# TODO / extraction: geometric part of Schlenker & Lamberton, "Iconological Semantics" (L&P 2024)

Page numbers are the journal-manuscript page numbers printed at the foot of each page in
`source/Iconological-Semantics.pdf` (txt line numbers in brackets refer to
`source/Iconological-Semantics.txt`).

## A. Definitions, formulas and claims to formalize

| # | Item | Location | Lean |
|---|------|----------|------|
| 1 | Signer frame r*: 3 orthogonal axes, right-handed (x right, y front, z up), origin at center of signing space; signer's chest at (0,-1,0) | §7.2, pp. 31-32 (83) [l. 1651-1665], (84c) p. 33 | `signerFrame` |
| 2 | Same coordinate system "from the addressee's position (0,1,0), x toward the addressee's left, y toward their back, z up" | p. 32 [l. 1665-1668] | `addressee_paper_frame_eq_signer`, `addressee_natural_frame` |
| 3 | Viewpoint pi determines frame r(pi), spatial scale s(pi) (>0), and temporal scale t(pi) | p. 32, (84b) p. 33 | `Viewpoint` |
| 4 | Classifiers and objects come with a center and an orientation = right-handed orthonormal triple | p. 32, (84a) | `Pose`, `Pose.Valid`, `rightHanded_isRot` |
| 5 | center(d,r) = triple of reals; orientation(d,r) = <u,v,w> coordinates of the 3 vectors | (84a)(i),(ii), p. 32-33 | `center`, `orientation` |
| 6 | fn 45: orientation better defined as triple of angles | p. 33 fn 45 | not formalized (rotation matrices used; equivalent parametrization) |
| 7 | (84d) lexical content word'_{t,w} | p. 33 | parameter `lex` |
| 8 | (85) proj(d,pi,t,w)=WORD iff (i) word'_{t,w}(d)=1 and (ii) a. center(d,r(pi)) = s(pi).center(WORD,r*), b. orientation(d,r(pi)) = orientation(WORD,r*) | p. 33 | `projStatic` |
| 9 | Example (86)-(89): Dalai Lama (0,0,0), Obama (50,0,0); orientations Obama <(0,1,0),(-1,0,0),(0,0,1)>, DL <(0,-1,0),(1,0,0),(0,0,1)>; scale 1/50; Obama classifier at (1,0,0); same orientations for the classifiers | pp. 33-35 | `obama`, `clObama`, `example_*` |
| 10 | fn 47: viewer at (0,-10,0) (signer-like) or (0,10,0) (addressee-like) | p. 33 fn 47 | covered by item 2 / `signer_addressee_equivalence` |
| 11 | Claim: (85)(i) makes the lexical condition boxed in (53) redundant | p. 35 [l. 1806-1810], fn 49 | `lexical_condition_redundant` (two-valued reading) |
| 12 | (90) dynamic projection: (i) word'_{t,w}(d)=1 and (ii) for each d' <= d, proj(d,pi,t+t(pi)d',w) = word-cl(d'); dynamic classifier = function [0,d] -> static classifiers | pp. 35-36 | `projDyn` |
| 13 | Flight example: 8 h shown as 2 s gives t(pi)=14,400 | p. 35 | `example : 8*60*60/2 = 14400` |
| 14 | (91): classifier moves 1 s, scaling factor 5, object moves for 5 s mirroring the classifier | p. 36 | `example_dynamic` |
| 15 | fn 64 (Chemla p.c.): objects and classifiers are both points+frames; classifier true of object iff there is a geometric transformation (translations, rotations, scalings) from one to the other | p. 49 fn 64 | `Sim`, `projSim`, `projStatic_iff` |
| 16 | App. I-E (126)-(128): loci; (127)=(85) copied; (128)(i) lexical = entity; center of locus = pointed part (head for person); orientation trivialized (128)(ii)b | pp. 48-49 | `projLocus`, `projLocus_iff_exists_orientation` |
| 17 | (129) numeric locus example: locus 30 cm above center, s=3, r(s(pi)) at 1.20 m => head at 2.10 m | p. 49 | `example_locus` |
| 18 | Remark p. 31: signer vs addressee viewpoint: scaling/marking can be done the same way | p. 31 [l. 1638-1645] | `signer_addressee_equivalence` |
| 19 | Rule (53)/(52): [[P_pi]](x) = 1 iff [[P]](x)=1 and proj(x,s(pi),t,w)=P | pp. 21-22 | absorbed in `lexical_condition_redundant` |

