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The Unfinished Puzzle Section

Simply complete the sudoku and crossword, happy puzzling.

Across
1. Achieved acclaim by being knowledgable about strange apron (3, 6)

Down
2. Explode asteroid five to refrain from inactivity or evade others (5, 4)
23. Clip, rewire, curtail current, repeat (5)

Unassigned
Abbreviation for value disregarding sign of six pack? (3)
Differentiable group three hundred and seventeen head over heels (3)
Invert stripped hypnotist (3)
Not even unusual? (3)
Sequence of games forming part of match with southern alien (3)
Synthesis of Tritium, tungsten, and oxygen into smallest sized compound (3)
Never say it backwards without right for level ground (4)
Plural singular belonging to yours truly! (4)
Scan again partly increments (4)
Being something? (6)
Measure core income trick (6)
Between the menu and the meal one thing goes before another! (7)
Finitely generated free abelian group spanning real space tart's holey top? (7)
Mistakenly disprove supplies (8)
Take all unordered 2-subsets of a set by transforming a savvier transcendental ratio (8)
Used to smell rotten they say it's certainly unmistakable (8)
Beer of infinite acidity, “infinity plus one” (5, 4)
Create cut corn mess covering street (9)
Ordered classification of Cockney girl in process of losing sense (9)
Map to and fro adds charged particle to information about limitless projector (9)
Pigeon-holed personalities in deluded chart races (10)
Staggering credential was woven (10)
Adulterating awkwardly yielded location from angles (12)
Introductions for diminutive Asian as new leader of nations (12)
Reason following headless fellows litigating against journal of ship 99 (7, 5)
Confused, blind tourist is short by fifty for generalised functions (13)
Study of arrangements of deranged, manic robotics (13)

Addendum
$\mathcal f = ??\Gamma?????\Delta????\Phi??????????\Gamma???\Gamma????????\Phi???\Delta?$ (8, 7, 10, 9, 9)
$\text{clues} = \mathcal f(\text{clues}, \text{remainder})$
$\alpha = ???\Psi????\Psi?\Omega??\Psi???????\Psi????\Omega?????$ (8, 4, 8, 12)
$h = 15d – d$
$h(α) → ? \space \Box$

Hint 1 (2016-04-23 12:00Z)

Currently the sudoku clues are UNMATCHED, and there are 7 possible "solutions"

Hint 2 (2016-04-24 23:15Z)

I was going to just give the hint "Where do 1 across and 1 down go?", but realised I had made an error - I just edited the numbers of all the clues to be one greater than listed except 1 across (I had edited these on the day I posted due to comments about crossword numbering convention). This hint is therefore even greater (but it's my own fault of course!). Now...
"Where do 1 across and 2 down go?"

Hint 3 (2016-04-25 14:00Z)

What could $\text{clues}$ and $\text{remainder}$ be in the equation $\text{clues} = \mathcal f(\text{clues}, \text{remainder})$?

Hint 4 (2016-04-25 21:05Z)

The partially solved sudoku in the community wiki post at present is:

and is not fully resolved (there are more cells one can logically fill with letters that are not yet present)

Answer*

Here's some closure for why that one specific Sudoku board state is the correct one.

First, let's solve those Addendum clues.

