Cost Unit From Minimality

The unit by minimality

In Recognition Science, a ledger (a discrete record of events) assigns a cost (a numerical charge) to each recognized distinction. The central cost function is J(x) = (x + 1/x)/2 - 1, forced by five plain conditions. The module UnitFromMinimality asks a natural question: among all powers of a base, which exponent gives the smallest cost? The answer, proved in the machine-checked library of formal theorems, is that the exponent 1 is the unique minimum. For any base x > 0 with x ≠ 1 and any integer n ≥ 2, the theorem jcost_lt_pow proves J(x) < J(x^n). In plain language, the first power is always cheaper than any higher power.

The same minimality selects the odd powers too. The theorem unit_is_selected_by_minimality proves that for any k ≥ 1, J(x) < J(x^(2k+1)). So among odd exponents, the exponent 1 is again the unique cheapest. The module then characterizes this uniqueness: the theorem anchor_iff_canonical proves that J(b^(2k+1)) = J(b) holds if and only if k = 0, meaning the only anchor where a higher odd power ties the base is the trivial exponent 1 itself. The theorem isLeast_iff_canonical states that the least odd power cost occurs exactly at k = 0. This is the unit: the base to the first power is the canonical minimal-cost distinction.

Why does this matter? It gives a principled origin for the unit of cost. The exponent 1 is not chosen by convention; it is forced by minimality over all powers. The theorem cost_of_the_first_distinction computes the actual value: J(2) = 1/4. So the first distinction, the jump from 1 to 2, costs exactly one quarter in the framework's native units. This is a concrete anchor for the cost scale, derived rather than postulated.

The module also proves a sharp contrast between discrete and continuous gauges. The theorem discrete_gauge_has_a_floor_and_continuous_gauge_does_not states that for integer exponents there is a floor (the exponent 1 is cheapest), but for real exponents there is no least member: for any l > 0, J(x^(l/2)) < J(x^l), so halving the exponent always reduces cost. The continuous gauge has no minimum, only a limit of zero as the exponent approaches 0. This is why the discrete structure matters: it is what makes a unit exist at all.

The zero exponent is special: exponent_zero_charges_nothing proves J(x^0) = 0, and exponent_zero_undercuts_everything shows this zero cost is always admissible. But this is the trivial case of no distinction, not a competing unit. The unit from minimality is the first nontrivial distinction, the exponent 1, with its cost of 1/4.

/-- **The unit is the cheapest member of the enlarged family.** For any base other than the
unit, `J` charges strictly less than the cost generated by any exponent above one. This is
`jcost_lt_odd_power` with the parity restriction removed, which is what the corrected
classification requires. -/
theorem jcost_lt_pow (x : ℝ) (hx : 0 < x) (hx1 : x ≠ 1) (n : ℕ) (hn : 2 ≤ n) :
    Jcost x < Jcost (x ^ n) := by
  rcases lt_trichotomy x 1 with hlt | heq | hgt
  · have hinv : 1 < x⁻¹ := one_lt_inv_iff₀.mpr ⟨hx, hlt⟩
    have h := jcost_lt_pow_of_one_lt x⁻¹ hinv n hn
    rwa [← Jcost_symm hx, ← jcost_pow_inv x hx n] at h
  · exact absurd heq hx1
  · exact jcost_lt_pow_of_one_lt x hgt n hn

/-- Packaging: on the discrete gauge, least cost is a selection principle. Every gauge
member other than `J` is strictly more expensive at every base that is not the unit. -/
theorem unit_is_selected_by_minimality :
    ∀ x : ℝ, 0 < x → x ≠ 1 → ∀ k : ℕ, 1 ≤ k → Jcost x < Jcost (x ^ (2 * k + 1)) :=
  fun x hx hx1 k hk => jcost_lt_odd_power x hx hx1 k hk

/-- **The cost of the first distinction.** The composition law fixes the output scale of
the cost, since rescaling a solution breaks it. The only freedom left was the input scale,
and minimality has just removed it. So this number is now determined rather than
stipulated: telling one from two costs a quarter. -/
theorem cost_of_the_first_distinction : Jcost 2 = 1 / 4 := by
  norm_num [Jcost]

/-- The two carriers, side by side. Least cost is a selection principle on the discrete
gauge and not on the continuous one, and completion to the real line is what turns the
first into the second.

The first clause is the countable carrier: `J` is strictly cheapest among the odd powers.
The second is the continuum: no scale is cheapest, because halving always undercuts. This
is a conjunction of the two facts, not a characterization of discreteness. -/
theorem discrete_gauge_has_a_floor_and_continuous_gauge_does_not :
    (∀ x : ℝ, 0 < x → x ≠ 1 → ∀ k : ℕ, 1 ≤ k → Jcost x < Jcost (x ^ (2 * k + 1)))
    ∧ (∀ x : ℝ, 0 < x → x ≠ 1 → ∀ l : ℝ, 0 < l → Jcost (x ^ (l / 2)) < Jcost (x ^ l)) :=
  ⟨fun x hx hx1 k hk => jcost_lt_odd_power x hx hx1 k hk,
   fun x hx hx1 l hl => no_least_gauge_member x hx hx1 l hl⟩

What this page does not claim

The module does not derive the value of the base x; it only shows that exponent 1 is minimal for any base. The continuous gauge having no least member does not imply the discrete gauge is the only possible structure. The unit of 1/4 is not claimed to be a fundamental physical constant outside the framework.

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/Cost/UnitFromMinimality.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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