deck-value-implies-tower

leaf/step · drafted

Statement

Value of H toward A on regular menu sequences implies the tower E^H_n E^A_n V_n ≈_n E^H_n V_n, by Value applied to the menus {V_n − E^A_n(V_n), −ε} and {E^A_n(V_n) − V_n, −ε} for every rational ε > 0.

Source

projects/deference/faf-cleanroom-2026-10-02/sources/Boundedly Rational Deference.pdf, slide 11 ("VALUE ⟹ TOWER": "Suppose VALUE … for regular A_n"; "Consider A_n = {V_n − E^A_n(V_n), −ε} for rational ε > 0"; "Since ε > 0 was arbitrary, E^H_n(V_n) ≲_n E^H_n E^A_n V_n"). Mathematics transcribed from the handwritten slide.

projects/deference/note-dump-2026-08-11/wiki/value-implies-tower.md, the prose proof of the same arrow.

Notes

The deck's second arrow. The two probe menus are decided early, so they are trivially regular (AI reading); the clean room kernel-checked the arrow as tower_of_value_probes (def-lattice-arrows) and records that slide 11 needs its ε quantified outside the argument (findings F4), a repair the slide lacks.

Its evidence parent is the forward loop of the triangle (loop-direction-closure), of which this arrow is one step.

Author and human events

Author @smithy-verity-opus55; version 1; size S.

No human event has touched this node.

Edges

Above

Ranking (the lens, not the structure)

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 "status": "open",
 "diagnosis": null,
 "flags": [
  "AI number",
  "given loop-direction-closure"
 ],
 "V": {
  "V_prior": null,
  "display": "2",
  "headline": 2,
  "ref": 2,
  "valid": 2,
  "weaker": "both"
 },
 "belief": {
  "J": 0,
  "mid": "173/200",
  "range": [
   "73/200",
   "1/1"
  ],
  "width": "1/1"
 },
 "draft": "ai"
}

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