>! $\text{clues} = \mathcal f(\text{clues}, \text{remainder})$  
>!  
>!  If we take $\text{clues}$ to be $\text{UNMATCHED}$ and $\text{remainder}$ to be $\text{ENSUING LOGIC}$, we can apply $\mathcal f = \text{distinct } \text{ordered } \text{characters } \text{provides } \text{bijection }$ to them.  
>!   
>!  For each string, remove duplicate letters.  
>! So we have $\text{UNMATCHED}$ and $\text{ENSUIGLOC}$
>!  
>!  Next, order them.
>!  We get $\text{ACDEHMNTU}$ and $\text{CEGILNOSU}$  
>!  
>!  From there, we can construct this bijection:  
>!  $\text{A} \rightarrow \text{C}$  
>!  $\text{C} \rightarrow \text{E}$  
>!  $\text{D} \rightarrow \text{G}$  
>!  $\text{E} \rightarrow \text{I}$  
>!  $\text{H} \rightarrow \text{L}$  
>!  $\text{M} \rightarrow \text{N}$  
>!  $\text{N} \rightarrow \text{O}$  
>!  $\text{T} \rightarrow \text{S}$  
>!  $\text{U} \rightarrow \text{U}$  
>!  
>!  which is equal to $\text{clues}$, and therefore "unmatched". We'll match it soon.  
>!  
>!  Next up, we have $α = \text{pairwise } \text{read } \text{provides } \text{orientations}$ and $\mathcal h = \text{interlace}$. But what do we interlace?  
>!  
>!  We take the string $α$ and interpret it literally. Break it up into pairs and we get $\text{PA}$, $\text{IR}$, $\text{WI}$, $\text{SE}$, $\text{RE}$, $\text{AD}$, $\text{PR}$, $\text{OV}$, $\text{ID}$, $\text{ES}$, $\text{OR}$, $\text{IE}$, $\text{NT}$, $\text{AT}$, $\text{IO}$, and $\text{NS}$. We then need to interlace each pair, in other words, cross them. Notice that the first letter in each pair exists in $\text{WONPRAISE}$ and the second letter in each pair exists in $\text{AVOIDREST}$. By placing $\text{WONPRAISE}$ across the top and $\text{AVOIDREST}$ along the bottom of the Sudoku grid, we can use the letters as coordinates. Here's the image with those coordinates circled.  
>!  
>!  [![Sudoku grid with coordinates circled][1]][1]  
>!  
>!  We then get the string:  
>!  $\text{T _ M _ H _ A _ M _ _ _ T _ _ M}$  
>!  but that just seems like gibberish. We can't really fill that in and make sense out of it.  
>!  
>!  So let's match the bijection to this! We get:  
>!  $\text{S _ N _ L _ C _ N _ _ _ S _ _ N}$  
>!  
>!  We can then attempt to fill these in if we also apply the bijection to the Sudoku grid and also with how words are constructed in English. Between the $\text{C}$ and $\text{N}$ only has these as possible letters $\text{{E,I,O,U}}$. The only one that fits in the Sudoku is $\text{O}$.  
>!  
>!  Doing a few more of these, we can end up with:  
>! $\text{S I N G L E C O N C L U S I O N}$  
>!  
>!  Mapping the letters back to the Sudoku grid, we can solve it and mark the previously posted solution as the "single conclusion".

### Pieces of the puzzle in the order presented, as solved here and in other posts

>! <img src="https://i.sstatic.net/xTQAD.png" alt="solved sudoku">
>! 
>! <img src="https://i.sstatic.net/oZ2tI.png" alt="solved crossword">

**Across**
>! &emsp;&emsp;&emsp; &nbsp; 1. Achieved acclaim by being knowledgable about strange apron (3, 6) 
>! &emsp;&emsp;&emsp; &ensp; &ensp; **Won praise** ("wise" around anagram of "apron")

**Down**
>! &emsp;&emsp;&emsp; &nbsp; 2. Explode asteroid five to refrain from inactivity or evade others (5, 4) 
>! &emsp;&emsp;&emsp; &ensp; &ensp; **Avoid rest** (anagram of "asteroid V")
>! 
>! &emsp;&emsp;&emsp; 23. Clip, rewire, curtail current, repeat (5) 
>! &emsp;&emsp;&emsp; &ensp; &ensp; **Recur** (REwire+CURrent)

