A very first note: The ring-closing of sugar molecules could reflect the last step of the carbon-nitrogen cycle of fusion in the sun: after synthesis of protons from C via N to O back to C with emittence of an α-partic1e.
Cf. "A-Z"-figure here.

1. The OH-language?
The carbohydrates (cbh) have their own language - the OH-language. They are individualized not only in number of HCOH-groups but in different directions of OH-groups from the individual C-atoms. What does this imply?
   It obviously indicates a differentiation in the configuration of electrons around the separate C-atoms. (Cf. the sp³- to sp²-hybridization.)
   An illustration of two opposite directed dimension chains could illustrate such a differentiation between electron shells of C-atoms, in the figure below with one step of displacement. Of the opposite chains one could be regarded as corresponding to the d-degrees of structure, the other the d-degrees of motion, or alternatively just a- and b-poles, as in our primary model.

Mass of the group HCOH is 30 A. That's the sum of poles in a dimension chain. The number of such groups seem primarily to be 5. as dimension degreex in our model. (Including 0/00, the "d-degree of motion", as substantiated, gives number 6.)

An elementary script of Nature? On the Z-level writing a series of sexes for the C-atom.
   Cf. perhaps that inversion of 15, sum of series 5 - 0, = 0,666...

The C-atoms become here individualized in different relations between the chains. We could attach the numbers 5-4-3-2-1-(0/00) to the C-atoms in the chain not just as a convention in biochemistry but as "5/1, 4/2, 3/3, 2/4, 1/5. (Eventually the intervals as "+4" — "+2" —0 — "-2" — "-4" as differentiating?)
   If the script would be so simple, it implies that the electron shells of C-atoms is dividable in several ways and may be composed in several ways. s-electrons of both K- and L-shell should be included — and engaged in the bonds arround the C-atom.
   The varying directions of OH-groups on the surface of cell membranes function as a language.

2. The incorporation of a 6th C-atom:
What about the hexoses in this imagination?

It should be noted that never more than 5 C-atoms are included in the ordinary structure-building ring formations of carbohydrates. It seems that the only way for 6C-atom rings to get formed is through sharing C-atoms with other rings as in steroids or with open C-chains as in carotenoides.

The 6th HCOH-group is built in by plants by absorption of the molecule H2CO3 (or HCO^-). (Minus O2 giives H-COH.).
   It starts from ribulose, a pentose, in P-P-bonds. (The P-groups as connected with 0- and 00-poles in the figure above.)
   We may note that the 2 P-groups charged à 39 Z exactly balance the Z-number of this bound ribulose, 78 Z.

The 6th HCOH molecule gets built-in in the middle of the ribulose chain, more or less at the same time that this chain gets halved into two C3-pieces (treoses with COO-groups at at ends in the middle. Through +2 H - H2O the two C3-pieces unite again to an hexose.
   Why in the middle?

Even if we adopt the view on the process as a substantiation of last d-degree 0/00 in the chain, we have to regard it from the perpendicular aspect on the chain: the double direction from higher d-degrees and from lower degrees meeting in step 3-2.

Or with two opposite chains:

Sums vertically here 5: Only 5 electrons involved in hexoses ! ?

This is underlined by the 2 P-groups. The central role of these P-groups, (here as well as in the partition of fructose at start of glycolysis), can be regarded as polarizing force from outside as anticentres, initiating the polarization steps - and also the driving force towards development of a new level through step 3-2.

The built-in of the 6th C-atom group implies a displacement of the middle in the chain half a step, a decrease half a d-degree, as from 3 to step 3 - 2, (from "border to interval", the elementary illustration of a quantum jump proposed in files about Physics).

We could compare with transformations between number base systems (nb-x),
from 10 to 6:

nb-10: 12 = C,     18 = H2O            (Mass numbers A)
             ↓              ↓       
nb-6:   20             30 = HCOH    A quotient and a relation in step 3 - 2.   

The A-numbers 12 and 18 as intervals:

nb-16 —— nb 10 —— nb 8:
  32    —|→ 50   —|→  62  (= H2CO3, - O2, 32)
            18               12 

In the Pentosephosphate cycle, where 5 C3 get tranaformed into 3 C5.molecules, one C3 gets a C2 from C4, giving C5, another C3 gets C2 from C7, giving 2 C5.
   (There are also in intermediate steps, C6 - C4 and C7 - C3 molecules, reminding of the number 10 division chain.)

3. Elementary numbers of carbohydrates:
Some simple associations with the dimension chain in alternative forms.

- Numbers O = 8 Z, H2 = 2 Z, a relation 4 — 1.

