Physics Tachyon Condensation From Jcost

Tachyon condensation and the cost floor

In particle physics, a tachyon is a hypothetical field whose mass squared is negative, written m² < 0. That sign is not a curiosity; it signals an instability. The field's potential energy has a maximum at zero, so the field rolls away from the symmetric state and condenses into a lower-energy configuration. This process, tachyon condensation, appears in string theory when unstable branes decay, and the endpoint is a stable vacuum with no tachyon.

The standard picture is classical. A scalar field with a negative mass squared has a potential V(φ) = -½|m²|φ² + λφ⁴, so the origin is unstable and the minimum sits at a nonzero field value. The condensation threshold is the magnitude of the negative mass squared, |m²|. In the framework of Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 plays the role of a potential: it is zero when its argument equals 1, and it grows as the argument moves away from 1. The framework models the threshold as |m²| = J(φ) × M_string², where φ is the golden ratio and M_string is the string scale. That identification is a modeling choice, not a derived equality.

In Recognition Science, the ledger, a discrete record of recognition events, assigns a cost to any ratio of two masses. The module under discussion defines domainCost as J(m/e), the cost of the ratio between a field mass m and a reference energy e. The machine-checked library proves three general facts about this cost. First, it vanishes when m equals e, so the cost is zero exactly at the rest point. Second, it is nonnegative for positive inputs, so the cost never goes below zero. Third, the constant φ - 3/2 is positive, which is a small numerical fact about the golden ratio.

These three facts are true, but they are not specific to tachyons. The definition of domainCost does not mention a tachyon field, a string scale, or any physical mechanism. The docstring in the source file says plainly that the module proves nothing specific to this subject. What would make it a theorem about tachyon condensation is a definition of m and e in the tachyon's own terms, for example a definition that ties the reference energy to the string scale. That definition does not exist in the module.

The honest summary is this: the framework has a general cost function with the right shape for a stability analysis, and it has a research note that proposes to identify the condensation threshold with a cost-scaled mass squared. The formal content is a proof that the cost function has a minimum at unity and is nonnegative, which is a necessary condition for any such identification but not sufficient. The physical bridge, from the cost function to the tachyon's mass squared, remains open.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove that tachyon condensation occurs, nor that the threshold equals J(φ) times the string scale squared. The framework does not derive the tachyon's negative mass squared from first principles in this module.

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

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