IndisputableMonolith.Verification.T5.ConstraintForcing
Module defining the abstract recognition cost between two positive values as the symmetric log-ratio, with exchange invariance and vanishing on the identity. It packages the T5 constraint-forcing interface: cost-function predicates, reciprocal symmetry, unit normalization, and curvature-as-gauge facts. Cited by anyone reconstructing J-uniqueness from ledger axioms rather than postulating the closed form. Argument structure is definitional plus forced equalities from upstream ledger symmetry.
claimThe recognition cost between positive values is the symmetric log-ratio cost $C(x,y)$, required to be exchange-invariant ($C(x,y)=C(y,x)$) and to vanish on the identity ($C(x,x)=0$). Cost-function predicates encode reciprocal symmetry $F(x)=F(1/x)$, unit normalization, and the statement that curvature is pure gauge (cancels in dimensionless ratios).
background
T5 in the Recognition Science forcing chain is J-uniqueness: the unique admissible cost is $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). Before uniqueness, one needs a clean interface for what a cost function is allowed to be.
Upstream, LedgerCost derives two T5 constraints from the T3 ledger: double-entry forces reciprocal symmetry $F(x)=F(1/x)$, and supplies a no-go certificate that the remaining constraint cannot be ledger-derived alone. Constants supplies the RS-native tick $\tau_0$.
This module sits on that base. It introduces the abstract recognition cost as the symmetric log-ratio between two values, together with named predicates (exchange invariance, identity recognition zero, reciprocal symmetry, unit normalization) and the curvature-as-gauge normalization used when passing to dimensionless observables.
proof idea
Definition-and-interface module rather than a single deep theorem. It declares the recognition log-cost, the cost-function predicate, and the structural axioms (exchange invariance, identity zero). Forced facts such as reciprocal symmetry and unit normalization are obtained by applying the ledger-derived constraints from LedgerCost, then recording that curvature cancels in dimensionless combinations. No independent analytic uniqueness proof lives here; the module stages the hypotheses and forced equalities that later T5 uniqueness arguments consume.
why it matters in Recognition Science
T5 J-uniqueness is a landmark of the forcing chain (T0–T8): once J is fixed, $\phi$ is forced as the self-similar fixed point (T6), the eight-tick octave follows (T7), and $D=3$ (T8). This module is the constraint-forcing layer that turns ledger symmetry into the cost-function interface those uniqueness proofs need.
It does not itself close J-uniqueness; it feeds the verification stack that does. Downstream consumers are the T5 uniqueness and RCL (Recognition Composition Law) developments that assume an exchange-invariant, identity-normalized, reciprocally symmetric cost and then pin the closed form. The curvature-cancels facts keep gauge artifacts out of dimensionless predictions (mass ladder, $\alpha$ band).
scope and limits
- Does not prove full J-uniqueness or derive the closed form of J.
- Does not derive the Recognition Composition Law or force phi.
- Does not claim all T5 constraints come from the ledger; upstream already has a no-go.
- Does not fix dimension, eight-tick structure, or physical constants.
- Does not supply numerical bounds on alpha or mass-ladder rungs.
depends on (2)
declarations in this module (16)
-
def
RecognitionLogCost -
theorem
recognition_exchange_invariance_axiom -
theorem
recognition_identity_axiom -
def
IsCostFunction -
theorem
reciprocal_symmetry_forced -
theorem
unit_normalization_forced -
theorem
curvature_is_gauge_normalization -
theorem
curvature_cancels_in_dimensionless -
def
ExchangeInvariant -
def
ReciprocalSymmetric -
def
IdentityRecognitionZero -
def
UnitNormalized -
theorem
t5_constraints_are_forced -
theorem
t5_constraints_forced_from_ledger -
theorem
t5_constraints_imply_reciprocal_from_ledger -
theorem
t5_constraints_implies_reciprocal_from_ledger