flowProduct_foldl
plain-language theorem explainer
Any ledger assembled from the empty list by repeated double-entry posts has flow product exactly 1 at every agent, once every posted ratio is nonzero. Cosmology proofs of σ conservation cite this as the fold-level invariant. The argument is induction on the number of posts, with the inductive step applying the single-post cancellation.
Claim. Fix an agent $a\in\mathbb{N}$ and a sequence of rational recognition events $f:\mathbb{N}\to\mathrm{QEvent}$ with every ratio nonzero. For every $n\in\mathbb{N}$, if $L_n$ is the ledger obtained by folding $n$ double-entry posts of $f(0),\ldots,f(n-1)$ onto the empty list, then the flow product of $L_n$ at $a$ equals $1$.
background
This module is the computable $\mathbb{Q}$-mirror of the real cosmogenesis dynamics in PreBigBang/FirstTick. A QEvent is a recognition event (source, target, rational ratio). Double-entry posting via addEvent prepends both the event and its reciprocal (swapped endpoints, inverted ratio).
The flow product at an agent is the product of per-event flow contributions; it is the multiplicative, kernel-checkable form of the conserved quantity $\sigma$, with conserved value $1$. The single-step lemma states that posting a paired event leaves the flow product unchanged at every agent, because the contributions multiply by $r\cdot r^{-1}=1$ whenever $r\neq 0$.
The local setting is an 8-tick ledger built by folding such posts; the present result lifts the one-step cancellation to an arbitrary finite fold from the empty ledger.
proof idea
Induction on $n$. The zero case is immediate: the empty fold is [], and flowProduct [] _ is an empty product, hence $1$.
For the successor, rewrite List.range (k+1) as range-append-singleton, simplify the fold, and obtain one extra addEvent of $f(k)$ on top of the $k$-fold ledger. Apply flowProduct_addEvent (using the nonzero-ratio hypothesis at $k$) to cancel that post, then finish with the inductive hypothesis.
why it matters
This is the fold-level conservation engine for the rational cosmogenesis ledger. Downstream, cosmogenesis_conserves is literally one application: the eight-tick cosmogenesis is the fold of length 8 of the cosmo-event sequence, and positivity of the seed supplies the nonzero-ratio side condition, so the flow product is $1$ at every agent after the full run.
In the Recognition framework this is the computable witness that double-entry posting preserves $\sigma$ through the eight-tick octave (T7). It closes the kernel-checked half of the module promise: conservation is a proved equality over $\mathbb{Q}$, not a runtime decide. The real-analytic cost story lives elsewhere; here the multiplicative invariant is what makes $\varphi$-emergence via the rational recurrence inspectable by #eval without breaking the ledger.
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