prc_shrunk_certificate
plain-language theorem explainer
The δ-program certificate packages seven proved headlines: recognition is one primitive, the cost form is forced up to a free gauge unit, RS physics and operations sit below the continuum, the forcing chain is fed by the calibrated δ-cost, the φ/eight-tick/D=3 scaffold lives in the countable RS field, and distinction is mandatory except for degeneracy. Foundation auditors cite this as the shrunk end-to-end certificate. Proof is a structure inhabitant wiring seven prior headlines.
Claim. There is a certificate of seven headlines: (A) recognition is one primitive (comparison is derived from the act); (B) the cost form is forced up to a free positive gauge unit $c$, and log-curvature $1$ selects $J(x)=\cosh(\log x)-1$; RS physics and operations live in a countable field strictly below the continuum; the RS forcing chain is fed by the calibrated $\delta$-cost with $\varphi$ in that field; the scaffold $(\varphi^{\mathbb{Z}}$, eight-tick, $D=3)$ sits in the RS field; any foundation with a reflexive expression order is either degenerate or realizes $\delta$.
background
Primitive Recognition Calculus packages the δ program: derive the Recognition Science cost and forcing chain from a single recognition act and a discrete distinction carrier, without continuum axioms at the foundation. The certificate structure collects the load-bearing headlines into one small Prop.
Upstream Calibration (Axiom 3) normalizes the second derivative of $F(\exp t)$ at the origin to $1$, fixing curvature rather than a family. The gauge theorem shows the δ-forced cost leaves a faithful one-parameter family whose only invariant is log-curvature $c^2$, with $c=1$ selecting $J$. The weld theorem asserts that $J$-cost has unit log-curvature, that the golden ratio $\varphi$ lives in the countable RS field, and that the field is countable. The distinction dichotomy states that any foundation with a reflexive expression order is either degenerate or realizes $\delta$.
proof idea
Term-mode structure construction with no new reasoning. Each certificate field is filled by a named upstream headline: one-primitive from the comparison-is-derived result; cost-form free unit from the calibration-unit-is-a-gauge theorem; below-continuum from the RS-physics-below-continuum result; chain-fed-by-δ from the weld delta-cost-feeds-rs-chain; scaffold-in-field as the triple of φ-powers, eight-tick, and dimension membership in the RS field; operations-below-continuum from the exp/log field result; distinction-not-optional from the distinction dichotomy. Pure assembly of already-proved components.
why it matters
End-of-module certificate for the δ program: seven proved headlines, no axioms, no sorry. It closes the foundation audit trail that recognition is one primitive, the cost $J$ is forced (forcing-chain T5: $J(x)=(x+x^{-1})/2-1=\cosh(\log x)-1$), the unit is a free gauge, and the chain runs on a countable carrier below the continuum through φ (T6), the eight-tick octave (T7), and $D=3$ (T8). Distinction is mandatory except for the foundation that distinguishes nothing. No downstream dependents yet; the object stands as the shrunk certificate for external citation and framework completeness at the foundation layer.
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