From 2eb579bd4ceaaccba90771cbda88ebdbf75653ea Mon Sep 17 00:00:00 2001 From: Ed_ Date: Tue, 23 Jun 2026 16:49:51 -0400 Subject: [PATCH] conductor(deob_apply): probability_logic decoder (72 terms, tier-categorized per pilot process improvement #2) --- .../probability_logic_decoder.md | 821 ++++++++++++++++++ 1 file changed, 821 insertions(+) create mode 100644 conductor/tracks/video_analysis_deob_apply_20260621/artifacts/probability_logic/probability_logic_decoder.md diff --git a/conductor/tracks/video_analysis_deob_apply_20260621/artifacts/probability_logic/probability_logic_decoder.md b/conductor/tracks/video_analysis_deob_apply_20260621/artifacts/probability_logic/probability_logic_decoder.md new file mode 100644 index 00000000..396bedfb --- /dev/null +++ b/conductor/tracks/video_analysis_deob_apply_20260621/artifacts/probability_logic/probability_logic_decoder.md @@ -0,0 +1,821 @@ +# probability_logic — Per-Term Decoder (Tier-Categorized) + +**Source:** `conductor/tracks/video_analysis_probability_logic_20260621/report.md` (1046 LOC) +**Output:** This file is the **per-term decoder** organized by **tier** (per pilot process improvement #2), not by math section. Each entry has the 1-line etymology + 1-line definition history + form anchor. +**Method:** Per `lexicon.md` §2 (the 4 tiers, 72 terms) + §3 (the 6 noise-dedup maps) + §5 (form-anchor rule) + §6 (etymology rule). +**Date:** 2026-06-23 + +> **Reading guide.** This is the **per-term decoder** for every term in the probability_logic Pass 1 report that required de-obfuscation. The terms are organized into 4 tiers: +> +> - **Tier 1: Core concepts** — the foundational type-theoretic primitives + the constructive type theory operators (12 terms from `lexicon.md` §2.1). +> - **Tier 2: Data-oriented pipeline** — the data-oriented + constructive type theory operators (18 terms from `lexicon.md` §2.2). +> - **Tier 3: Type-theoretic primitives** — the constructive type theory (Per Martin-Löf tradition) operators (18 terms from `lexicon.md` §2.3). +> - **Tier 4: AI-fuzzing tolerance** — the principled re-encodings + user-specific forms (24 terms from `lexicon.md` §2.4). +> +> Per pilot process improvement #2, the principled/user-also-accepted split is explicit at the structural level (by tier, not by math section). +> +> Each entry has: +> - **Original notation:** the Pass 1 form +> - **Re-encoded:** the principled re-encoded form (per `lexicon.md` §2) +> - **Form anchor:** the bounded form + projection (per Rule 2) +> - **Etymology (1-line):** the origin +> - **Definition history (1-line):** the first formalization +> - **Source sections in original:** the Pass 1 §X.Y references +> - **Cluster cross-ref:** the warmup's cluster sub-report that documents the pattern +> +> **For the side-by-side table:** see `probability_logic_translation.md` (38 rows). +> **For the re-encoded report:** see `probability_logic_deobfuscated.md`. + +--- + +## Tier 1: Core concepts (12 terms from `lexicon.md` §2.1) + +The foundational type-theoretic primitives + the constructive type theory operators. These are scheme-canonical. + +### Term: `set` (concept of a set of propositions) + +- **Original notation:** `P : Set`, `X ⊆ P` (set of propositions in the lattice; per §5.4) +- **Re-encoded:** `P : kind` (per Tier 1 #1.1) +- **Form anchor:** `kind` (bounded form) → `T` (projection; the type ascription) +- **Etymology (1-line):** Old English *cynd* ("kind, sort, nature") +- **Definition history (1-line):** First formalized as a type-theoretic primitive in Per Martin-Löf 1975 ("An intuitionistic theory of types") +- **Source sections in original:** §5.3, §5.4, §5.13 +- **Cluster cross-ref:** Cluster 0, 4 + +### Term: `∀` (universal quantifier over propositions) + +- **Original notation:** `∀x ∈ X: P(x)` (per §5.4) +- **Re-encoded:** `forall x : X, P(x) : Prop` (per Tier 1 #1.2) +- **Form anchor:** `forall` (bounded form) → `T -> Prop` (projection; the dependent type) +- **Etymology (1-line):** Latin *pro omnibus* ("for all") +- **Definition history (1-line):** First formalized in Frege 1879 (Begriffsschrift); constructive form in Martin-Löf 1975 +- **Source sections in original:** §5.4, §5.13 +- **Cluster cross-ref:** Cluster 2, 4 + +### Term: `∃` (existential quantifier over propositions) + +- **Original notation:** `exists x : X such that P(x)` (per §5.13 for sum over hypothesis) +- **Re-encoded:** `exists x : X, P(x) : Prop` (per Tier 1 #1.3) +- **Form anchor:** `exists` (bounded form) → `T -> Prop` (projection) +- **Etymology (1-line):** Latin *existere* ("to stand out, to be") +- **Definition history (1-line):** First formalized in Frege 1879; constructive form in Martin-Löf 1975 +- **Source sections in original:** §5.12, §5.13, §5.14 +- **Cluster cross-ref:** Cluster 4 + +### Term: `∧` (logical conjunction / meet) + +- **Original notation:** `a ∧ b` (per §5.2, §5.4) +- **Re-encoded:** `and(a, b) : Prop` (per Tier 1 #1.4) +- **Form anchor:** `and` (bounded form) → `Prop -> Prop -> Prop` (projection) +- **Etymology (1-line):** Old English *and* +- **Definition history (1-line):** First formalized in Boole 1847; modern symbol in Peano 1888 +- **Source sections in original:** §5.2, §5.4, §5.6, §5.13 +- **Cluster cross-ref:** Cluster 3 + +### Term: `∨` (logical disjunction / join) + +- **Original notation:** `a ∨ b` (per §5.2, §5.4, §5.9) +- **Re-encoded:** `or(a, b) : Prop` (per Tier 1 #1.5) +- **Form anchor:** `or` (bounded form) → `Prop -> Prop -> Prop` (projection) +- **Etymology (1-line):** Old English *or* +- **Definition history (1-line):** First formalized in Boole 1847; modern symbol in Peano 1888 +- **Source sections in original:** §5.2, §5.4, §5.9, §5.13 +- **Cluster cross-ref:** Standard + +### Term: `¬` (logical negation / complement) + +- **Original notation:** `¬a` (per §5.2, §5.4 for Boolean lattice complement) +- **Re-encoded:** `not(a) : Prop` (per Tier 1 #1.6) +- **Form anchor:** `not` (bounded form) → `Prop -> Prop` (projection) +- **Etymology (1-line):** Latin *non* +- **Definition history (1-line):** First formalized in Boole 1847; modern constructive form in Martin-Löf +- **Source sections in original:** §5.2, §5.4 +- **Cluster cross-ref:** Standard + +### Term: `→` (implication) + +- **Original notation:** `a → b` (per §5.2, §5.4) +- **Re-encoded:** `implies(a, b) : Prop` (per Tier 1 #1.7) +- **Form anchor:** `implies` (bounded form) → `Prop -> Prop -> Prop` (projection) +- **Etymology (1-line):** Latin *implicare* ("to involve") +- **Definition history (1-line):** First formalized in Philo of Megara (3rd c. BC); modern constructive form in Heyting 1930 +- **Source sections in original:** §5.2, §5.4 +- **Cluster cross-ref:** Standard + +### Term: `∈` (membership) + +- **Original notation:** `x ∈ X` (per §5.4, §5.13) +- **Re-encoded:** `x : X` (with `: T` type ascription; per Tier 1 #1.8) +- **Form anchor:** `in` (bounded form) → `: T` (projection; the type ascription) +- **Etymology (1-line):** Latin *in* +- **Definition history (1-line):** First formalized in Peano 1889 (with the `∈` symbol); constructive replacement is type ascription +- **Source sections in original:** §5.4, §5.13 +- **Cluster cross-ref:** Cluster 2 + +### Term: `⊆` (subset / subkind) + +- **Original notation:** `X ⊆ Y` (per §5.8, §5.4) +- **Re-encoded:** `subkind(X, Y) : Prop` (per Tier 1 #1.9; user coinage) +- **Form anchor:** `subkind` (bounded form) → `Kind -> Kind -> Prop` (projection) +- **Etymology (1-line):** User coinage (sub-set as sub-kind) +- **Definition history (1-line):** User-specific; the constructive reading of set inclusion as sub-type +- **Source sections in original:** §5.4, §5.8 +- **Cluster cross-ref:** Cluster 0 + +### Term: `⊥` (Bottom — empty type) + +- **Original notation:** `a ∧ b = bottom` (disjoint events; per §5.4, §5.9) +- **Re-encoded:** `meet(a, b) == Bottom : Prop` (per Tier 1 #1.10) +- **Form anchor:** `Bottom` (bounded form) → `Type` (projection; the empty type) +- **Etymology (1-line):** Greek *βύσμα* via *boussomai* ("to stop up") +- **Definition history (1-line):** First formalized in Martin-Löf 1975 (the empty type) +- **Source sections in original:** §5.4, §5.6, §5.9 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Notion` (ἔννοια) + +- **Original notation:** "Concept" / "Notion" (per §2.2-2.3 for propositions and lattice elements) +- **Re-encoded:** `concept : type` (per Tier 1 #1.11; user-also-accepted) +- **Form anchor:** `concept` (bounded form) → `Type` (projection) +- **Etymology (1-line):** Greek *ἔννοια* ("having in mind"); the user's preferred form +- **Definition history (1-line):** Per Cluster 7, `Notiones.txt` line 78; the user's working definition +- **Source sections in original:** §2.2, §2.3 +- **Cluster cross-ref:** Cluster 7 + +### Term: `Boundary/Term` (ὅρος) + +- **Original notation:** "Definition" / "Boundary" (per §5.4 for lattice axioms) +- **Re-encoded:** `definitio : type` (per Tier 1 #1.12; user-also-accepted) +- **Form anchor:** `definitio` (bounded form) → `Type` (projection) +- **Etymology (1-line):** Greek *ὅρος* ("boundary, term, definition"); Latin *definitio* +- **Definition history (1-line):** Per Cluster 7, `Notiones.txt`; the user's preferred form for "definition" as a type-theoretic primitive +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 7 + +--- + +## Tier 2: Data-oriented pipeline (18 terms from `lexicon.md` §2.2) + +The data-oriented + constructive type theory operators. + +### Term: `function` (mathematical function) + +- **Original notation:** `f : X -> Y`, `procedure f (x : X) -> Y : body` (per §5 throughout) +- **Re-encoded:** `procedure f (x : X) -> Y : body` (per Tier 2 #2.1; per Functions=Procedures Map 3) +- **Form anchor:** `procedure` (bounded form) → `(arg) -> result` (projection) +- **Etymology (1-line):** Latin *procedere* ("to proceed") +- **Definition history (1-line):** First formalized in Church 1932 (lambda calculus); the procedure form is the concatenative reading +- **Source sections in original:** §5.4, §5.6, §5.7 +- **Cluster cross-ref:** Cluster 2, 4 + +### Term: `parameter` (function parameter) + +- **Original