Physics Quark Coordinate Reconciliation

Two coordinate schemes, one framework

In physics, a quark's mass can be calculated from a ladder of values. The ladder is built from the golden ratio, φ ≈ 1.618. Each rung on the ladder multiplies the mass by φ. The question is which rung a quark sits on. Two different answers emerged inside the Recognition Science framework. One answer places quarks on whole-number rungs. The other places them on quarter rungs. For a time, this looked like a contradiction that needed to be resolved by proving one answer right and the other wrong.

The framework's machine-checked library of formal theorems now settles the matter. It does not prove one scheme correct. Instead, it assigns each scheme to a different layer of modeling. The whole-number rungs belong to the core model. This layer claims to derive particle masses from geometry alone, with no fitting to experiment. The quarter-rung scheme belongs to the hypothesis layer. This layer aims for phenomenological accuracy, meaning it matches measured quark masses closely, especially for heavy quarks. The library records this separation as a formal theorem: the two schemes are not meant to be mathematically equivalent.

The core model places up-type quarks (u, c, t) on rungs 4, 15, and 21. Down-type quarks (d, s, b) use the same rungs. The mass formula multiplies a sector-specific yardstick by φ raised to a power. The hypothesis layer instead measures quark positions relative to the electron's structural mass. Its positions are quarter steps: top at 5.75, bottom at -2.00, charm at -4.50, strange at -10.00, down at -16.00, up at -17.75. The library proves these positions differ from the core rungs. It also proves that rounding the quarter positions to the nearest integer gives -18 for the up quark, not 4, showing the two schemes cannot be reconciled by rounding.

This resolution is intentional. The framework does not claim its core model achieves sub-percent accuracy for quark masses. The quarter-ladder is an exploratory refinement. It might later be derived from first principles, replaced by a better model, or promoted to the core layer if a geometric rationale emerges. For now, the library's formal record keeps the two schemes separate and clearly labeled. The gap in the framework's development, once flagged as an open problem, is now closed by layer separation rather than by equivalence.

/-- The quarter-ladder is a hypothesis, not core -/
theorem quarter_ladder_is_hypothesis :
    convention_layer .QuarterLadder = .Hypothesis := rfl

/-- The core model uses integer rungs -/
theorem core_uses_integer_rungs :
    convention_layer .IntegerRung = .Core := rfl

/-- The two conventions assign DIFFERENT rung values to the top quark.
    This formally demonstrates they are not equivalent coordinate systems. -/
theorem conventions_differ_top_quark :
    (core_up_rungs.t : ℚ) ≠ hypothesis_positions.top := by
  simp only [core_up_rungs, hypothesis_positions]
  norm_num

/-- The conventions also differ for charm quark -/
theorem conventions_differ_charm :
    (core_up_rungs.c : ℚ) ≠ hypothesis_positions.charm := by
  simp only [core_up_rungs, hypothesis_positions]
  norm_num

/-- The conventions also differ for bottom quark -/
theorem conventions_differ_bottom :
    (core_down_rungs.b : ℚ) ≠ hypothesis_positions.bottom := by
  simp only [core_down_rungs, hypothesis_positions]
  norm_num

/-- Nearest integer rungs from quarter positions -/
theorem nearest_int_positions :
    quarter_to_nearest_int hypothesis_positions.top = 6 ∧
    quarter_to_nearest_int hypothesis_positions.bottom = -2 ∧
    quarter_to_nearest_int hypothesis_positions.charm = -4 ∧
    quarter_to_nearest_int hypothesis_positions.strange = -10 ∧
    quarter_to_nearest_int hypothesis_positions.down = -16 ∧
    quarter_to_nearest_int hypothesis_positions.up = -18 := by
  simp only [quarter_to_nearest_int, hypothesis_positions]
  constructor <;> native_decide

What this page does not claim

The core integer-rung model achieves sub-percent accuracy for quark masses. The quarter-ladder scheme is derived from first principles. The two coordinate conventions are mathematically equivalent.

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

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