IndisputableMonolith.Foundation.Euler_Number_e_RS
Foundation module that packages the Recognition-Science Euler number together with a domain cost and a strictly positive canonical threshold. Supplies nonnegativity of the cost, positivity of the threshold, and an inhabited certificate. Anyone needing an RS-native base for exponential or log comparisons cites this file. The file is mostly definitions plus short algebraic positivity lemmas over the imported cost and constants layers.
claimThe module introduces a domain cost $C$, proves $C \ge 0$ and evaluates it at equality cases, defines a canonical threshold $\theta > 0$, and packages the RS Euler number $e_{\mathrm{RS}}$ with an inhabited certificate of the resulting structure.
background
Recognition Science works in RS-native units fixed by the Constants layer ($\tau_0 = 1$ tick, and the usual identifications $c=1$, $\hbar=\varphi^{-5}$). The Cost import supplies the J-cost $J(x)=(x+x^{-1})/2-1$ that appears throughout the forcing chain (T5 J-uniqueness and the Recognition Composition Law).
This module sits one level above those primitives. It defines a domain-level cost built from that J-data, records elementary evaluation and nonnegativity facts, and isolates a canonical positive threshold used to pin down an RS Euler number. The sibling names (domainCost, canonicalThreshold, EulerNumberERS, cert) mark the objects; no external physics input is required beyond the already-forced $\varphi$ and J.
The local setting is purely foundational: prepare a named, certified real that later layers can treat as the base of the natural exponential without re-deriving positivity or cost inequalities.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity arguments that reduce to the imported Cost lemmas and elementary real arithmetic. EulerNumberERS packages the resulting number; cert and cert_inhabited supply a trivial inhabited certificate so downstream code can depend on a single structure rather than a loose collection of facts. No deep tactic scripts or multi-step rewrites appear.
why it matters in Recognition Science
Gives the foundation layer a single named RS Euler number and a positive threshold tied to the domain cost, so later modules need not re-prove $C\ge 0$ or $\theta>0$. The forcing chain already fixes $J$ (T5) and $\varphi$ (T6); this file simply exports the corresponding exponential base in the same units. No downstream used_by edges are recorded yet, so the module is currently a leaf that closes a small definitional gap rather than a step inside a larger proved theorem. It does not itself touch T7 (eight-tick) or T8 ($D=3$), nor the mass ladder or $\alpha$ band.
scope and limits
- Does not derive e from the Recognition Composition Law or from T5–T8.
- Does not prove uniqueness of the Euler number beyond the packaged certificate.
- Does not connect e_RS to mass rungs, alpha, or eight-tick dynamics.
- Does not supply numerical bounds or decimal expansions.
- Does not claim any downstream theorem currently depends on this module.