feat: dashboard Second Brain, essai 7 jours et vérification e-mail

Rendre le dashboard actionnable (inbox, peek, carte mentale), aligner la facturation sur l’essai 7 jours, et bloquer le login e-mail tant que l’adresse n’est pas confirmée.

Co-authored-by: Cursor <cursoragent@cursor.com>

memento-note/lib/ai/services/interactive-page-llm.service.ts

@@ -31,7 +31,7 @@ import {
 import { catalogForPrompt } from '@/lib/simulators'
 import thermoFixture from '@/lib/interactive-page/fixtures/thermo-page.json'
 
-const MAX_ATTEMPTS = 2
+const MAX_ATTEMPTS = 3
 
 // ── Shared helpers ───────────────────────────────────────────────────────────
 
@@ -124,19 +124,19 @@ function fixtureDemo(panelType: string): string | null {
 
 // ── 1. Page plan ─────────────────────────────────────────────────────────────
 
-const DEMO_KINDS = ['svg-scene', 'chart', 'heatmap-matrix', 'simulation', 'none'] as const
+const DEMO_KINDS = ['steps', 'svg-scene', 'chart', 'heatmap-matrix', 'simulation', 'none'] as const
 export type PagePlanDemoKind = (typeof DEMO_KINDS)[number]
 
 const planSectionSchema = z.object({
   title: z.string().min(1),
   goal: z.string().min(1),
   demoKind: z.enum(DEMO_KINDS),
-  demoGoal: z.string().optional(),
+  demoGoal: z.string().nullish(),
 })
 
 const pagePlanSchema = z.object({
   heroTitle: z.string().min(1),
-  heroSubtitle: z.string().optional(),
+  heroSubtitle: z.string().nullish(),
   overviewLead: z.string().min(1),
   overviewCards: z
     .array(
@@ -144,7 +144,7 @@ const pagePlanSchema = z.object({
         badge: z.string().min(1),
         title: z.string().min(1),
         body: z.string().min(1),
-        intent: z.enum(INTENT_IDS).optional(),
+        intent: z.enum(INTENT_IDS).nullish(),
       })
     )
     .min(INTERACTIVE_PAGE_CAPS.minOverviewCards)
@@ -170,7 +170,7 @@ SCHEMA:
   "heroTitle": string,          // the SUBJECT of the note, never a generic title
   "heroSubtitle": string,       // one sentence, autoportant
   "overviewLead": string,       // the central idea in ONE self-contained paragraph (may use $KaTeX$ inline)
-  "overviewCards": [ { "badge": string, "title": string, "body": string, "intent?": ${JSON.stringify(INTENT_IDS)} } ],  // 2–4 key concepts, small-caps badges (ex. PROBLEM / APPROACH / RESULT)
+  "overviewCards": [ { "badge": string, "title": string, "body": string, "intent?": ${JSON.stringify(INTENT_IDS)} } ],  // 3–4 key concepts, small-caps badges (ex. PROBLEM / APPROACH / RESULT)
   "sections": [ { "title": string, "goal": string, "demoKind": ${JSON.stringify(DEMO_KINDS)}, "demoGoal?": string } ]  // 2–5
 }
 
@@ -181,6 +181,7 @@ COUVERTURE (règle n°1):
 - If the content does not benefit from an interactive page, REFUSE: return { "error": "unsuitable_content", "reason": "…" } instead.
 
 MATCHING CONTENU → DÉMO (choose demoKind per section, "none" when no visual helps):
+- Dérivation, démonstration, résolution d'équation, calcul pas à pas (maths, physique) → "steps" — JAMAIS de svg-scene pour du contenu mathématique
 - Catégories × propriétés (comparatif, matrice, échanges) → "heatmap-matrix"
 - Loi / relation / courbe / évolution chiffrée → "chart"
 - Processus / flux / cycle / architecture → "svg-scene"
@@ -298,8 +299,14 @@ Block types (discriminated by "type"):
 - { "type": "table", "columns": string[], "rows": string[][], "caption?": string }   // every row.length === columns.length
 - { "type": "demo", "demo": InteractiveDemoV1, "caption?": string }
 - { "type": "sim", "sim": SimRef, "caption?": string }   // interactive simulation with sliders
+- { "type": "steps", "title?": string, "steps": [{ "tex": string, "rule?": string, "speak?": string }] (2–12), "caption?": string }   // step-by-step derivation: tex = KaTeX of the equation state, rule = transformation applied (short, e.g. "on sépare les variables")
 IntentId: ${JSON.stringify(INTENT_IDS)}
 
+BLOCK "steps" (Symbolab-style derivation) — MANDATORY for math/derivation content:
+- Each step = the equation state AFTER applying the rule; steps must chain logically (each follows from the previous).
+- rule = the transformation applied to reach THIS state (short verb phrase). speak = 1 sentence teacher narration.
+- FORBIDDEN: using "demo" svg-scene (boxes with arrows) for equations, derivations, proofs, or calculus content — always "steps" instead.
+
 SimRef — TWO forms:
 (A) CATALOG simulator (PREFERRED when the section matches one): { "simId": "<id from catalog>", "title?": string, "preset?": { "<paramId>": number }, "disclaimer?": string }
     → You ONLY pick the simId and preset values (from the note's real numbers within the allowed ranges). The app runs the simulation.
@@ -347,13 +354,16 @@ function buildSectionUserPrompt(
 
 SECTION_GOAL: ${section.goal}
 ${
-  section.demoKind === 'simulation'
-    ? `SIMULATION_REQUIRED: include ONE "sim" block. Prefer a catalog simulator if the section matches one (bind the note's real values into "preset"); otherwise "generic-formula" with the section's key relation.
+  section.demoKind === 'steps'
+    ? `STEPS_REQUIRED: include ONE "steps" block — the step-by-step derivation of this section's key result. Real equations from the note, logically chained, a short "rule" per step.
+STEPS_GOAL: ${section.demoGoal || section.goal}`
+    : section.demoKind === 'simulation'
+      ? `SIMULATION_REQUIRED: include ONE "sim" block. Prefer a catalog simulator if the section matches one (bind the note's real values into "preset"); otherwise "generic-formula" with the section's key relation.
 SIM_GOAL: ${section.demoGoal || section.goal}`
-    : section.demoKind !== 'none'
-      ? `DEMO_REQUIRED: include ONE "demo" block of kind "${section.demoKind}".
+      : section.demoKind !== 'none'
+        ? `DEMO_REQUIRED: include ONE "demo" block of kind "${section.demoKind}".
 DEMO_GOAL: ${section.demoGoal || section.goal}`
-      : `NO demo/sim block for this section — rich prose/formula/callout/stats/table only.`
+        : `NO demo/sim/steps block for this section — rich prose/formula/callout/stats/table only.`
 }
 
 EXCERPT_START
@@ -396,8 +406,10 @@ function validateSectionCandidate(
     hero: { kicker: 'CHECK', title: 'Section check' },
     sections: [
       typeof candidate === 'object' && candidate !== null
-        ? { id: sectionId, ...(candidate as Record<string, unknown>) }
+        ? { ...(candidate as Record<string, unknown>), id: sectionId }
         : candidate,
+      // schema requires ≥2 sections — inert filler for the wrap check
+      { id: 's99', title: '—', blocks: [{ type: 'prose', md: '—' }] },
     ],
   }
   const normalized = normalizeInteractivePageCandidate(wrapped, lang)