IndisputableMonolith.Foundation.UniversalForcing.ReciprocalGenerator
Module packaging the reciprocal involution ι(x)=x⁻¹ on the positive reals as the dual-recognition symmetry of Recognition Science. It proves involutivity, J-cost symmetry, the link from fixed points to zero cost, and the recip-shift whose unique positive fixed point is φ. Downstream PublicSpine and the dual forcing surface import the certificate that reciprocal action pins unit and scale.
claimOn $\mathbb{R}_{>0}$, the map $\iota(x)=x^{-1}$ is an involution. The J-cost is symmetric under $\iota$, and $\iota(x)=x$ if and only if the cost vanishes. A reciprocal shift has unique positive fixed point $\varphi$, and the reciprocal generator certificate asserts that $\iota$ forces both the cost functional and the scale unit.
background
Recognition Science treats dual recognition as the map sending a positive scale factor to its inverse. The cost functional of the framework is the unique continuous solution of the Recognition Composition Law with $J(1)=0$ and $J(x)=J(x^{-1})$; explicitly $J(x)=(x+x^{-1})/2-1$ (T5). Thus reciprocity is not an optional symmetry: it is built into the cost.
This module sits in Foundation.UniversalForcing and imports Cost together with golden-ratio support lemmas (φ²=φ+1, the fixed-point identity φ=1+1/φ, uniqueness of the positive root of $x^2=x+1$). The local objects are the bare reciprocal recip, a shifted reciprocal used to isolate φ, and a Boolean-style certificate ReciprocalGeneratorCert that packages the generation claims for the public spine.
The theoretical setting is the dual forcing surface: reciprocity generates both the cost landscape and the self-similar scale, before the eight-tick and dimension steps of the full chain.
proof idea
Definition layer first: recip is $x\mapsto x^{-1}$ on positives. Involutivity and the fixed-point criterion are elementary field algebra. J-cost symmetry is the identity $J(x^{-1})=J(x)$ read off the closed form (or the RCL axioms). Zero-cost iff fixed under recip follows because $J(x)=0$ forces $x+x^{-1}=2$, hence $x=1$ on $\mathbb{R}_{>0}$.
The recip-shift is the standard rearrangement whose fixed-point equation is equivalent to $x=1+1/x$; uniqueness of the positive solution is the golden-ratio uniqueness lemma from PhiSupport. The certificate reciprocalGeneratorCert_holds assembles these facts into a single discharged Prop for PublicSpine consumption: reciprocal action pins the unit (cost zero only at 1) and the scale (φ as the self-similar fixed point).
why it matters in Recognition Science
PublicSpine imports this module as part of the dual forcing surface (the public δ-stratified counterpart of UnifiedForcingChain). Without a pinned reciprocal generator, the J-uniqueness step (T5) and the forcing of φ as self-similar fixed point (T6) lack their symmetry substrate.
The certificate is the honest packaging of “reciprocity generates cost and scale” for the Boolean/certificate spine kept for loop compatibility and pedagogy. Downstream δ-only tower material and classical continuum extension sit above this foundation; they do not re-prove involution or φ-uniqueness.
Framework landmarks touched directly: T5 (J from $x+x^{-1}$), T6 (φ fixed by the self-similar reciprocal shift), and the RCL symmetry $J(x)=J(x^{-1})$.
scope and limits
- Does not derive J-uniqueness from RCL; that lives in the T5 forcing chain.
- Does not prove eight-tick periodicity or D=3; those are later chain steps.
- Does not construct continuum measure or classicalExtension; PublicSpine handles δ stratification.
- Does not claim numerical values for c, ħ, G, or α; only unit and scale pinning via ι and φ.
- Does not address mass ladder rungs or Berry threshold; those sit outside this module.
used by (1)
depends on (2)
declarations in this module (12)
-
def
recip -
theorem
recip_involutive -
theorem
recip_fixed_iff -
theorem
jcost_recip_symmetric -
theorem
recip_fixed_iff_cost_zero -
def
recipShift -
theorem
phi_is_recipShift_fixed -
theorem
recipShift_fixed_iff -
theorem
recip_generates_cost_and_scale -
theorem
recip_pins_unit_and_scale -
structure
ReciprocalGeneratorCert -
def
reciprocalGeneratorCert_holds