Foundation Gauge Symmetry3 From Jcost

Gauge symmetry from cost

Gauge symmetry is a principle in physics where a theory's equations stay the same under certain transformations that change the description but not the physical state. In the standard model, the electromagnetic force arises from a U(1) gauge symmetry, and the weak force from a larger SU(2) x U(1) symmetry. The foundation gauge symmetry3 file in the framework's machine-checked library takes the J-cost function, a measure of recognition cost, and defines a new function on it.

The file defines domainCost as J(m/e), where m and e are real numbers. It then proves three general facts about this function: it equals zero when m and e are equal, it is never negative when both inputs are positive, and the number phi minus 3/2 is positive. These are all true for any positive real inputs, because they follow from the properties of J-cost itself. The file packages these three facts into a structure called RSGaugeSymm3Cert and shows that this structure is inhabited, meaning the facts are consistent.

The library's own documentation states clearly what this file does not do. It proves nothing specific to gauge symmetry, because domainCost is defined as J(m/e) without any reference to a physical system. The paragraph about U(1) sigma x U(1) Theta from Noether charges, and the photon as a diagonal subgroup, is a research note recording where the idea was meant to go, not a result. What would make this file a theorem about gauge symmetry is a definition of m and e in that subject's own terms.

In plain language, the file establishes that a certain cost function has three basic mathematical properties: it vanishes at equality, it is nonnegative, and a related threshold is positive. These are prerequisites, not the gauge symmetry itself. The gauge symmetry story remains open, awaiting a bridge that connects the abstract cost function to the concrete charges and groups of particle physics.

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

This module does not prove that gauge symmetry exists or that U(1)_EM is the diagonal subgroup. The three proved facts are about a generic function and do not by themselves establish any particle physics. The research note about Noether charges is not a formal result.

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/Foundation/GaugeSymmetry3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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