Pyramid reduction hypothesis

The pyramid reduction hypothesis is a credible explanation of the origin of bird fingers. It is "developmentally plausible, and is also consistent with the phalangeal reduction pattern seen in basal birds."
It is consistent with a pterosaur to bird theory.
It is inconsistent with a dinosaur to bird theory.

http://www.ivpp.ac.cn/qt/papers/201403/P020140314320497349455.pdf
http://aracnologia.macn.gov.ar/st/biblio/Xu%20and%20Mackem%202013%20Tracing%20the%20Evolution%20of%20Avian%20Wing%20Digits.pdf

(Xu and Mackem 2013)
Good description of pyramid reduction hypothesis even though Xu and Mackem do not agree with it. 

The ‘pyramid reduction hypothesis’ assumes II-III-IV identities
for neornithine manual digits and postulates the existence
of a conservative five-digit pattern with a gradual,
bilateral reduction of phalanges and metacarpals in avian
evolution [9]. One proposed mechanism postulates that an
elevation in peripheral BMPs, signaling factors that modulate
cell survival and proliferation [60,61], drove bilateral medial
and lateral digital reduction [9]. This hypothesis is developmentally
plausible, and is also consistent with the phalangeal
reduction pattern seen in basal birds [9,23]. However, it predicts
that the direct avian ancestor had a five-fingered hand
with dominant digits II, III, and IV [9] which is inconsistent
with the digital reduction data from basal theropods (e.g.,
all known basal theropods, including ceratosaurs, have a
vestigial digit IV) [5,62–64].

In fact, the pyramid reduction hypothesis implies that either birds are not descended from theropod dinosaurs, or that some as yet to be discovered
basal theropods were five-fingered with dominant digits II, III, and IV.

And concerning the alternative homeotic shift (frameshift):

Also, it is difficult to identify plausible selective pressures that would drive this type of homeotic shift, considering that the post-frameshift adult hand would be morphologically identical to the pre-frameshift condition [69].

The frame shift and lateral shift hypotheses adopt different criteria for assessing primary homologies: the frame shift hypothesis relies on morphological information and selective spatial expression patterns of developmental genes, whereas the lateral shift hypothesis emphasizes positional information. However, the two hypotheses are similar, as both require genes that regulate digit pattern to shift their expression domains for the developing hand and the re-emergence of a fully functional digit in position 4, but differ in proposed mechanism of homeosis. 

Xu and Mackem note that the Pyramid Reduction Hypothesis "is inconsistent with the digital reduction data from basal theropods". But it is only inconsistent if birds evolved from dinosaurs. If birds did not evolve from dinosaurs, then there is no inconsistency. 
In fact, as Xu and Mackem acknowledge, one possible implication of the Pyramid Reduction Hypothesis is that "birds are not descended from theropod dinosaurs".

And here is another point:
Xu and Machem conclude that for the Pyramid Reduction Hypothesis to be correct:
"the direct avian ancestor had a five-fingered hand with dominant digits II, III, and IV".
They have overlooked the possibility that the ancestor had digits I-II-III-IV and lost digit I in the transition to primitive bird. Which is what I suggest occurred in the transition from pterosaur to primitive bird.

https://www.academia.edu/11078361/Pentadactyl_pattern_of_the_avian_wing_autopodium_and_pyramid_reduction_hypothesis

ABSTRACT
We report herein that a pentadactyl developmental pattern is evident in early wing morphogenesis of Gallus (chicken) and Struthio (ostrich). Five avascular zones (spatially predestined locations of contiguous metacarpal and phalangeal aggregation) and four interdigital vascular spaces are established by the regression patterns of autopodial vasculature. Transient vestiges of the first and fifth metacarpals are confirmed histologically and histochemically. They lie within the preaxial-most and postaxial-most avascular zones, respectively. These observations reveal conservative patterning of the avian hand and corroborate a II-III-IV metacarpal interpretation, argue for II-III-IV identity of ossified digits in birds, and favour a simple reduction rather than a homeotic shift in terms of the phenotype expressed by Hox genes in the phylogeny of the avian manus. We suggest that gradual, bilateral reduction of phalanges and metacarpals, via apoptosis mediated by BMP, occurred during the evolution of birds (Pyramid Reduction Hypothesis). This is congruent with the establishment of a central wing axis that became co-opted for coordinated movements. On the basis of evidence presented here, the direct avian ancestor is predicted to have been five-fingered with dominant digits (+ metacarpals) as follow: II, III, IV.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3986427/
Frame shift  and the rabbit

SUMMARY:
For years, people have been striving to reconcile the I-II-III digits of dinosaurs with the II-III-IV digits of birds.
Why?
Because they believe that birds evolved from dinosaurs.
Why do they believe that?
Because the cladistic analyses seem to support that idea.
How so?
Because they show oviraptors and alvarezsaurs as intermediates between dinosaurs and primitive birds (Paraves).
BUT recent studies show that oviraptors and alvarezsaurs are WITHIN Paraves.
So there is no cladistic analysis evidence to support the dino to bird theory.
Therefore there is no reason to strive to reconcile the I-II-III digits of dinosaurs with the II-III-IV digits of birds.
Which is good, because there is no credible way to reconcile them in the first place.


For reference:
https://www.researchgate.net/publication/260714879_Reply_to_Limusaurus_and_bird_digit_identity
Reply to “Limusaurus and bird digit identity”
Xing Xu1, James Clark2, Jonah Choiniere2, David Hone1 & Corwin Sullivan1

Morphological data from extinct theropods, even without considering Limusaurus and ceratosaurs, clearly contains two contradictory signals for the identification of tetanuran manual digits. Thus, neither our hypothesis nor the frameshift hypothesis is able to avoid a substantial number of homoplasies.