## B. Issues spotted in the original (see RESULTS.md for status)

1. **Scale direction contradicts the worked example.** (85)(ii)a says center(d, r(pi)) = s(pi) . center(WORD, r*), i.e. s multiplies classifier coordinates to get object coordinates. The example (p. 34) takes s = 1/50 with Obama at 50 and the classifier at 1: that satisfies center(WORD) = s . center(d), the opposite direction. With (85) as written s must be 50. (The temporal scale in (90)/(91), 5 and 14,400, does go in the (85) direction, so (85) and (90) are mutually consistent and the example is the odd one out.) Lean: `example_obama_s50`, `example_obama_paper_scale_fails`.
2. **r* and r(pi) swapped in the prose of the example** (p. 34): "the frame of reference notated as r* in (85)" is used for the scene's frame (that is r(pi)), and "the signer's frame of reference (notated as r(pi) in (85))" (that is r*).
3. **Orthonormality/handedness is only in prose.** (84a)(ii) says "right-handed triple of orthogonal vectors of unit length" but nothing in (85) uses this; (85) is well defined and a fortiori satisfiable for arbitrary triples. Consequences: (a) the reconstructed characterization "exists a rotation carrying classifier to object" needs it (`no_sim_to_mirror_pose`, `no_sim_to_nonunit_pose`); (b) a left-handed object can never project on a right-handed classifier: mirror images are not handled (the "mirror-image version" of the Obama picture in fn 43 is only about the figure).
4. **Which vector is the "front"?** The triple <u,v,w> has no stated meaning (front/side/up); the example is consistent with u being "front"-ish but nothing in (84) fixes it, and the classifier's own orientation convention is left to the lexicon.
5. **r(pi) and r* are unrelated a priori**, and r(pi) is free (any frame): existential quantification over pi makes (85) trivially satisfiable for valid poses (`exists_sim_of_valid`). The constraint only bites because one viewpoint pi is shared by all classifiers of a scene (`shared_viewpoint_distance`, `shared_viewpoint_relative_orientation`). Same for fn 64: read as a plain existential over transformations it is vacuous.
6. **No time/world argument in center/orientation.** (84a) center(d,r), orientation(d,r) have no t,w parameters although proj(d,pi,t,w) is evaluated at t,w and (90) requires the object's pose at times t+t(pi)d'. Also center(WORD,r*) of a dynamic classifier depends on d'. (Formalized by making `obj`, `cl` functions of time.)
7. **proj is a relation, written as a function.** "proj(d,pi,t,w) = WORD" only makes sense as a relation: two classifier tokens with the same pose (or same word in different loci, cf. plane-cl_i, p. 22) would make it multi-valued. Argument order also differs: proj(d,pi,t,w) in (85) vs proj(pi,t,w,d) in (90).
8. **Notation clashes in (90).** d = the object in (85) but the duration in (90) (and (90) header uses d as object argument while (ii) uses d' <= d as a time); t is the evaluation time, `t(pi)` the temporal scale, and word-cl_t (subscript t) marks dynamic classifiers; word'_{t,w} is again t. The range "for each d' <= d" omits d' >= 0. Fixed in Lean: `dur`, `ts`, `0 <= d' <= dur`.
9. **(90)(i) is redundant** (take d' = 0 in (ii), since (ii) re-imposes (85)(i) at t+t(pi).0 = t): `projDyn_i_redundant`. Conversely (85)(i) is evaluated at time t+t(pi)d' for each d', not only at t; harmless but the "lexical" condition thus must hold throughout (e.g. a sitting person must remain sitting while moving).
10. **Redundancy claim (p. 35, fn 49)** is true only in the two-valued reading: (53) also handles the presupposition failure case [[P]](x)=#, which (85) (a condition on proj) does not (fn 49 second remark says similar for assignments).
11. **Uniform scale only; centers only scaled about the origin of r*.** Fine given a similarity, but note that the formulation implicitly puts r*'s origin and r(pi)'s origin in correspondence; the elegant version makes the translation explicit (`Viewpoint.toSim`).
12. **Signer/addressee remark.** The paper says the frame from the addressee's position (x = addressee's left, y = addressee's back, z up) is "the very same coordinate system". True: the axes coincide (`addressee_paper_frame_eq_signer`). The addressee's *natural* (right, front, up) axes are a 180 degree rotation about z of r* (`addressee_natural_frame`). Using them requires changing r* and r(pi) together (`projStatic_rot`); changing only one flips the verdict (`example_one_sided_change_fails`).
13. fn 45's "triple of angles" is not a complete invariant (Euler angles have gimbal-lock/non-uniqueness); rotation matrices (or unit quaternions) are the clean choice.