phase equals J times the geometric Berry phase. The module does not define m and e in physical terms. The module does not derive the value 0.118 from first principles.

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/AharonovAnanda3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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