module module moderate

`IndisputableMonolith.Foundation.RecognitionSignatureGauge`

Defines observational equivalence of states under a full family of recognition observables, i.e. equality of the induced recognition signature. It packages the T0-language gauge: two states are identified exactly when every admitted observable agrees. Downstream core material cites this for the signature quotient and completeness of scalar cost. The module is mostly definitions plus short equivalence and injectivity lemmas tying the signature kernel to the forced quotient.

claimTwo states $x,y$ have the same recognition signature under a family $\mathcal{F}$ of observables when $\forall f\in\mathcal{F},\, f(x)=f(y)$. The module treats this relation as the T0 observational gauge, relates it to the forced indistinguishability quotient, and records when a scalar cost is complete for that signature (its kernel equals signature equality).

background

At the T0 layer of the forcing chain, physics begins with a recognizer and Boolean (or scalar) observables rather than a pre-given manifold. A single Boolean bit is atomic but not complete: the full recognition signature is the joint evaluation of an admitted family of observables. States that agree on every member of the family are observationally equivalent.

The upstream quotient-selection material supplies the forced indistinguishability quotient of the recognizer. This module names the signature equality relation in that language and connects it to descent of observables and injectivity of the signature projection when the family separates points.

Sibling notions include a gauge certificate (the signature relation is the correct observational equivalence), scalar-cost completeness (the cost kernel coincides with signature equality), and elementary pair-of-bits state models used as test carriers for first- and second-bit families.

proof idea

Definition-heavy module with short supporting lemmas rather than a single deep proof. Same-signature is introduced as pointwise agreement on the observable family. Equivalence and iff lemmas relate that relation to the forced quotient and to descent of observables along the projection. Injectivity of the signature map is proved under a separating-family hypothesis. Completeness of a scalar cost is the equality of its kernel with signature equality; the certificate is a bundled Prop recording that the gauge story holds for the chosen data.

why it matters in Recognition Science

Feeds the public RecognitionCore aggregator at the T0/T4 recognizer and signature layer. Core exposes the recognizer, its indistinguishability quotient, the full recognition signature, and the completeness condition under which the signature determines all physically relevant states; this module is the gauge formalization of that signature equality.

Without a named observational equivalence, later claims that cost or geometry descends to physical states remain ambiguous. The completeness side (scalar cost kernel equals signature) is the precise sense in which "a single Boolean observable is atomic, not complete" is repaired by a separating family. Landmark contact is T0 of the forcing chain: the recognizer and signature before J-uniqueness (T5), phi (T6), eight-tick (T7), and D=3 (T8).

scope and limits

Does not construct the unique physical observable family from first principles.
Does not prove J-uniqueness, phi-forcing, eight-tick structure, or D=3.
Does not assert every scalar cost is complete; completeness is a named hypothesis.
Does not identify signature equality with metric or topological equality beyond observables.
Does not supply numerical constants (c, hbar, G, alpha) or mass-ladder claims.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.RecognitionCore module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.QuotientSelection module_import

declarations in this module (20)

def SameRecognitionSignature L41
theorem signature_forced_quotient_iff L46
theorem signature_observable_descends L52
theorem signature_projection_injective_of_separating L58
structure RecognitionSignatureGaugeCertificate L66
theorem recognitionSignatureGaugeCertificate_holds L76
def ScalarCostCompleteFor L87
theorem scalar_cost_kernel_eq_signature_of_complete L93
abbrev PairBoolState L103
def firstBit L106
def secondBit L109
def firstBitFamily L112
def pairBitFamily L115
theorem one_boolean_coordinate_not_complete L120
def firstBitScalarCost L132
theorem first_bit_scalar_cost_not_complete L137
theorem pairBitFamily_separating L144
theorem pairBitFamily_projection_injective L161
structure BooleanShadowCompletenessBoundary L166
theorem booleanShadowCompletenessBoundary_holds L185