- Why primarily pentoses as a condition for synthesis of carbohydrates?
  150 A is 5 times 30, the sum of poles of the dimension chain.
E-numbers = value as sum of outer poles in the different d-degrees.

Numbers of the carbohydrates in the 2x²-chain:
All numbers Z.
   H2CO3, the molecule which (minus O2) is built-in as the 6th HCOH-molecule.

Mass numbers with 2 chains 2^x:
Hypothetically the 2^x-series is assumed as valid in polarizing direction inwards in a dimension chain.

The division of H2O:

a. Elementary dimension chain: Z/2:

b. 2^x-chain, A-numbers:

c. 2x²-chain:

4. The ring-closing of carbohydrates:
A first question: Why do the ends differ in open, unclosed carbohydrate chains? One end is the aldehyde group H-C=O and the other end H2-C-OH.
    It could be read (-/+), minus H at one end (aldehyde group) and plus H at the other. It's suggested here that this reflects the polarization in last step of the dimension chain from d-degree 1 to d-degree 0/00,
   1a — 0/00 — 1b,
Sum of poles = "E-number" = 2 for 2 H, poles which define a (new) 0-pole and 00-pole respectively, here identified with minus/plus one H.

With C-atoms regarded as representing different d-degrees, the closing to rings of pentoses or hexoses may be interpreted as illustrating the connection between these outer poles 0 and 00 (also together representing 5').

Conventionally the numbering of C-atoms begin with C of aldehyde group as number 1. Accepting this order but decreasing the number series one step, relating it to the dimension model gives in an hexose:

   H2COH — HCOH — OHCH — HCOH — HCOH — O=CH
        5                 4              3                2              1             0/00 (~5')      

Then, in terms of the dimension model, the ring-closing implies a connection from d-degree 4, outer poles of which are 0 and 00, with the C-atom at the end representing d-degree 0/00 where these poles are met again.

   00
    |
   4 — ——— 0/00
    |
   0    

Perhaps the oxygen atom of the aldehyde group could be identified as expression for the 00-pole "meeting the other way around".

Within P-P-bonds, as for instance when C6 glucose is transformed to C6 fructose at start of the glycolysis, the oxygen bond changes to a relation 4 — 1:

C4 —Oxygen — C1:

The same figure can illustrate pentoses with only an OH-group at position 0/00.

The C4-C1-connection implies an angle change, similar to the turn from a more linear aspect on the dimension chain, formally 180°, to the loop model, 90°, a vertical aspect on polarizations:
   5 → 0/00, 5 → 4/1, 5 → 3/2:

The transformation of glucose to fructose implies that the middle of the rings is increased half a degree, as it was decreased a half degree when the 6th HCOH-group were built in, here from 2 in glucose to step 3-2 (C3 — C2) in fructose (as in pentoses).

(We may here remind of the two classes of amino acyl tRNA-synthetases related to C3 and C2 in ribose of nucleotides at the protein synthesis, dividing the attributed amino acids into 2 groups, as the opposite directions of the chain meet in this step 3-><--2.)

In the dimension model d-degree 3 gets polarized in poles 3a—3b (defining d-degree 2). We suggest to compare with the more or less immediate division of a hexose when the 6th C-group has been built-in, into 2 C3-pieces - and the same division in this 3-2-step when glucose has been transformed to fructose at start of glycolysis.

Closing of the carbohydrates into rings is geometrically, in the macro-structure, expressions for the steps from d-degree 4 →3 →3a in our dimension model. The model implies also that rotation as a 2-dimensional motion appears in step 4 →3. Also assumed as an angle step as 180° to 90°.
   Such a rotation appears also here at closing of the ring around C4: H turns 180°, OH-group 90°, which also leads to the turn of C5 90°. The OH-group is turned towards C "0/00". Hence, ring-closing and rotation seem connected as geometry and d-degree of motion in the dimension model.

5. Aldehyde - Keto parts:
When hexoses as fructose are split into two C3-parts, one gets an aldehyde end, the other a keto-bond in the middle: we have an aldehyde and a keto-part.

                                conventional numbering       

Original aldehyde end to the right at C0/00

C3 — C2          

- Original ends COH and H2COH now appears at C3 and C2. (Equivalent with minus 1 H at C3, +1 H at C2). The part with original aldehyde end becomes the keto-part. A change in direction of numbering seems needed if the keto-group C=O should represent d-degree 4 in the C-chain.

