ContinuumLimitCert
plain-language theorem explainer
A master certificate packaging every continuum-limit claim needed to pass from discrete RS ledger sites to a Lorentzian 3+1 manifold. Gravity and foundations workers cite it when asserting that signature, light cone, J-cost spatial metric, ADM flatness, Laplacian convergence, κ_RS, and D=3 are all in place. As a structure it has no proof body; the companion theorem fills each field with a named lemma.
Claim. A continuum-limit certificate is a record asserting: (i) Minkowski form $\eta=-t^2+x^2+y^2+z^2$ has Lorentzian signature and causal trichotomy; (ii) lightlike vectors obey $t^2=x^2+y^2+z^2$ (so $c=1$); (iii) $J_{\log}\varepsilon=\cosh\varepsilon-1$ is isotropic with $J''(1)=1$ and quartic remainder bound near $0$; (iv) unit-lapse ADM interval equals $\eta$; (v) the second-difference Laplacian converges at $O(a^2)$ for $C^4$ test functions, and lattice spacing $L/N$ can be arbitrarily fine; (vi) the zero-defect weak-field interval is $\eta$; (vii) $\kappa_{\mathrm{RS}}=8\varphi^5>0$; (viii) the eight-tick period equals $2^3$.
background
The module builds the zero-parameter bridge from discrete Recognition Science ledger sites to continuum Lorentzian spacetime: J-cost lattice → quadratic cost → Laplacian → Lorentzian interval → Minkowski flat limit → curved metric from defect → Einstein equations. Unlike the phenomenological ILG time-kernel, every geometric ingredient here is forced or derived.
The Minkowski form is $s^2=-t^2+x^2+y^2+z^2$. Causal type is the sign of that form (timelike / spacelike / lightlike). The recognition cost is $J(x)=(x+x^{-1})/2-1$, equivalently $J_{\log}(t)=\cosh t-1$ in log coordinates; its second derivative at the identity fixes the spatial metric scale. The eight-tick period is the $2^D$ hypercube period at the forced spatial dimension $D=3$ (DimensionForcing / T8). The RS gravitational coupling $\kappa_{\mathrm{RS}}$ comes from ZeroParameterGravity.
Upstream, $J$ is the unique cost solving the Recognition Composition Law (T5); $\varphi$ is the self-similar fixed point (T6); $c=1$ is one voxel per tick.
proof idea
No proof body: this is a structure whose fields are propositions. Instantiation is deferred to the companion theorem continuum_limit_certificate, which assigns each field a previously proved lemma (signature_temporal, signature_spatial_x/y/z, causal_trichotomy, light_cone_speed_limit, the J-log Taylor bound, second-derivative normalization of Jcost, ADM identity, standard $C^4$ Laplacian remainder, Archimedean resolution, weak-field flat limit, the ZeroParameterGravity coupling identity, and spatial_dim_is_3 : eight_tick = 2^3 by rfl).
why it matters
This is the Part 11 master certificate of ContinuumManifoldEmergence: the single object that says every ingredient for "N→∞ ledger sites with J-cost interactions yield a Lorentzian manifold" is present. Downstream, continuum_limit_certificate constructs an inhabitant, discharging the whole bundle at once.
It locks several forcing-chain landmarks into the gravity layer: Lorentzian signature from tick/voxel asymmetry; spatial metric from $J''(1)=1$ (RCL → $J=\cosh-1$); light cone from $c=1$ voxel/tick; $D=3$ via eight_tick $=2^3$ (T7/T8); and $\kappa_{\mathrm{RS}}=8\varphi^5$ from ZeroParameterGravity. That package is what lets later curved-metric / EFE steps start from a forced flat Lorentzian background rather than an assumed continuum spacetime.
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