# Validation of date

## Mon, April 26, 2010 8:00 pm Paul Ballonoff

From: <paul@ballonoff.net> To: Douglas.White@uci.edu

Doug,

I verify I have read, downloaded, and printed a document from the indicated website, at the cuurent date as indicated on the attached file "Dravidian_kinship_con.pdf", and that for verification of authenticity I note the only error observed by me lies at Page 3 of the attached, in the "Proof of 2", the initial words "Suppost M" are an obvious typographical error of "Suppose M", and do not affect the substance of the proof offered. The nearly 11 hour difference in date/time stamp on the file and in the file properties, occurs since you are in the time zone of Los Angeles, California and I am (until Friday this week) in the time zone of Kabul Afghanistan.

As to publication sequence, the way this astonishing exchange is going overall, I see no reason why you can not add to your initial paper at any time. I now expect two related papers (not Comments), apart from your comment, will be or are under review shortly, and when review completed, would also be offered for your and other's Comment. You might therefore wish to wait on that opportunity. But I see no other structural reason except your own choice, to do so. The earlier you insert this discussion to the sequence, the earlier others can both respond to or otherwise acknowledge it's existence.

I save other comments for a separate note to follow.

Paul

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On Tue, Apr 27, 2010 at 6:37 AM, <drwhite@uci.edu wrote:

http://intersci.ss.uci.edu/wiki/index.php/Dravidian_kinship_controversy#Theorems

Today I proved three theorems about the relationship among viri-sidedness, uxori-sidedness and same-generation marriage, on the following web site. I would like to add these theorems and proofs in the next round of revision of my comments or a separate comments. I would like you to check and date the completion of these proofs since I would like to claim priority in case anyone else should be my motivation in the comments sent out as to the relations between U, V and M, but did so incorrectly.

thanks

Doug White

## Theorems and proofs

This is the corrected version verified by Paul as of Doug 07:23, 27 April 2010 (PDT)

### Theorems

The presence of both viri-sides (condition V) and uxori-sides (condition V), for a network of consanguineal marriages, logically entails implicit generational moieties (condition M) in a bicomponent of a marriage network. This includes the possibility of an ego at generation marrying someone at where the absolute difference |i - j| ia an even number, e.g., +2 or -2 generations. Further, the presence of a same-generation marriage and either viri-sides and uxori-sides logically entails the complementary type of sidedness.

**Theorem 1**. U and V, and consanguineal messages => M, including implicit alternate-generational moieties.**Theorem 2**. U and M => V**Theorem 3**. V and M => U

### Proofs

**Proof of 1.**

- Suppose U and V and consanguineal marriages. For a subnetwork in which every marriage is between consanguines (blood marriages), then marriages in which husband and wife have marriage cycles with ancestors that are both U an V, then the cycle will have even number of female links, even number of male links, and an even number of total links. If the ancestral graph is drawn from parent to child in successive generations then either husband and wife are of the same generation or one is an even number of generations above the other. This will apply to all such marriages.
**Q.E.D.**This proof will generalize to Australian section systems.

**Counterexample in the case of nonconsanguineal marriage. **

- Suppose U and V and a network with nonconsanguineal marriages one of which is a man who marries a BDHSD. Then the number of males is even (4) as is the number of females (2), qualifying for U and V.

**Proof of 2.**

- Suppose M. In these cases M may apply so that the generation number of each marriage is one more than the generation number of the parents of the husband and that of the wife. This entails that any marriage cycle will have an even number of links.
- Suppose U and M. U in addition to M requires that any marriage cycles with have an even number of male links, thus an even number of female links in order to add up to an even number of male links. Hence the network is V.
**Q.E.D.**

**Proof of 3.**

- Suppose V and M. Exchanging U and V above: U, V and M entail U.
**Q.E.D.**

Q.E.D.s for theorems and proofs: Doug 18:58, 26 April 2010 (PDT)
**Validation of date**