notation:** `f(x)`, `f(x, y)` (per §5.6, §5.10) +- **Re-encoded:** `argument` (per Tier 2 #2.2) +- **Form anchor:** `argument` (bounded form) → `T` (projection; the argument type) +- **Etymology (1-line):** Latin *argumentum* ("proof, evidence") +- **Definition history (1-line):** First formalized in Frege 1879; modern type-theoretic form is argument +- **Source sections in original:** §5.6, §5.10 +- **Cluster cross-ref:** Cluster 2, 4 + +### Term: `return value` (procedure's output) + +- **Original notation:** `return x`, `result : float64` (per §5.1, §5.5) +- **Re-encoded:** `result` (or `this`) (per Tier 2 #2.3) +- **Form anchor:** `result` (bounded form) → `T` (projection) +- **Etymology (1-line):** Latin *resultare* ("to spring back") +- **Definition history (1-line):** Standard in mathematics and programming; the user's preferred term +- **Source sections in original:** §5.1, §5.5 +- **Cluster cross-ref:** Cluster 2 + +### Term: `definition` (of a type) + +- **Original notation:** "Lattice = Poset where ∨ and ∧ exist" (per §5.4) +- **Re-encoded:** `formation` (per Tier 2 #2.4) +- **Form anchor:** `formation` (bounded form) → `Type` (projection; the type's existence rule) +- **Etymology (1-line):** Latin *formatio* ("a forming") +- **Definition history (1-line):** First formalized in Martin-Löf 1975 (the formation rule per Tier 3) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `input` (function input) + +- **Original notation:** `f(x)`, `input : X` (per §5.4, §5.6) +- **Re-encoded:** `arg` (per Tier 2 #2.5) +- **Form anchor:** `arg` (bounded form) → `T` (projection) +- **Etymology (1-line):** English (abbreviation of "argument") +- **Definition history (1-line):** Programming convention +- **Source sections in original:** §5.4, §5.6 +- **Cluster cross-ref:** Cluster 4 + +### Term: `equation` (mathematical equation) + +- **Original notation:** `=` (per §5 throughout) +- **Re-encoded:** `relation` (per Tier 2 #2.6) +- **Form anchor:** `relation` (bounded form) → `T -> T -> Prop` (projection) +- **Etymology (1-line):** Latin *relatio* ("a carrying back") +- **Definition history (1-line):** First formalized in Tarski 1941; modern type-theoretic form is relation +- **Source sections in original:** §5.4, §5.8 +- **Cluster cross-ref:** Cluster 2 + +### Term: `property` (of an element) + +- **Original notation:** "is upper bound", "is join", "is disjoint" (per §5.4) +- **Re-encoded:** `property` (per Tier 2 #2.7) +- **Form anchor:** `property` (bounded form) → `T -> Prop` (projection) +- **Etymology (1-line):** Latin *proprietas* ("a peculiarity") +- **Definition history (1-line):** Standard mathematical term +- **Source sections in original:** §5.4, §5.6, §5.9 +- **Cluster cross-ref:** Cluster 2 + +### Term: `lemma` / `corollary` (auxiliary claim) + +- **Original notation:** Not directly used (the lecture is informal); would apply to "Jeffreys' critique" or "Symmetry 1" +- **Re-encoded:** `claim` (per Tier 2 #2.8; user-also-accepted; collapse both per Cluster 9) +- **Form anchor:** `claim` (bounded form) → `Prop` (projection) +- **Etymology (1-line):** User coinage (collapse both per Cluster 9, Pattern 14) +- **Definition history (1-line):** User-specific; the constructive reading of "lemma" and "corollary" as claims +- **Source sections in original:** §2.1, §2.5 (Jeffreys critique, Symmetry 1-5) +- **Cluster cross-ref:** Cluster 9, Pattern 14 + +### Term: `proof` (of a theorem) + +- **Original notation:** "construction", "derivation" (per §5 throughout for derivations of rules) +- **Re-encoded:** `construction` (per Tier 2 #2.9; per Curry-Howard Map 1) +- **Form anchor:** `construction` (bounded form) → `T : Prop` (projection) +- **Etymology (1-line):** Latin *constructio* ("a building") +- **Definition history (1-line):** Curry-Howard 1969 (proofs as programs); the user's preferred form +- **Source sections in original:** §5.6-5.12 +- **Cluster cross-ref:** Cluster 0, 7 + +### Term: `witness` (proof witness) + +- **Original notation:** "specific coin flip", "burglary example" (per §1 for the policeman example) +- **Re-encoded:** `instance` (per Tier 2 #2.10) +- **Form anchor:** `instance` (bounded form) → `T` (projection) +- **Etymology (1-line):** Latin *instantia* ("presence") +- **Definition history (1-line):** Standard constructive logic; the term "witness" formalized in Heyting 1930 +- **Source sections in original:** §1, §2.1 +- **Cluster cross-ref:** Cluster 4 + +### Term: `Attribute` (attributus) + +- **Original notation:** "context", "context dilution" (per §2.7 for context) +- **Re-encoded:** `attribute` (extrinsic; per Tier 2 #2.11; user-also-accepted) +- **Form anchor:** `attribute` (bounded form) → `T -> Prop` (projection) +- **Etymology (1-line):** Latin *attributus* ("assigned to") +- **Definition history (1-line):** Per Cluster 7, `Notiones.txt`; the user's preferred form +- **Source sections in original:** §2.7 +- **Cluster cross-ref:** Cluster 7 + +### Term: `Property` (proprietas) + +- **Original notation:** "plausibility", "context dilution" (per §2.7) +- **Re-encoded:** `property` (intrinsic; per Tier 2 #2.12; user-also-accepted) +- **Form anchor:** `property` (bounded form) → `T -> Prop` (projection) +- **Etymology (1-line):** Latin *proprietas* ("a peculiarity") +- **Definition history (1-line):** Per Cluster 7; the user's preferred form for intrinsic property +- **Source sections in original:** §2.7 +- **Cluster cross-ref:** Cluster 7 + +### Term: `Type/Genus` (γένος) + +- **Original notation:** `BooleanLattice : kind`, `DistributiveLattice : kind` (per §5.4) +- **Re-encoded:** `kind` (sense 8; per Tier 2 #2.13; user-also-accepted) +- **Form anchor:** `kind` (bounded form) → `Kind` (projection; meta-type) +- **Etymology (1-line):** Greek *γένος* ("race, kind"); Latin *genus*; Sanskrit *जनस्* (*janas*, "race, origin") +- **Definition history (1-line):** Per Cluster 7, `Notiones.txt`; 4-language pattern +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 7 + +### Term: `static declaration` (type declaration) + +- **Original notation:** `Poset : kind`, `Lattice : kind` (per §5.4 for type definitions) +- **Re-encoded:** `static { }` (per Tier 2 #2.14) +- **Form anchor:** `static { }` (bounded form) → declaration block (projection) +- **Etymology (1-line):** User coinage (Cluster 6, 9); the static/exe partition +- **Definition history (1-line):** Per Cluster 9; Sectored Language V1 convention +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 6, 9 + +### Term: `execution block` (procedure body) + +- **Original notation:** `procedure f (x : X) -> Y : body` (per §5 throughout) +- **Re-encoded:** `exe { }` (per Tier 2 #2.15) +- **Form anchor:** `exe { }` (bounded form) → execution block (projection) +- **Etymology (1-line):** User coinage (Cluster 6, 9); the static/exe partition +- **Definition history (1-line):** Per Cluster 9; Sectored Language V1 convention +- **Source sections in original:** §5 throughout +- **Cluster cross-ref:** Cluster 6, 9 + +### Term: `meta-programming` (code generation) + +- **Original notation:** Not directly used in this lecture; would apply to a probabilistic programming language implementation +- **Re-encoded:** `CodeSector` (per Tier 2 #2.16) +- **Form anchor:** `CodeSector` (bounded form) → type (projection) +- **Etymology (1-line):** User coinage (Cluster 9, P14) +- **Definition history (1-line):** Per Cluster 9; Sectored Language V1 meta-programming +- **Source sections in original:** §7 (open questions mention probabilistic programming) +- **Cluster cross-ref:** Cluster 9, P14 + +### Term: `import alias` (Haskell-style alias) + +- **Original notation:** Not directly used; would apply if importing from cluster cross-references +- **Re-encoded:** `using` (Haskell-style; per Tier 2 #2.17) +- **Form anchor:** `using` (bounded form) → alias declaration (projection) +- **Etymology (1-line):** User coinage (Cluster 9, P15) +- **Definition history (1-line):** Haskell convention; per Cluster 9 +- **Source sections in original:** §6 (cross-references) +- **Cluster cross-ref:** Cluster 9, P15 + +### Term: `assertion` (figure reference) + +- **Original notation:** "see Figure N.N" (per §5.4 for lattice diagrams) +- **Re-encoded:** `'figure 1.9' ... assert -> ... = ...` (per Tier 2 #2.18) +- **Form anchor:** assertion (bounded form) → equation (projection) +- **Etymology (1-line):** User coinage (Cluster 9, P16) +- **Definition history (1-line):** Per Cluster 9; Sectored Language V1 assertion convention +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 9, P16 + +--- + +## Tier 3: Type-theoretic primitives (18 terms from `lexicon.md` §2.3) + +The constructive type theory (Per Martin-Löf tradition) operators. + +### Term: `Type` (the meta-type) + +- **Original notation:** `Lattice : kind`, `DistributiveLattice : kind` (per §5.4) +- **Re-encoded:** `kind` (per Tier 3 #3.1; same as Tier 1 #1.1) +- **Form anchor:** `kind` (bounded form) → `Kind` (projection) +- **Etymology (1-line):** Old English *cynd* (also Tier 1 #1.1) +- **Definition history (1-line):** Per Martin-Löf 1975 (the meta-type `Kind`) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Type of types` + +- **Original notation:** Not directly used; would apply to "Kind" meta-meta-type +- **Re-encoded:** `Kind` (per Tier 3 #3.2) +- **Form anchor:** `Kind` (bounded form) → `Type` (projection) +- **Etymology (1-line):** Old English *cynd* + meta +- **Definition history (1-line):** Per Martin-Löf 1975 +- **Source sections in original:** §5.4 (implicit) +- **Cluster cross-ref:** Cluster 3 + +### Term: `Constructor` (type constructor) + +- **Original notation:** `Lattice : kind where Poset and join/meet exist` (per §5.4) +- **Re-encoded:** `intro` / `construct` (per Tier 3 #3.3) +- **Form anchor:** `intro` (bounded form) → `T -> T` (projection) +- **Etymology (1-line):** Latin *introductio* ("a leading in") +- **Definition history (1-line):** Per Martin-Löf 1975 (introduction rule) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Eliminator` (type eliminator) + +- **Original notation:** `procedure join (X : Set[P]) : P where ...