metaForcedArithmeticInvariance_self
Researchers cite this result when closing the reflexive loop of the Universal Forcing meta-theorem. It states that the canonical equivalence between the forced arithmetic of any realization and itself is exactly the identity on its orbit. The argument reduces the claim to the uniqueness property of the natural-number-object recursor for that orbit and verifies that the identity map satisfies the required zero and successor conditions.
claimLet $R$ be a meta-carrier (the type of all $LogicRealization.{0,0}$ instances). The forced-arithmetic-invariance map supplied by the universal-forcing meta-theorem satisfies $forcedArithmeticInvariance(R,R) = id_{R.Orbit}$.
background
The module UniversalForcingSelfReference records the structural self-reference of the Universal Forcing meta-theorem. MetaCarrier is the type of all $LogicRealization.{0,0}$ instances and therefore lives in Type 1. The meta-cost between two such instances is zero precisely when they are propositionally equal. The local setting is that the meta-theorem itself supplies the forced-arithmetic-invariance condition required by the Law-of-Logic shape, without instantiating the full orbit/step coherence axioms of the heavy $LogicRealization$ structure.
proof idea
Apply Equiv.ext to reduce to pointwise equality. Unfold metaForcedArithmeticInvariance and universal_forcing_via_NNO to expose the recursor. Construct the identity map on the orbit and verify it sends orbitZero to itself and commutes with orbitStep. Invoke the recursor_unique lemma from realizationOrbit_isNNO R, then use symmetry and simpa to obtain the required equality with the identity.
why it matters in Recognition Science
The theorem supplies the forced-arithmetic-invariance clause inside metaRealizationCert, thereby completing the three Aristotelian conditions (identity, non-contradiction, totality) on the meta-cost. It therefore shows that the framework is reflexively closed at the meta-level. The construction stays within the explicit Type-1 convention already fixed by UniversalForcing.NaturalNumberObject and avoids any claim to full heavy-structure instantiation.
scope and limits
- Does not prove Gödel-style self-reference of the meta-theorem.
- Does not establish the full orbit/step coherence axioms for the meta-carrier.
- Does not claim to instantiate the heavy LogicRealization structure at the meta-level.
- Does not address universe lifting beyond the fixed Type-1 convention.
formal statement (Lean)
119theorem metaForcedArithmeticInvariance_self (R : MetaCarrier) :
120 metaForcedArithmeticInvariance R R = Equiv.refl R.Orbit := by
proof body
Term-mode proof.
121 -- Both sides are the canonical NNO equivalence from R to itself,
122 -- which by uniqueness is the identity.
123 apply Equiv.ext
124 intro n
125 -- The NNO equivalence applied at n satisfies the universal property
126 -- of the recursor: it is the unique map R.Orbit → R.Orbit sending
127 -- orbitZero to orbitZero and intertwining orbitStep. The identity
128 -- is one such map. By uniqueness, the canonical equivalence equals
129 -- the identity.
130 unfold metaForcedArithmeticInvariance universal_forcing_via_NNO
131 IsNaturalNumberObject.equiv
132 simp only [Equiv.refl_apply, Equiv.coe_fn_mk]
133 -- Use the recursor uniqueness: the recursor with target (R.orbitZero, R.orbitStep)
134 -- is the identity.
135 have h_id_zero : (id : R.Orbit → R.Orbit) R.orbitZero = R.orbitZero := rfl
136 have h_id_step : ∀ k, (id : R.Orbit → R.Orbit) (R.orbitStep k) =
137 R.orbitStep ((id : R.Orbit → R.Orbit) k) := fun _ => rfl
138 have huniq := (realizationOrbit_isNNO R).recursor_unique
139 R.orbitZero R.orbitStep
140 (id : R.Orbit → R.Orbit) h_id_zero h_id_step n
141 -- huniq : id n = (realizationOrbit_isNNO R).recursor R.orbitZero R.orbitStep n
142 -- Goal : (realizationOrbit_isNNO R).recursor R.orbitZero R.orbitStep n = n
143 -- `id n` reduces to `n`.
144 simpa using huniq.symm
145
146/-! ## Meta-Realization Certificate
147
148The structural properties that a "meta-realization" of the Law-of-Logic
149framework would carry, all proved.
150-/
151
152/-- **Meta-Realization Certificate.**
153
154The Universal Forcing Meta-Theorem fits the Law-of-Logic structural shape:
155the meta-cost is identity, non-contradiction, total; and the meta-theorem
156itself supplies the forced-arithmetic-invariance condition that completes
157the structure.
158
159This is the reflexive-closure content of the framework. We do not claim
160to instantiate the heavy `LogicRealization` structure with all its
161orbit/step coherence axioms; instead, we record that every structural
162property the heavy structure would require has been independently
163proved. -/