---
author: 4NK Team
date: 2026-02-17
scope: v0/livre.md
type: fix
---

# Fix: LaTeX subscript corruption (`*{...}` / `*+`) in `v0/livre.md`

## Problem

Several mathematical expressions in `v0/livre.md` used a non-LaTeX marker `*` where a subscript `_` is expected, for example:

- `((x_t)*{t\ge 0})` instead of `((x_t)_{t\ge 0})`
- `x*{t+1}` instead of `x_{t+1}`
- `\mathbb{R}*+` instead of `\mathbb{R}_+`
- `t*{n+1}` instead of `t_{n+1}`

This corruption is not present in `v0/chapitre4.md` and breaks the intended mathematical notation.

## Impacts

- LaTeX rendering and readability are degraded (subscripts are no longer syntactically correct).
- Risk of semantic confusion in definitions (e.g., orbit definition and time accumulation recursion).
- Divergence between the source chapter (`v0/chapitre4.md`) and the compiled book (`v0/livre.md`) on notation.

## Cause

`v0/livre.md` contains occurrences of `*{...}` and `*+` in places where LaTeX subscripts are expected.

## Root cause

Unidentified transformation step or manual editing introduced `*` in place of `_` in parts of `v0/livre.md`.
At this stage, no build/compile rule has been identified that would legitimately require `*{...}` as an intermediate syntax.

## Fix applied

Corrections were applied **only** in `v0/livre.md` (source chapter files are unchanged):

- Chapter 2: sequence notation in the detailed repetition bound block
  - `((x_t)*{t\ge 0})` → `((x_t)_{t\ge 0})`
  - `x*{t+1}` → `x_{t+1}`
- Chapter 3: orbit definition
  - `\((x_t)*{t\ge 0}\)` → `\((x_t)_{t\ge 0}\)`
  - `\(x*{t+1}=f(x_t)\)` → `\(x_{t+1}=f(x_t)\)`
- Chapter 4: weighted clock section
  - `\mathbb{R}*+` → `\mathbb{R}_+`
  - `t*{n+1}` → `t_{n+1}`
- Chapter 5: subscripts in partitions and transmission definitions
  - `\{F_a\}*{a\in A}` → `\{F_a\}_{a\in A}`
  - `\bigsqcup*{a\in q(X)}` → `\bigsqcup_{a\in q(X)}`
  - `\mathcal{I}*\gamma` → `\mathcal{I}_\gamma`
  - `S*\gamma` → `S_\gamma`
- Chapter 6: subscripts in multi-segments and memory updates
  - `\{[i_p,j_p]\}*{p=1}^k` → `\{[i_p,j_p]\}_{p=1}^k`
  - `S*\gamma` → `S_\gamma`
  - `S'*t` → `S'_t`
  - `S'*{t+1}` → `S'_{t+1}`
  - `S*{m_t,t}` → `S_{m_t,t}`
- Chapter 7: subscripts in filters and norms
  - `\mathbf{1}*{M(a,b)\ge \theta}` → `\mathbf{1}_{M(a,b)\ge \theta}`
  - `F*\theta` → `F_\theta`
  - `\|M_{\mathcal{T}}\|*1` → `\|M_{\mathcal{T}}\|_1`
  - `\sum*{a,b}` → `\sum_{a,b}`
  - `\sum*{i\in V}` → `\sum_{i\in V}`
  - `n*{\max}` → `n_{\max}`
- Chapter 8: subscripts in semi-flow, basin entropy, and Markov metrics
  - `\{\Phi_t\}*{t\ge 0}` → `\{\Phi_t\}_{t\ge 0}`
  - `\Phi*{t+s}` → `\Phi_{t+s}`
  - `\{B(C_i)\}*{i=1}^K` → `\{B(C_i)\}_{i=1}^K`
  - `H*{\mathrm{bassins}}` → `H_{\mathrm{bassins}}`
  - `h|*C` → `h|_C`
  - `h|*{X\setminus (B\cup C)}` → `h|_{X\setminus (B\cup C)}`
  - `\mathbb{E}*x[\tau_B]` → `\mathbb{E}_x[\tau_B]`
  - `\sum*{y\in B}` → `\sum_{y\in B}`
  - `(X_t)*{t\ge 0}` → `(X_t)_{t\ge 0}`
  - `X*{t+\tau}` → `X_{t+\tau}`
- Chapter 9: subscripts in selection expectation
  - `\mathbb{E}*{S_w p}[w]` → `\mathbb{E}_{S_w p}[w]`
- Later occurrence (selection/fitness proposition)
  - `\mathbb{R}*+` → `\mathbb{R}_+`

## Affected pages

- `v0/livre.md`

## Deployment / regeneration considerations

- If `v0/livre.md` is regenerated by a build script (e.g., concatenation from `v0/chapitre*.md`), these manual fixes can be overwritten.
- To make the fix persistent, the generating pipeline should be audited to ensure it does not introduce `*` in place of `_`.

## Analysis / verification steps

- Search for corrupted markers in `v0/livre.md`:
  - `*{` (brace-form subscripts)
  - `*+`, `*\infty`, `*<digit>`, `*<letter>` (non-brace subscripts)
- Compare with corresponding source chapters (`v0/chapitre*.md`) to confirm whether the corruption is book-only or source-level.