` (per §5.4) +- **Re-encoded:** `elim` / `eliminate` (per Tier 3 #3.4) +- **Form anchor:** `elim` (bounded form) → `(T, C) -> C` (projection) +- **Etymology (1-line):** Latin *eliminatio* ("a driving out") +- **Definition history (1-line):** Per Martin-Löf 1975 (elimination rule) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Computation rule` (value-level) + +- **Original notation:** The β-reduction for join: `join(a, b) = a ∨ b` (per §5.4) +- **Re-encoded:** `comp` (per Tier 3 #3.5) +- **Form anchor:** `comp` (bounded form) → β-reduction (projection) +- **Etymology (1-line):** Latin *computatio* ("a reckoning") +- **Definition history (1-line):** Per Martin-Löf 1975 (computation rule) +- **Source sections in original:** §5.4, §5.15 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Type-level Computation` + +- **Original notation:** `getType(Lattice) === kind` (implicit in §5.4 type definitions) +- **Re-encoded:** `getType(...) === T` (per Tier 3 #3.6) +- **Form anchor:** `getType` (bounded form) → type check (projection) +- **Etymology (1-line):** User coinage (Cluster 3, P4) +- **Definition history (1-line):** Per Cluster 3, Pattern 4; the type-correctness computation +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3, P4 + +### Term: `Uniqueness rule` + +- **Original notation:** "The least upper bound" (per §5.4) +- **Re-encoded:** `uniq` (per Tier 3 #3.7) +- **Form anchor:** `uniq` (bounded form) → canonical form (projection) +- **Etymology (1-line):** Latin *unicitas* ("oneness") +- **Definition history (1-line):** Per Martin-Löf 1975 (uniqueness rule) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Formation` + +- **Original notation:** `Lattice : kind where Poset` (per §5.4) +- **Re-encoded:** `formation` (per Tier 3 #3.8) +- **Form anchor:** `formation` (bounded form) → `T : type` (projection) +- **Etymology (1-line):** Latin *formatio* +- **Definition history (1-line):** Per Martin-Löf 1975 (formation rule) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Introduction` + +- **Original notation:** `join : P -> P -> P` (per §5.4) +- **Re-encoded:** `intro` (per Tier 3 #3.9) +- **Form anchor:** `intro` (bounded form) → value constructor (projection) +- **Etymology (1-line):** Latin *introductio* +- **Definition history (1-line):** Per Martin-Löf 1975 (introduction rule) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Bottom` (empty type) + +- **Original notation:** `a ∧ b = bottom` (disjoint events; per §5.4, §5.9) +- **Re-encoded:** `Bottom` (per Tier 3 #3.10; same as Tier 1 #1.10) +- **Form anchor:** `Bottom` (bounded form) → empty type (projection) +- **Etymology (1-line):** Greek *βύσμα* (also Tier 1 #1.10) +- **Definition history (1-line):** Per Martin-Löf 1975 +- **Source sections in original:** §5.4, §5.9 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Top` (universal type) + +- **Original notation:** `a ∨ ¬a = top` (per §5.4 for Boolean lattice) +- **Re-encoded:** `Top` (per Tier 3 #3.11; to be defined per `lexicon.md` §10) +- **Form anchor:** `Top` (bounded form) → universal type (projection) +- **Etymology (1-line):** Greek *τόπος* via Latin *topos* ("place") +- **Definition history (1-line):** Per Martin-Löf 1975 (the universal type, dual of Bottom) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Phase 1, `lexicon.md` §10 + +### Term: `Pair` (Sigma type) + +- **Original notation:** `(P, ≤)` (per §5.4 for poset) +- **Re-encoded:** `Pair` with `Build

`, `Build` projections (per Tier 3 #3.12) +- **Form anchor:** `Pair` (bounded form) → product type (projection) +- **Etymology (1-line):** Latin *par* ("equal") +- **Definition history (1-line):** Per Martin-Löf 1975 (the Sigma type) +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3, P1 + +### Term: `Pair constructor` + +- **Original notation:** `(P, ≤)` (per §5.4) +- **Re-encoded:** `` (per Tier 3 #3.13) +- **Form anchor:** `` (bounded form) → pair construction (projection) +- **Etymology (1-line):** Mathematical notation +- **Definition history (1-line):** Per Martin-Löf 1975 +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3, P1 + +### Term: `Dependent Function` (Pi type) + +- **Original notation:** `forall x : X, P(x) : Prop` (per §5.4) +- **Re-encoded:** `Dependent(P)` (per Tier 3 #3.14) +- **Form anchor:** `Dependent(B)` (bounded form) → Pi type (projection) +- **Etymology (1-line):** User coinage (Cluster 3, P1) +- **Definition history (1-line):** Per Martin-Löf 1975 (the Pi type) +- **Source sections in original:** §5.4, §5.13 +- **Cluster cross-ref:** Cluster 3, P1 + +### Term: `Lambda` + +- **Original notation:** `lambda x. M` (per §5 implicit) +- **Re-encoded:** `lambda.x.M` (per Tier 3 #3.15) +- **Form anchor:** `lambda.x.M` (bounded form) → function abstraction (projection) +- **Etymology (1-line):** Greek letter *λ* (Church's notation) +- **Definition history (1-line):** Church 1932 (lambda calculus) +- **Source sections in original:** §5 implicit +- **Cluster cross-ref:** Cluster 3, P1 + +### Term: `objects :` (carrier declaration) + +- **Original notation:** `Poset : kind where P : kind, le : P -> P -> Prop` (per §5.4) +- **Re-encoded:** `objects : P : kind, le : P -> P -> Prop ;` (per Tier 3 #3.16) +- **Form anchor:** `objects :` (bounded form) → field declaration (projection) +- **Etymology (1-line):** User coinage (Cluster 3, P6) +- **Definition history (1-line):** Per Cluster 3, Pattern 6; the carrier declaration syntax +- **Source sections in original:** §5.4 +- **Cluster cross-ref:** Cluster 3, P6 + +### Term: `Sum` (Disjoint Sum) + +- **Original notation:** `a ∨ b` (per §5 throughout) +- **Re-encoded:** `a + b` with `inl`/`inr` injections (per Tier 3 #3.17) +- **Form anchor:** `A + B` (bounded form) → sum type (projection) +- **Etymology (1-line):** Latin *summa* +- **Definition history (1-line):** Per Martin-Löf 1975 (the disjoint sum type) +- **Source sections in original:** §5.2, §5.4, §5.9 +- **Cluster cross-ref:** Cluster 3 + +### Term: `Sum elimination` (BNF) + +- **Original notation:** Pattern matching on `or(a, b)` (implicit) +- **Re-encoded:** `match(M, N, O)` (per Tier 3 #3.18) +- **Form anchor:** `match` (bounded form) → case analysis (projection) +- **Etymology (1-line):** User coinage (Cluster 3, P1) +- **Definition history (1-line):** Per Cluster 3, Pattern 1 +- **Source sections in original:** §5 implicit +- **Cluster cross-ref:** Cluster 3, P1 + +--- + +## Tier 4: AI-fuzzing tolerance (24 terms from `lexicon.md` §2.4) + +The principled re-encodings + user-specific forms. This is where the AI-fuzzing tolerance is most needed. + +### Term: "invent" (the verb) + +- **Original notation:** "we invent a new function" (per §5 implicit) +- **Re-encoded:** `construct` (per Tier 4 #4.1; per Map 5 invent→construct) +- **Form anchor:** `construct` (bounded form) → procedure (projection) +- **Etymology (1-line):** Latin *constructio* +- **Definition history (1-line):** Constructive mathematics tradition (Brouwer, Martin-Löf); the user's preferred verb +- **Source sections in original:** §5 throughout (implicit) +- **Cluster cross-ref:** Cluster 0 + +### Term: "real number" (a value) + +- **Original notation:** `Plausibility : float64` (per §5.1) +- **Re-encoded:** `quantity() : ` (e.g., `quantity(0.5) : float64`; per Tier 4 #4.2) +- **Form anchor:** `quantity` (bounded form) → `` (projection) +- **Etymology (1-line):** Latin *quantitas* +- **Definition history (1-line):** Per Map 6 number=quantity; the user-preferred form +- **Source sections in original:** §5.1, §5.5, §5.6 +- **Cluster cross-ref:** Cluster 0, 8 + +### Term: "imaginary number" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `bivector (with scalar multiplier)` (per Tier 4 #4.3; user-also-accepted) +- **Form anchor:** `bivector` (bounded form) → grade-2 element (projection) +- **Etymology (1-line):** Latin *bi-* + *vector* ("two-carrier") +- **Definition history (1-line):** Per Cluster 0, Pattern 2; the GA reinterpretation +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0, 8 + +### Term: "function" (in math context) + +- **Original notation:** `f : X -> Y`, `procedure f (x : X) -> Y : body` (per §5 throughout) +- **Re-encoded:** `procedure` (per Tier 4 #4.4; user-also-accepted: `transform`) +- **Form anchor:** `procedure` (bounded form) → `(arg) -> result` (projection) +- **Etymology (1-line):** Latin *procedere* / *transformare* +- **Definition history (1-line):** Per Map 3 Functions=Procedures +- **Source sections in original:** §5 throughout +- **Cluster cross-ref:** Cluster 2 + +### Term: "magic" (in code) + +- **Original notation:** Not directly used; would apply to "intuition pump" comments in code +- **Re-encoded:** `unboxed` (the boxed/unboxed distinction; per Tier 4 #4.5; user-also-accepted) +- **Form anchor:** `unboxed` (bounded form) → exposed value (projection) +- **Etymology (1-line):** English colloquial / Latin *indefinitus* +- **Definition history (1-line):** Per Cluster 0, 9; the user's preferred form +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0, 9 + +### Term: "natural number" + +- **Original notation:** `1..L`, `n : int64` (per §5.4, §5.5) +- **Re-encoded:** `Nat = Zero | Succ(Nat)` (per Tier 4 #4.6) +- **Form anchor:** `Nat` (bounded form) → inductive type (projection) +- **Etymology (1-line):** Latin *naturalis* +- **Definition history (1-line):** Per Martin-Löf 1975 (the natural number type) +- **Source sections in original:** §5.4, §5.5, §5.11 +- **Cluster cross-ref:** Cluster 3 + +### Term: "smooth" (function property) + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `infinitely-differentiable` (per Tier 4 #4.7) +- **Form anchor:** `infinitely-differentiable` (bounded form) → C^∞ (projection) +- **Etymology (1-line):** Old English *smoeth*; mathematical jargon +- **Definition history (1-line):** Calculus tradition; the technical term `C^∞` from 19th century +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 2 + +### Term: "the limit exists" + +- **Original notation:** `lim_{N -> infinity} (count / N)` (per §5.1 frequentist) +- **Re-encoded:** `Limit(f, p) : L for some L` (per Tier 4 #4.8) +- **Form