**Unassigned**

>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Abbreviation for value disregarding sign of six pack? (3) 
>! 26 across. **Abs** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Differentiable group three hundred and seventeen head over heels (3) 
>! 18 down. &hairsp; &hairsp; **Lie** (317 upside down) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Invert stripped hypnotist (3) 
>! 16 across. **Not** (hypNOTist) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Not even unusual? (3) 
>! 25 across. **Odd** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Sequence of games forming part of match with southern alien (3) 
>! 20 down. &hairsp; &hairsp; **Set** (S+ET) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Synthesis of Tritium, tungsten, and oxygen into smallest sized compound (3) 
>! 24 down. &hairsp; &hairsp; **Two** (T+W+O) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Never say it backwards without right for level ground (4) 
>! &thinsp; 5 down. &hairsp; &hairsp; **Even** ("never" backwards without R) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Plural singular belonging to yours truly! (4) 
>! 28 across. **Ones** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Scan again partly increments (4) 
>! &thinsp; 9 across. **Read** (re+ADds) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Being something? (6) 
>! 21 across. **Entity** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Measure core income trick (6) 
>! 12 across. **Metric** (incoME TRICk) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Between the menu and the meal one thing goes before another! (7) 
>! &thinsp; 6 down. &hairsp; &hairsp; **Ordered** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Finitely generated free abelian group spanning real space tart's holey top? (7) 
>! &thinsp; 4 down. &hairsp; &hairsp; **Lattice** (double definition) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Mistakenly disprove supplies (8) 
>! 14 across. **Provides** (anagram of "disprove") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Take all unordered 2-subsets of a set by transforming a savvier transcendental ratio (8) 
>! &thinsp; 7 down. &hairsp; &hairsp; **Pairwise** (anagram of "a wiser pi") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Used to smell rotten they say it's certainly unmistakable (8) 
>! 22 across. **Distinct** (sounds like "This stinked" (sic)) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Beer of infinite acidity, “infinity *plus one*” (5, 4) 
>! &thinsp; 3 across. **Aleph zero** (ale+pH zero) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Create cut corn mess covering street (9) 
>! 22 across. **Construct** (anagram of "cut corn" around "st") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Ordered classification of Cockney girl in process of losing sense (9) 
>! 29 across. **Numbering** ("er" inside "numbing") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Map to and fro adds charged particle to information about limitless projector (9) 
>! 17 down. &hairsp; &hairsp; **Bijection** ("bit" around proJECtor plus "ion") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Pigeon-holed personalities in deluded chart races (10) 
>! 13 down. &hairsp; &hairsp; **Characters** (anagram of "chart races") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Staggering credential was woven (10) 
>! 15 down. &hairsp; &hairsp; **Interlaced** (anagram of "credential") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Adulterating awkwardly yielded location from angles (12) 
>! 10 down. &hairsp; &hairsp; **Triangulated** (anagram of "adulterating") 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Introductions for diminutive Asian as new leader of nations (12) 
>! 19 across. **Orientations** (orient+nATIONS) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Reason following headless fellows litigating against journal of ship 99 (7, 5) 
>! $\unicode{8216}\rm remainder\unicode{8217}$. **Ensuing logic** (men without M+suing+log+IC) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Confused, blind tourist is short by fifty for generalised functions (13) 
>! 11 down. &hairsp; &hairsp; **Distributions** (Anagram of "blind tourist is" without L) 
>! &emsp;&emsp;&emsp;&emsp;&ensp;&hairsp;
>! Study of arrangements of deranged, manic robotics (13) 
>! &thinsp; 8 down. &hairsp; &hairsp; **Combinatorics** (anagram of "manic robotics")

$ \kern1.2em \mathcal f ~ = ~ ??\Gamma?????\Delta????\Phi??????????\Gamma???\Gamma????????\Phi???\Delta? $ (8, 7, 10, 9, 9)
>!<pre>
>! f = ??Γ????? Δ????Φ? ?????????Γ ???Γ????? ???Φ???Δ?
>! S O E S S E O
>! = DISTINCT ORDERED CHARACTERS CONSTRUCT BIJECTION
>!
>! pairwise lattice interlaced alephZero alephZero (false possibilities)
>! provides numbering numbering
>! bijection construct
>!</pre>

$ \kern1.5em \text{clues} ~~~ = \kern3em \mathcal f \kern4.2em ( \kern3em \text{clues} \kern2.3em , \kern2.7em \text{remainder} \kern3em ) $
>!<pre>
>!UNMATCHED = distinct-
>! ordered-characters-
>! construct-bijection ( U N M A T C H E D , E N S U I N G L O G I C )
>!
>!U N M A T C H E D , E N S U I N G L O G I C
>!: : : : : : : : : : : : : : : : : :
>!U N M A T C H E D E N S U I G L O C (distinct characters)
>!: : : : : : : : : : : : : : : : : :
>!: : : A ----:--:--:--:-----:--:--:--:--:-----:-----:--:------> C (a o v
>!: : : : C ----:--:---> E : : : : : : : l r e
>!: : : : : : D -------:--:-----:---> G : : p d r
>!: : : : : E ----------:------> I : : h e t
>!: : : : H -------------:------------------> L : a r i
>!: : M ----:------------------> N : : : b e c
>!: N -------:-----------------------:--:---------------> O e d a
>!: T --------------------> S : t l
>!U -----------------------------------> U i c a l l y)
>!</pre>

