SelfSimilarMemory
plain-language theorem explainer
Packages a self-similar memory kernel by a fractional exponent α with 0<α<1 and the two-scale ledger constraint α=(1-φ⁻¹)/2. Anyone deriving the ILG a-priori α prediction cites this bundle. It is a pure structure definition: the physics sits in the field constraints, not in a proof body.
Claim. A self-similar memory datum is a real number $\alpha$ such that $0 < \alpha < 1$ and $\alpha = (1 - \varphi^{-1})/2$, where $\varphi$ is the golden ratio. The equality encodes the two-scale ledger decomposition: a loop at scale $\ell s$ splits into sub-loops at $\ell$ and $\ell/s$, forcing $s=\varphi$ and equal half-contributions to the incomplete fraction $1-\varphi^{-1}$.
background
The module closes the paper's post-hoc gap: SPARC fits treated $(A,\alpha,r_0)$ as free, then noticed golden-ratio matches. Recognition Science claims $\alpha$ and $C$ are forced a priori from self-similarity; this file completes that chain.
The memory kernel is a fractional integral $\rho_{\mathrm{rec}}(t)=I_t^\alpha\rho_{\mathrm{baryon}}$. Under scale $\varphi$ it transforms as $\rho_{\mathrm{rec}}(\varphi\cdot t)\sim\varphi^\alpha\rho_{\mathrm{rec}}(t)$. The forcing chain (RCL $\to$ unique $J$ $\to$ self-similar ledger) already forces the scale ratio $\varphi$ via $\varphi^2=\varphi+1$ (T6 / PhiForcing).
Paper §II.F supplies the two-scale picture: total scale equals sum of sub-scales, each sub-loop contributes equally (factor $1/2$), and the incomplete recognition fraction is $1-\varphi^{-1}$. Hence $\alpha=(1-\varphi^{-1})/2\approx 0.191$.
proof idea
No proof body: this is a structure definition. The four fields are the data of a self-similar memory kernel. Positivity and $\alpha<1$ mark a proper fractional (not integer) memory. The field two_scale_constraint hard-wires the paper identity $\alpha=(1-\varphi^{-1})/2$ as a Prop on the carrier. Downstream theorems simply project that field (e.g. self_similarity_forces_alpha is intro M; have h := M.two_scale_constraint; simp).
why it matters
This is the carrier type for the a-priori $\alpha$ claim. The parent theorem self_similarity_forces_alpha states that every inhabitant has $\alpha$ equal to the locked constant $\alpha_{\mathrm{Lock}}=(1-\varphi^{-1})/2$. That feeds APrioriPrediction and the ILG a-priori certificate, separating prediction from SPARC validation.
In the module derivation chain: RCL $\to$ $J$-uniqueness (T5) $\to$ ledger self-similarity $\to$ $\varphi$ forced (T6) $\to$ memory kernel $\varphi$-structure $\to$ $\alpha=(1-\varphi^{-1})/2$ and $C=\varphi^{-2}$. The structure is the formal home of the last arrow on $\alpha$. It turns the paper's conditional FC5-style self-similarity step into a typed hypothesis that later certificates can discharge before any fit.
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