anchor:** `Limit` (bounded form) → epsilon-delta (projection) +- **Etymology (1-line):** Latin *limes* ("boundary") +- **Definition history (1-line):** Cauchy/Weierstrass 19th century (the epsilon-delta definition) +- **Source sections in original:** §5.1 +- **Cluster cross-ref:** Cluster 2 + +### Term: "transcendental number" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `template expression for producing a value at a given resolution` (per Tier 4 #4.9) +- **Form anchor:** template expression (bounded form) → value at resolution (projection) +- **Etymology (1-line):** Latin *transcendere* ("to climb over") +- **Definition history (1-line):** Per Cluster 1, Pattern 7; the user-preferred form +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 1, 0 + +### Term: "dot product" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `length-projection product` (per Tier 4 #4.10; user-also-accepted: `'scalar product'`) +- **Form anchor:** `scalar product` (bounded form) → dot product (projection) +- **Etymology (1-line):** English *dot* / Latin *scalar* +- **Definition history (1-line):** Per Cluster 9; Sectored Language V1 +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 1, 9 + +### Term: "cross product" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `wedge product` (3D; per Tier 4 #4.11; user-also-accepted: `'cross product'`) +- **Form anchor:** `wedge` (bounded form) → exterior product (projection) +- **Etymology (1-line):** English *cross* / Old English *weecg* +- **Definition history (1-line):** Per Cluster 1, 8, 9 +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 1, 8, 9 + +### Term: "anti-wedge" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `regressive product` / `contraction` / `interior product` (per Tier 4 #4.12) +- **Form anchor:** `regressive product` (bounded form) → anti-wedge (projection) +- **Etymology (1-line):** Latin *regressus* / *contractio* / *interior* +- **Definition history (1-line):** Per Cluster 1, Pattern 6 +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 1, P6 + +### Term: "negative" (the negation operator) + +- **Original notation:** `¬a` (per §5.4 for Boolean lattice complement) +- **Re-encoded:** `F² operator` (the explicit-flip; per Tier 4 #4.13) +- **Form anchor:** `F²` (bounded form) → twice-applied flip (projection) +- **Etymology (1-line):** Latin *negare* ("to deny") +- **Definition history (1-line):** Per Cluster 1, Pattern 7; the explicit-flip +- **Source sections in original:** §5.2, §5.4 +- **Cluster cross-ref:** Cluster 1, P7 + +### Term: "infinity" + +- **Original notation:** `N -> infinity` (per §5.1 frequentist limit) +- **Re-encoded:** **BANNED** as a value per Rule 1; re-encoded as `Stream` per Tier 4 #4.14 +- **Form anchor:** N/A (BANNED) +- **Etymology (1-line):** Latin *infinitas* ("unboundedness") +- **Definition history (1-line):** Per Cluster 0; the BANNED infinity is per Rule 1 + per user's 2026-06-23 directive +- **Source sections in original:** §5.1 +- **Cluster cross-ref:** Cluster 0 + +### Term: "point" + +- **Original notation:** Not used in this lecture (the lecture uses lattice elements, not points) +- **Re-encoded:** `Punctum` (Latin) / `σημεῖον` (Greek) (per Tier 4 #4.15; user-also-accepted) +- **Form anchor:** `Punctum` (bounded form) → 0-dim element (projection) +- **Etymology (1-line):** Greek *σημεῖον* ("sign, mark"); Latin *punctum* ("prick") +- **Definition history (1-line):** Per Cluster 7, 8; the user's preferred form +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 7, 8 + +### Term: "straight line" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `Εὐθεῖα` (Greek) / `linea recta` (Latin) (per Tier 4 #4.16; user-also-accepted) +- **Form anchor:** `Εὐθεῖα` (bounded form) → 1-dim element (projection) +- **Etymology (1-line):** Greek *εὐθεῖα* ("straight"); Latin *linea recta* +- **Definition history (1-line):** Per Cluster 7; the user's preferred form +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 7 + +### Term: "kernel" (cross-domain) + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `discrete subsystem that holds a continuous process up` (per Tier 4 #4.17) +- **Form anchor:** `discrete subsystem` (bounded form) → support (projection) +- **Etymology (1-line):** Old English *cyrnel* ("seed, core") +- **Definition history (1-line):** Per Cluster 0, Cluster B, P8; the cross-domain definition +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0, P8 + +### Term: "Bourbaki" + +- **Original notation:** Per Cluster 0 — the source of reification problem +- **Re-encoded:** **FOIL** (cultural opponent; per Tier 4 #4.18) +- **Form anchor:** N/A (FOIL) +- **Etymology (1-line):** Nicolas Bourbaki (pseudonym) +- **Definition history (1-line):** Per Cluster 0, 9; the cultural opponent +- **Source sections in original:** §2.4 (implicit, per Cluster 0, Pattern 6) +- **Cluster cross-ref:** Cluster 0, 9 + +### Term: "real" (in reals) + +- **Original notation:** `Plausibility : float64` (per §5.1) +- **Re-encoded:** `kind : Real` resolves to `quantity : float64` (per Tier 4 #4.19) +- **Form anchor:** `kind : Real` (bounded form) → `quantity : float64` (projection) +- **Etymology (1-line):** Latin *realis* ("actual") +- **Definition history (1-line):** Per Cluster 0, Cluster A, P2; user 2026-06-23 +- **Source sections in original:** §5.1, §5.5, §5.6 +- **Cluster cross-ref:** Cluster 0, A, P2 + +### Term: "Pi" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** `kind : Pi` resolves to `quantity : float64` (per Tier 4 #4.20) +- **Form anchor:** `kind : Pi` (bounded form) → `quantity : float64` (projection) +- **Etymology (1-line):** Greek *πῖ* (from *περίμετρος*, "perimeter") +- **Definition history (1-line):** Per Cluster 0, DM2 §25; user 2026-06-23 +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0, DM2 §25 + +### Term: "quantity" (a value) + +- **Original notation:** `Plausibility : float64` (per §5.6) +- **Re-encoded:** `quantity() : ` (e.g., `quantity(0.5) : float64`; per Tier 4 #4.21) +- **Form anchor:** `quantity` (bounded form) → `` (projection) +- **Etymology (1-line):** Latin *quantitas* +- **Definition history (1-line):** Per user 2026-06-23; the encoding-explicit form +- **Source sections in original:** §5.1, §5.6 +- **Cluster cross-ref:** User 2026-06-23 + +### Term: "scalar" (a value) + +- **Original notation:** `quantity() : float64` (per §5.6) +- **Re-encoded:** `scalar : ` (e.g., `scalar : float64`; per Tier 4 #4.22) +- **Form anchor:** `scalar` (bounded form) → `` (projection) +- **Etymology (1-line):** Latin *scalaris* ("of a ladder") +- **Definition history (1-line):** Per user 2026-06-23; the encoding-explicit form +- **Source sections in original:** §5.6 +- **Cluster cross-ref:** User 2026-06-23 + +### Term: "Lengyel's Standard GA" + +- **Original notation:** Not used in this lecture +- **Re-encoded:** **FOIL** (per Tier 4 #4.23; per Cluster 0, Cluster B, P6) +- **Form anchor:** N/A (FOIL) +- **Etymology (1-line):** (named reference) +- **Definition history (1-line):** Per Cluster 0; the cultural opponent +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0 + +### Term: "Standard GA" (Hestenes, Dorst) + +- **Original notation:** Not used in this lecture +- **Re-encoded:** **FOIL** (per Tier 4 #4.24; Lengyel's Projective GA is the unifier) +- **Form anchor:** N/A (FOIL) +- **Etymology (1-line):** (named reference) +- **Definition history (1-line):** Per Cluster 0; the cultural opponent +- **Source sections in original:** Not applicable +- **Cluster cross-ref:** Cluster 0 + +--- + +## Decoded: encoding-explicit re-encodings (per Rule 5) + +The following terms have explicit `encoding:` attributes per Rule 5: + +| Term | Encoding | Conventional → Re-encoded | +|---|---|---| +| `Plausibility : float64` | `float64` | "real number" / "value" → `quantity() : float64` | +| `quantity(0.5) : float64` | `float64` | "plausibility value" → `quantity : float64` | +| `quantity(1) : float64` | `float64` | "indicator 1" → `quantity : float64` | +| `quantity(0) : float64` | `float64` | "indicator 0" → `quantity : float64` | +| `count : int64` | `int64` | "count" → `Count : int64` (exact integer) | +| `n : int64` (chain length, world-state length) | `int64` | "length" → `Length : int64` | + +--- + +## Decoded: FOILs and BANNED (per `lexicon.md` §2.4 Tier 4) + +- **`Bourbaki`** is a FOIL (per Cluster 0, Pattern 6). Not directly referenced in this lecture, but relevant to the foundational critique of reification in standard math notation. +- **`"infinity"` (in §5.1 frequentist limit)** is BANNED as a value per Rule 1. Re-encoded as `Stream Plausibility = nat -> Plausibility` (the indefinite process). +- **`Standard GA`** is a FOIL (per Cluster 0, Cluster B, P6). Not referenced. +- **`Lengyel's Standard GA`** is a FOIL. Not referenced. + +--- + +## Verification (per `lexicon.md` §12) + +- [x] **Lossless** — 72 terms decoded (12 Tier 1 + 18 Tier 2 + 18 Tier 3 + 24 Tier 4) +- [x] **Bounded** — no `∞_val`. The "infinity" in §5.1 is re-encoded as `Stream Plausibility`. +- [x] **Encoding-explicit** — every value-bearing term has `encoding:` (default `float64`; `int64` for exact integers). +- [x] **Constructively typed** — every expression has a type signature. +- [x] **Etymology-cited** — every term has 1-line origin + 1-line definition history. +- [x] **Form-anchored** — every re-encoding has a form anchor. +- [x] **No esoteric content** — secular sanitization preserved. +- [x] **Tier-categorized** — per pilot process improvement #2 (organized by Tier 1-4, not by math section). + +--- + +## See also + +- `lexicon.md` (the codified operational spec) — see §2.4 Tier 4 entries 4.1-4.24 +- `dedup_map.md` (the 6 noise-dedup maps) +- `probability_logic_translation.md` (the side-by-side table) — 38 rows +- `probability_logic_deobfuscated.md` (the re-encoded report) — 15 math sections + +--- + +*End of `probability_logic_decoder.md`. Total: 72 terms decoded across 4 tiers. The shape of the re-encoding, not the verbatim content of any specific sample. Per pilot process improvement #2: tier-categorized (not by math section).* \ No newline at end of file