Firm not rigid.

Firm and soft/fluid are complimentary not contradictory. 

Firm to allow energy transfer.

We have to be firm for the energy to transfer smoothly through the kinetic chain.
Rigidity will block the flow, being too loose will cause us to be disconnected and also interfere with the flow of energy from segment to segment.

We are not only firm at kime but throughout the technique, even though the muscles activation is different at moving and at kime.

Aiko San and sensei Nishiyama constantly told me “keep the very inside (around the sacrum) strong and the outside soft and flexible." Of course, to be able to apply and get the feel of this we need proper training methods and good feedback.

Sport research confirms this, for force transfer between the legs to torso to arms and vice versa (through the sacrum), the smaller, local muscles around the sacrum have to activate and stabilize the sacrum.

Connection and disassociation.

We have to connect all segments of the body, but if we brace and over tense we cannot disassociate.

For example, to increase energy between the sacrum to the thoracic spine, the thoracic spine has to move slightly more the the sacrum and lumbar, this is disassociation. If we brace, they will move as one peace, if we are too loose they are not connected and energy will not transfer.

When someone is too rigid and brace at the shoulder or hip, we cannot disassociate and energy will not transfer smoothly at that joint.

Also, disassociation allows us for subtle adjustment at the hip (or shoulder) that are so important for smooth weight shifting and footwork as well as force production in technique.

Also, firmness allows for relaxation, for example, if the lower spine is not stabilized by the smaller, inner unit muscles, the bigger, global, movement muscles, have no working foundation and will tend to be to stiff.
When the inner unit muscles activate optimally to stabilize the spine, the bigger muscles  can be used to their full rate and potential, contraction/expansion.

Our body is the most complex machine ever created, so we must understand the complexity in order to achieve simplicity, at the end one does not think of any of these, when I face my opponent, I just want to make chance for finish technique, every thing else should just happen because you train hard and correctly.

Requirements for "whole body cooperating to one line of energy".

Whenever Sensei was teaching a seminar he used to describe karate technique: “the whole body must cooperate to one direction”, or “even (if) one finger moves, feet make top technique.”, than he used to say “total energy must be delivered in shortest amount of time”, or else he used to say: “karate is not arms and legs training but whole body training.”

What are the conditions required for the whole body to cooperate effectively to one line of energy in order to make Todome (finishing blow)?

Optimal posture - moving from optimal posture and keeping optimal posture dynamically.

An optimal posture can be defined as the neuro-muculo-skeletal relationships which optimize joint motion and muscular action, trigger automatic stabilizing activity and minimize structural stress on the body.

Body center on top of base of support
Efficiency of using ground reaction forces, which means optimal stance and its use.

All action is reaction, indirect power.

Main power is from body center (using ground reaction).
Body center is intention and action center.
Intention and breath control power from body center by means of body dynamics and muscles action.

Sequence and timing of body segments.
Each segment achieve maximum velocity and than next distal segment moves to achieve maximal total amount of energy.
If a segment moves too soon, such as arm before body action, than body action energy is lost. If each segment is not used to its full range, than total amount of energy is smaller. In basic technique we would like to move through the full range, but in application we must make best use within available space.

Joint as action center 

Each joint serve as a fix point to allow maximum transfer of energy, and to allow optimal use of the muscles in next segment (muscles need stable base for optimal contraction).

Linkage between body segments
Connection between all body segments to avoid leak of energy.

Rate of muscles contraction
Maximum muscles contraction at shortest time to technique line of energy (the necessary muscles for the purpose) at kime and total relaxation (due amount of tension) in between action.
What said above gives us the right image, but to be more precise “the right amount of activation at the right time”.

Functional range and mobility
Restricted motion in one joint will cause compensation and over use in another. 

Stability
This has few implications, we must have balance between stability and fluidity, stability is not rigidity but rather firmness and mobility is not collapse or motion without direction. Firmness allows for fluent movement and transition and transfer of energy.
The small muscles around the sacrum and lumbar spine have to activate reflexively and subconsciously before any kick or punch to stabilize the low spine so it can provide a base from which we deliver energy from, and also to allow force transfer through the sacrum from the legs to the arms and vice versa. This stability also reduces stresses on the spine. We cannot talk about moving from the body center if the sacrum and lumbar are not stable.
Also, at the right moment each joint has to be stabilized to allow transfer of energy between segments and to allow optimal muscles activation as well as to protect the joint. (see above joint as action center).

Optimal stability help us avoid introducing other movement directions than the intended direction.

The power of karate is in the brain, not in the muscles, says recent Oxford university research.

Group of healthy controls was compared with a group of karate black belts, who are able to perform rapid, complex movements that require years of training.

Researchers chose to investigate karate experts’ ability to generate extremely high impact forces as this ability is not replicable by novices, and the mechanism used to achieve this feat not fully understood.

The research investigated weather the ability to control ballistic movement is associated with difference in white matter microstructure in the brain.

Early studies found that although karate experts were able to generate higher impact forces than controls, isometric muscle force and velocity measurements of individual joints were not significantly different.

Karate experts demonstrated higher peak acceleration in ballistic elbow extensions, but this was not related to activity in the biceps or triceps as measured by electromyography.

Karate experts are better able than novices to coordinate the timing of inter-segmental joint velocities.

Further research demonstrated that these individuals are also better able to maintain body stability by reducing the amount of backward replacement during punching to produce higher impact forces.

It was interesting to the researchers because karate punching is rapid, ballistic movement, yet performance was not determined by muscular strength, but rather by timing and coordination, specifically, the relative timing of different joint velocities.

There was significant differences between groups in the microstructure of white matter in the superior cerebellar peduncles (SCPs) and primary motor cortex – brain regions that are critical to the voluntary control of movement.
These findings suggest a role of the white matter pathways of the PCPs in motor expertise.

This study examined the behavioral and brain basis of expert motor control in karate experts. It was found that these individuals are able to repeatedly coordinate certain actions with a level of skill that novices are unable to reproduce. We argue that these abilities may be due primarily to changes to white matter structure in the SCPs, allowing the synchronization of movements of the upper limbs and trunk with a high degree of accuracy. This is the first example of a link between human cerebellar white matter and motor control measures in an elite sporting group. This has implications for our understanding of the role of white matter connectivity in motor coordination, the relationship between measures of white matter microstructure and elite performance, and how brain changes may be related to the stage of development in which learning begins.