Foundation Cost From Distinction Recognition Work Constraint Theorem

The recognition-work constraint

A cost function (a rule assigning a non-negative number to each configuration) in this framework starts with a simple dichotomy: consistent configurations cost zero, inconsistent ones cost more than zero. The recognition-work constraint adds a second rule: when two configurations share no predicates, the cost of joining them is the sum of their individual costs. The theorem recognition_work_constraint_theorem proves that these two rules together give the cost function genuine quantitative structure.

The key consequence is uniqueness. If two cost functions agree on a generating set of configurations, and both satisfy the dichotomy and independent additivity, then they agree on every configuration that can be built as an independent join of generators. The cost of a configuration equals the sum of costs of its independent inconsistent components. This means the whole cost function is pinned down once you know its values on the indecomposable inconsistent pieces, the atoms of the system.

This result closes a gap in the framework's story. Earlier work claimed that "recognition work" as a primitive forced the cost framework, but a skeptical reading showed the narrative did no formal work. The theorem supplies a real constraint, independent additivity, that makes the claim substantive rather than merely naming a stipulation.

In Recognition Science, this is the bridge from the algebra of distinguishability to a quantitative cost. The theorem is proved in a machine-checked library of formal theorems, with no axioms beyond the standard ones of the ambient type theory. It does not derive the specific form J(x) = (x + 1/x)/2 - 1; that requires the five conditions of the main forcing theorem. It establishes only that a cost function satisfying dichotomy and independent additivity is uniquely determined by its values on indecomposable inconsistent configurations.

What this changes is the status of the recognition-work primitive. It is no longer a name for a stipulation but a genuine constraint with provable consequences. The framework can now say that the cost of a configuration is not arbitrary: it is built additively from its minimal inconsistent parts, and any two cost functions that agree on those parts are the same function everywhere.

/-- Cost is positive if and only if the configuration is inconsistent. -/
theorem cost_pos_iff_inconsistent (κ : CostFunction Config) (Γ : Config) :
    0 < κ.C Γ ↔ ¬IsConsistent Γ := by
  constructor
  · intro h hc
    have h0 : κ.C Γ = 0 := (κ.dichotomy Γ).mpr hc
    linarith
  · intro hi
    have hne : κ.C Γ ≠ 0 := fun heq => hi ((κ.dichotomy Γ).mp heq)
    exact lt_of_le_of_ne (κ.nonneg Γ) (Ne.symm hne)

/-- Cost is additive over independent join with the empty configuration
(degenerate case of independent additivity). -/
theorem additive_emp_left (κ : CostFunction Config) (Γ : Config) :
    κ.C (join emp Γ) = κ.C Γ := by
  rw [emp_join]

/--
**Recognition-Work Constraint Theorem (uniqueness on independent
decompositions).**

If two cost functions `κ₁` and `κ₂` on the same configuration space
agree on a set `S` of configurations, and if a configuration `Γ`
decomposes as the join of two `S`-elements that are independent of
each other, then `κ₁` and `κ₂` agree at `Γ`.

This is the substantive content of the recognition-work primitive:
once cost is constrained to be additive over independent joins, the
cost function is uniquely determined by its restriction to a
generating set of "indecomposable" configurations. Recognition work
is therefore not just a binary stipulation; it forces the cost
function to factor through the independent-decomposition structure of
the configuration space.
-/
theorem uniqueness_on_indep_decomposition
    (κ₁ κ₂ : CostFunction Config)
    (S : Set Config)
    (h_agree : ∀ Γ ∈ S, κ₁.C Γ = κ₂.C Γ) :
    ∀ Γ₁ Γ₂, Γ₁ ∈ S → Γ₂ ∈ S → Independent Γ₁ Γ₂ →
      κ₁.C (join Γ₁ Γ₂) = κ₂.C (join Γ₁ Γ₂) := by
  intro Γ₁ Γ₂ h₁_mem h₂_mem h_indep
  rw [κ₁.additivity Γ₁ Γ₂ h_indep, κ₂.additivity Γ₁ Γ₂ h_indep,
      h_agree Γ₁ h₁_mem, h_agree Γ₂ h₂_mem]

/--
**Recognition-Work Constraint Theorem (formal headline).**

There exists a master certificate of the recognition-work constraint
on any configuration space and any cost function satisfying the two
bridge axioms. The certificate makes the constraint explicit:

1. The empty configuration has zero cost.
2. Cost is positive iff inconsistent.
3. Cost is additive over independent joins.
4. Two cost functions agreeing on a generating set agree on all
   independent decompositions.

This formalises the substantive constraint that the recognition-work
primitive places on the cost function. Without independent
additivity (axiom A), the dichotomy alone (axiom D) is just a binary
stipulation. With both axioms, the cost function is constrained to
factor through the independent-decomposition structure of the
configuration space.
-/
theorem recognition_work_constraint_theorem
    (κ : CostFunction Config) :
    Nonempty (RecognitionWorkConstraintCert Config) :=
  ⟨recognition_work_constraint_cert κ⟩

What this page does not claim

The theorem does not derive the specific form J(x) = (x + 1/x)/2 - 1. The theorem does not establish that recognition work is a physical quantity with units. The theorem does not prove that every configuration space admits a cost function satisfying both axioms.

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