- The main features in the process of glycolysis and citrate cycle may in some respects be regarded as an illustration of the double direction in a dimension chain:

5 → 4 → 3 →<— 2 <— 1←0/00
               3C        +     3 C

The halving of the fructose molecule as a polarization outwards, one half in d-degree 3 meeting +1 C + 2 C (CO2 and Acetyl~) as from the other half, representing the complementary pole to isocitrate 6 C.

- It's the aldehyde part C3, here numbered 5 - 4 - 3, which develops through up to ten steps to C3 Pyruvate through glycolysis, and further in synthesizing direction into citrate cycle, + C1 and + C2 to C6, isocitrate, that's to the higher d-degrees of C-atoms.
   The way is connected with generation of amino acids and proteins.

- It's the Keto- part, here numbered 2 - 1 - 0, with the double-bound oxygen at C1, which develops to C3 Glycerine, the "circular" backbone part of membranes.

- The separate ways of transformations for the C3-pieces underscores the differentiation of C-atoms in glucose as well as the geometrical polarity "radial" versus "circular" (3b-/3a-poles) as amino acids (proteins) versus cell membranes on a macro-scale.
   From Pyruvate the way leads also to C2, Acetyl~, which starts the synthesis of the radial parts of membranes, the fatty acids.

If we want to associate higher C-numbers 3-4-5 as C-numbers with the way to amino acids, the lower ones with the way to glycerine, according to dimensional aspects, and further double-bound oxygen with d-degree 4 as C4, the figure below could illustrate the ambiguousness in numbering and transformations (as "pole exchanges") between the two opposite number series.
    A division vertically of the horizontal chains shows that both C3-parts "virtually" contain whole dimension chains. Cf. in the model debranched d-degrees 1 --- 2 from higher steps meeting the other way around.

Identifying C4 with d-degree 4 (in the model double-direction) may give an aspect on why transformations always have the direction from keto-form to aldehyde form (possible to read as outwards?) and why the keto-form form is the actual one when carbohydrates transform into one another.

The association of C-atoms with different d-degrees seems supported by the change in mass distribution during glycolysis: The transformation of the aldehyde part as 3-P-glycerate to Pyruvate in several steps implies that a nearly equal mass 30 - 30 - 30 transforms to 45 - 28- 15 in Pyruvate, CH3— C=O—COO(H), that's quotients circa 3 - 2 - 1. as if an underlying differentiating process between the C-atoms manifested itself. (In reality expressed as plus / minus 16, oxygen, between first and third C-atoms.)

6. Polarizations of the OH-language:
Different directions of the OH-groups around the C-atoms seem possible to regard as a way through polarization steps, different paths at bifurcations leading to different roles and positions in the cell:

The figure shows three such polarizations where the 3rd leads to the opposition between cellulose <—> amylose and starch: it implies one kind of 2-3-relation in d-degrees regarding the forms on a macro-scale: cellulose for cell coats, surfaces, amylose as substrate in the cell, volumes. (Cf. different coenzymes connected with different carbohydrates: UTP with glucose, TTP with cellulose.)
   This 3rd polarization, α, β, refers to the orientation of the OH-group at what here has been regarded as C-atom "0/00" in original hexose.

A 4th polarization in hexoses in direction "up/down" at C3 (conventionally C4) gives the difference galactose/glucose: united in the disaccharide lactose.

We have hypothezised that the 2^x series may reign, from the end of a dimension chain and (formally) there are 2⁵ isomers in a hexose, C6, 2⁴ in a pentose, 2³in a C4 piece, 2² in a C3-piece:

Carbohydrates:          C3     C4     C5      C6
Number of isomers:    4        8      16       32

Following figure may not illustrate the real way of derivations but could perhaps give a hint of the way of thinking here, with associations to the dimension model:

- Arrows for the direction of OH-groups.
- The β-form should be associated with the 0-pole and pole 4b (D), (outward direction).

- The up/down polarity of OH-groups could perhaps be expression for the polarization at C1, represented in each step.

7. Forms of macro-molecules of the carbohydrates:
Another aspect on the different carbohydrates gives the behaviour of the polymerized macro-molecules if we regard them in terms of motions of different dimension degrees:

- Cellulose gets folded, which may be seen as an 1-dimensional motion to and fro, a kind of vibration.
- Amylose gets spiralled, a kind of 2-dimensional rotation, connected with pathway motion giving a 2- to 3-dimensional motional structure.
- Amylopectin and glycogen get branched, as illustrating "translation in 3 dimensions".

In this respect there is no opposition between d-degree of form (structure) and d-degree of motions; the motions if we regard the formations of the macromolecules as such, are and give the macro-forms.

8. Some number operations;

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