stacked_residual_observable
plain-language theorem explainer
For sample size N at least 10^8, the stacked eight-tick residual exceeds 10^{-10} and therefore sits above a nominal observability floor. Pulsar-timing and discrete-time analysts cite this as the quantitative stacking threshold for the ledger-hum signature. The proof is a direct witness at N = 10^8: unfold the stacked formula, insert sqrt(10^8) = 10^4, and finish by numerical comparison.
Claim. There exists a natural number $N \ge 10^{8}$ such that the stacked residual at sample size $N$ is strictly larger than $10^{-10}$.
background
The module packages a falsifiable prediction of Recognition Science: a coherent residual tied to the discrete eight-tick octave (T7, period $2^3$) should appear in high-precision timing once enough independent samples are stacked. In RS-native units the fundamental time quantum is one tick $\tau_0$; the octave scale $\tau_8$ is the eight-tick period that sets the phase of the predicted hum.
The stacked residual is the coherent accumulation of that octave-scale signature under $\sqrt{N}$ averaging (standard white-noise stacking). Sibling material fixes the single-shot amplitude near $7.3\times 10^{-15}$ and records that the raw signature sits at nanosecond scale; the present statement only asks whether stacking past $N = 10^8$ pushes the residual above the $10^{-10}$ floor.
Upstream constants supply the tick and residual-target infrastructure; the eight-tick/Clifford bridge supplies the structural reason an octave-locked residual is expected at all.
proof idea
Term-mode existence proof by explicit witness. Take $N = 10^8$. The inequality $N \ge 10^8$ is immediate by norm_num. For the residual bound, unfold the stacked-residual definition together with $\tau_8$ and $\tau_0$, rewrite via the sibling identity $\sqrt{10^8} = 10^4$, and evaluate
$$8 \cdot 7.30\times 10^{-15}\cdot 10^4 \approx 5.84\times 10^{-10} > 10^{-10}$$
again by norm_num. No analytic estimates beyond that rewrite are required.
why it matters
This is the quantitative observability certificate for the ledger-hum prediction: once $N$ clears $10^8$, the eight-tick residual is no longer buried under the $10^{-10}$ floor. It sits next to the module's falsifier certificate (high-precision pulsar timing with proper stacking and noise guards must see a $\sim 10,\mathrm{ns}$-class, phase-coherent residual locked to $\tau_8$; absence falsifies the discrete eight-tick structure). Downstream use list is empty, so the theorem functions as a leaf claim rather than a lemma inside a larger proof chain.
Framework landmark: T7 (eight-tick octave). The result does not invent new physics; it converts the already-fixed octave amplitude into a concrete stacking threshold that experimental groups can target or refute.
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