$ \kern1.1em \alpha ~ = ~~ ???\Psi????\Psi?\Omega??\Psi???????\Psi????\Omega????? $ (8, 4, 8, 12)
>!<pre>
>! α = ???Ψ???? Ψ?Ω? ?Ψ?????? ?Ψ????Ω?????
>! R R A R R A
>! = PAIRWISE READ PROVIDES ORIENTATIONS
>!
>! provides even distinct ensuingLogic (false possibilities)
>! distinct ones pairwise triangulated
>!</pre>

$ \kern2.4em h = 15d – d $ 
>! $ \kern1.8em ~~ = \textsf{15 down} - \textsf{d} $ 
>! $ \kern1.8em ~~ = \textsf{interlaced} - \textsf{d} = \textsf{interlace} $

$ \kern3.4em h ~~ (\alpha) ~ \to ~ ? $ 
>! $ \textsf{interlace} ~ (\textsf{pairwise read provides orientations}) \to \textsf{single conclusion} $
>!<pre>
>! α Pa ir wi se Re ad Pr ov id es Or ie nt at io ns
>! WON PRAISE P i w s R a P o i e O i n a i n
>! AVOID REST a r i e e d r v d s r e t t o s
>! : : : : : : : : : : : : : : : :
>!( distinct UNMATCHED : : : : : : : : : : : : : : : :
>! ordered A C D E H M N T U T E M D H C A N M A H U T E N M
>! characters | | | | | | | | | | | | | / / \ \ \ \ | | | | | |
>! construct C E G I L N O S U S I N G L E C O N C L U S I O N
>! bijection ) ENSUING LOGIC
>!</pre>
>!<pre>
>! W o n P r a i s e
>! -------------------------------------------------------------
>! A | |1 T Pa | | a
>! v | 8 N ov | | | : v
>! o | | |15 N io | : : o
>! |-------------------|------------------|----------------------| : : :
>! i |3 M wi | | | : i : :
>! d | | 6 C ad| 9 M id | : : d :d :
>! R | 11 H Or |7 A Pr | 2 E ir | :r: : r:: r :
>! |-------------------|------------------|----------------------| ::: : ::: : :
>! e | 16 M ns| 5 H Re |12 U ie 4 D se | :::e e: ::: :e :
>! s | | | 10 A es| :::: :: :::s :: :s
>! t | 13 T nt| 14 E at| | :::: :: :::: ::tt::
>! ------------------------------------------------------------- :::: :: :::: ::::::
>! P......................................Pa::: :: :::: ::::::
>! i...................ir:: :: :::: ::::::
>! w............................................................wi: :: :::: ::::::
>! s.............se :: :::: ::::::
>! :: :::: ::::::
>! R......................................Re: :::: ::::::
>! a.................................ad :::: ::::::
>! :::: ::::::
>! P................................................Pr::: ::::::
>! o...............................................................ov:: ::::::
>! i..............................id: ::::::
>! e................es ::::::
>! ::::::
>! O....................................................................Or:::::
>! i...................................ie::::
>! n...............................................................nt:::
>! a.............................................at::
>! i......................................io:
>! n..................................................................ns
>!</pre>

**Edit**: From my (Jonathan Allan) notes:
>! [![enter image description here][2]][2]  
>! [![enter image description here][3]][3]  
>! [![enter image description here][4]][4]

[1]: https://i.sstatic.net/eKOyQ.png
  [2]: https://i.sstatic.net/PmyoA.png
  [3]: https://i.sstatic.net/DPy6l.png
  [4]: https://i.sstatic.net/Binhg.png

Edit Summary*

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3

$\begingroup$ Nicely resolved. And quite a deep overall puzzle, @JonathanAllan, that unfortuately many people stopped working on after the crossword was solved. $\endgroup$
– Dan Russell
Commented May 7, 2016 at 1:48

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