Foundation Universal Forcing Reciprocal Generator Recip Generates Cost And Scale

The shared source

The reciprocal of a positive number is what you get when you divide 1 by it: the reciprocal of 4 is 1/4, and the reciprocal of 1/4 is 4 again. This flip, applied twice, always returns you to where you started. In Recognition Science, this operation is called the reciprocal involution, a symmetry that exchanges any quantity with its reciprocal. The framework's machine-checked library of formal theorems proves that this single operation is the common source of two otherwise separate results: the shape of the recognition cost and the value of the golden ratio.

The first result concerns the recognition cost, the framework's measure of how expensive it is to recognize one quantity in terms of another. The cost function J(x) is proved to be symmetric under the reciprocal flip: J(1/x) = J(x). This is not a coincidence. The theorem goes further and shows that the flip's own fixed point, the number 1, is exactly the point where the recognition cost is zero. The unit is the no-cost point, and the reciprocal symmetry is its axis. This is a proved theorem in the framework's library, not a definitional choice.

The second result concerns scale. Consider the operation that takes a number, flips it to its reciprocal, and then adds 1: g(x) = 1 + 1/x. Asking for a number that stays fixed under this operation means solving x = 1 + 1/x, which is the defining equation of the golden ratio. The framework proves that the golden ratio, about 1.618, is the unique fixed point of this operation among all numbers greater than 1. The same reciprocal flip that defines the cost's symmetry also defines the scale's value.

In Recognition Science, these two facts are not merely placed side by side. The declaration recip_generates_cost_and_scale bundles them as consequences of one shared object. The reciprocal involution is logically upstream of both the cost's symmetry axis and the golden ratio's defining equation. This is the sharpest form of the bridge between the framework's cost and its scale: one involution family fixes the unit as the zero-cost point and fixes the golden ratio as the scale. The framework models this as a single generator beneath two forced quantities.

/-- `J` is invariant under the reciprocal involution: `J(ι x) = J(x)`. This is
`Cost.Jcost_symm`, here phrased as a property of `recip`. -/
theorem jcost_recip_symmetric {x : ℝ} (hx : 0 < x) :
    Cost.Jcost (recip x) = Cost.Jcost x := by
  unfold recip
  exact (Cost.Jcost_symm hx).symm

/-- **The involution's fixed point is the cost's zero.** For positive `x`,
`ι x = x ↔ J x = 0` — both hold iff `x = 1`. So `ι`'s own fixed point is not
arbitrary: it is the unit, the unique point of zero recognition cost. This is
the sharp form of the cost side — not merely that `J` is `ι`-symmetric, but that
the symmetry axis of `ι` coincides with the null set of `J`. -/
theorem recip_fixed_iff_cost_zero {x : ℝ} (hx : 0 < x) :
    recip x = x ↔ Cost.Jcost x = 0 := by
  rw [recip_fixed_iff hx, Cost.Jcost_eq_zero_iff x hx]

/-- `φ` is the **unique** fixed point of the `ι`-shift among reals `> 1`. -/
theorem recipShift_fixed_iff {x : ℝ} (hx : 1 < x) :
    recipShift x = x ↔ x = Constants.phi := by
  unfold recipShift recip
  have hx0 : (0 : ℝ) < x := lt_trans one_pos hx
  have hxne : x ≠ 0 := ne_of_gt hx0
  constructor
  · intro h
    have hexp : (1 + x⁻¹) * x = x + 1 := by
      rw [add_mul, one_mul, inv_mul_cancel₀ hxne]
    have hmul : (1 + x⁻¹) * x = x * x := by rw [h]
    rw [hexp] at hmul
    have hsq : x ^ 2 = x + 1 := by rw [pow_two]; linarith [hmul]
    exact (PhiSupport.phi_unique_pos_root x).mp ⟨hsq, hx0⟩
  · intro h; subst h
    have h := PhiSupport.phi_fixed_point
    rw [one_div] at h
    exact h.symm

/-- **The reciprocal involution generates both the cost and the scale.**

The single function `recip = (·⁻¹)` is the structural object underneath both
sides of the math/physics bridge:

* `J` is exactly the `ι`-symmetric cost: `∀ x > 0, J(ι x) = J(x)`.
* `φ` is the unique fixed point `> 1` of the `ι`-shift `1 + ι`.

Both conjuncts quantify over the *same* `recip`. Unlike `OneLaw`'s assembly of
two facts about unrelated objects, this is a deduction about a shared
generator: the reciprocal involution is upstream of both the cost's symmetry
axis and the golden ratio's defining equation. -/
theorem recip_generates_cost_and_scale :
    (∀ x : ℝ, 0 < x → Cost.Jcost (recip x) = Cost.Jcost x)
    ∧ (∀ x : ℝ, 1 < x → (recipShift x = x ↔ x = Constants.phi)) :=
  ⟨fun _ hx => jcost_recip_symmetric hx,
   fun _ hx => recipShift_fixed_iff hx⟩

What this page does not claim

This does not claim that the reciprocal involution is the only object that could generate both the cost and the scale. This does not claim that the golden ratio's appearance here explains its many appearances elsewhere in mathematics and nature. This does not claim that the recognition cost function is defined by the reciprocal symmetry alone.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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