This is coming to you from Yiwei LI (PhD, Applied math), Taiyuan University of Science and Technology  (TYUST) Taiyuan, China

It's going on here for the third round of learning of Birkar's BAB-paper (v2), with scenarios of chess stories. No profession implications. 

When one tries to understand a mathematical proof, one actually tries to recover what has happened from the "traces".

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Th 2.15    Th 1.8            ♖    ♘

    ↓   ↖     ↓

Th 1.1      Th 1.6            ♔    ♗

    Mathematics vs Palace stories.(v2)

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Note: technical theorem is not on the board.

♙ ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁺⁻⁰ 1

(continued) Overall, it is intended to find an increment, so that the desired properties are (locally) kept with this increment attached. This increment is associated with a natural number n, subjected to a set of constraints. So gives the formal name of "n-complement" to such an increment. Here, the unstated object refers to something called "pair", with the property labelled as "projective lc", while the increment is non-negative as an additional requirement. If one checks the definition of "n-complement", one will find the increment falls into a mode ——

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B + d/n

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where d/n is the increment, with "d"* as a temporary notation for some divisor. What has been done from move two to move six is just toward a specific construction of d. If one denotes the increment as R, one has R = d/n, or nR = d. That's why one sees the expression of nR:= G in the proof, as G plays the role of d in that context. Another notable expression is of nB, which presents as the third item in the definition of n-complement. The natural number n serves somewhat like a physical unit. By multiplying n to the mode above, one has ——

*I assume one would confuse the (ab)use of "d" here and the use of "d" as the dimension of X ---- In mathematics, one judges a symbol by its role in context, instead of its bare "look".(TOM)

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nB + d.

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In view of physical unit, one may expect d to take the form of "n·divisor", compatible with the expression nR = d. One can verify this unstated philosophy fulfilled in the construction of L', a variant of n·c(M), i.e. the prototype; see move three. So, c(M), or explicitly M - (Kx + B), is an essential invention. But, how could the author know to let nR: = G, as no one tells him to do so? —— When setting out to construct L', the author held the vision for L' as a "pre-construction" of d. That is, by construction of L', the author just intended  —— to let nR:= the right form of L'. That right form is represented by G. (It is beneficial to remember that, R is only a nominal existence before this definition of nR:= G). In retrospect, the topic of the present theorem (Th 1.9) is of n-complement which serves as the source of vision in the first place. Another philosophy is that, in mathematics, the "target construction" (here G) is out of indirect approach.

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In summary (high view), the mode suggests the form of n·divisor for the target construction (G), while n·c(M) serves as the prototype of the pre-construction (L') ——

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fom ~> pro

  |            ↓

tar  <~  pre

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Ground framework for mathematical construction.

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↑↓ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ⌈ ⌉ ∨∧∞Φ⁺⁻⁰ 1

Calling graph for the technical theorem (Th1.9) ——

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Th1.9

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[5, 2.13(7)]   Lem 2.26   Pro4.1   Lem2.7

                                                          |

............................................... ......Lem2.3   

Note: Th1.9 is only called by Pro.5.11, one of the two devices for Th1.8, the executing theorem.

Pro4.1                                                    

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[5, ?]   [37, Pro3.8]   [5, Lem3.3]   Th2.13[5, Th1.7]   [16, Pro2.1.2]  [20]  [25, Th17.4]

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Special note: Original synthesized scenarios in Chinese for the whole proof of v1 Th1.7, the technical theorem.

*It's now largely revised* due to new understandings.

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See also: Earlier comments in Chinese* (v1).

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It is my hope that this action would not be viewed from the usual perspective that many adults tend to hold.