Foundation Primitive Recognition Calculus Delta Real Width Real Bound

The width bound

A real number such as π is usually introduced as an infinite decimal, but a working mathematician often treats it as a nested stack of rational intervals. At stage n the stack offers a lower bound and an upper bound, both rational, and the next stage fits inside the previous one. The classical fact is that if the interval widths tend to zero, exactly one real number lies in every interval. This is how the real numbers are built from the rationals, and it is the same idea behind the framework's ledger, a discrete record of events that the framework uses to represent real quantities.

The declaration width_real_bound is the framework's version of the shrinking condition. It says that for every natural number n, the width of the n-th interval is at most 1/(n+1). The width is simply the upper bound minus the lower bound. Because the bound shrinks as n grows, the intervals close in on a single value. The framework proves that this value is unique: if some real number y lies in every interval, then y must equal the value the framework assigns to the stack. The proof runs by comparing y with the assigned value and using the width bound to show the difference is smaller than any positive number, hence zero.

This uniqueness result is what makes the whole construction usable. The framework defines addition, negation, and subtraction on these interval stacks, and it proves that the value operation respects all of them: the value of a sum is the sum of the values, and likewise for negation and subtraction. It also proves that every real number arises from some stack, so the construction does not leave any real numbers out. The width bound is the load-bearing condition that turns a merely nested collection of intervals into a genuine representation of a real number.

In Recognition Science, this is the primitive calculus on which later structure is built. The framework models real quantities as these shrinking interval stacks, and the width bound is what guarantees that the model is faithful. What the declaration does not claim is any physical content: it does not say that the universe actually keeps such a ledger, nor that recognition events are literally intervals. It is a piece of pure mathematics, a proved theorem about a formal construction. The physical interpretation is a separate step that the framework takes elsewhere, and this declaration is silent on it.

/-- A Delta-real: a nested family of rational intervals with width controlled by
`1/(n+1)` at precision `n`. The intended quantity is the unique real common to
all the intervals. -/
structure Protocol where
  approx : ℕ → RatInterval
  nested : ∀ n, (approx (n + 1)).Subset (approx n)
  width_bound : ∀ n, (approx n).width ≤ 1 / (n + 1)

/-- Squeeze: any real in every interval equals the value. -/
theorem value_unique (x : Protocol) (y : ℝ)
    (hy : ∀ n, x.lo n ≤ y ∧ y ≤ x.hi n) : y = x.value := by
  have hbound : ∀ n : ℕ, |y - x.value| ≤ 1 / (n + 1) := by
    intro n
    obtain ⟨h1l, h1r⟩ := hy n
    obtain ⟨h2l, h2r⟩ := x.value_mem n
    have hw := x.width_real_bound n
    rw [abs_le]
    constructor <;> linarith
  have : |y - x.value| = 0 := tiny_le_zero (abs_nonneg _) hbound
  have := abs_eq_zero.mp this
  linarith

/-- **Phase 1 headline.** The classical real line is the forgetful value of a
Delta-real protocol. Concretely:

1. every real is the value of a protocol (`value_surjective`);
2. the rational embedding `ofRat` has `value (ofRat q) = q`, so ℚ sits inside;
3. observational equality of protocols is exactly equality of value
   (`obsEq_iff_value`), so the presentation is faithful;
4. the protocol operations are native interval rules whose value is a ring
   homomorphism (`value_add`, `value_neg`, `value_sub`).

So ℝ is not a primitive completed object here; it is recovered as the value
display of refinement protocols, and nothing in analysis needs more than the
protocol that produces rational data to any requested precision. -/
theorem display_real_forgetful :
    Function.Surjective Protocol.value
      ∧ (∀ q : ℚ, (ofRat q).value = (q : ℝ))
      ∧ (∀ x y : Protocol, ObsEq x y ↔ x.value = y.value)
      ∧ (∀ x y : Protocol, (add x y).value = x.value + y.value)
      ∧ (∀ x : Protocol, (neg x).value = -x.value)
      ∧ (∀ x y : Protocol, (sub x y).value = x.value - y.value) :=
  ⟨value_surjective, value_ofRat, obsEq_iff_value, value_add, value_neg, value_sub⟩

/-- `value` is surjective onto ℝ. -/
theorem value_surjective : Function.Surjective Protocol.value :=
  fun r => ⟨canonical r, value_canonical r⟩

What this page does not claim

The declaration does not claim that physical reality is literally made of rational intervals. The declaration does not claim that the framework's real-number construction is the only possible one.

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

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