DEF
Let $\mathcal F_0$ denote THE FIRST SENSATION.

Let $Q>0$ be its fixed primitive amplitude, and let $\theta$ indicate the phase position in one complete reading progression:

$0\leq\theta\leq 2\pi.$

$\theta$ is here a dimensionless phase coordinate. It tells us where in the progression the reading is situated. The connection between this progression and manifest time is addressed in point 5.4.

Starting at an upward zero crossing, the impulse is represented as:

$\mathcal F_0(\theta)=Q\sin\theta.$

Amplitude and current readout value are different:

$\max_{\theta\in[0,2\pi]}|\mathcal F_0(\theta)|=Q,$

while $\mathcal F_0(\theta)$ varies throughout the cycle.

The mathematical rotational representation is:

$\mathbf u(\theta)=Q(\cos\theta,\sin\theta).$

With the component readout

$\pi_2(x,y):=y$

we obtain:

$\mathcal F_0(\theta)=\pi_2\bigl(\mathbf u(\theta)\bigr).$

$\mathbf u$ and $\pi_2$ are representation notation. They do not denote a physical object rotating in a pre-given space, nor a transfer from KNOWING to the manifest side.

Reading point denotes the current phase in the ongoing reading. It is not a spatial position to which FOCUS must move, and it is also present between the qualitative boundary events.

Absolute continuity denotes that the reading of FOCUS is not interrupted between reading phases, during examinations against KNOWING or at the transition between completed and new impulse progressions. The term is used here ontologically. It should not automatically be identified with the specific concept of «absolute continuity» for mathematical functions in analysis.

REL — the conditions that determine the sinusoidal form
The sine argument uses the following explicit conditions:

1. Cycle: The primitive progression is represented as one complete closed phase cycle, $S^1$.
2. Phase homogeneity: No phase position is structurally privileged in the cycle itself. The phase shifts are represented by a uniform rotational action.
3. Minimal representation: The phase shifts are expressed through a continuous, faithful real linear representation of the lowest possible dimension. We use an orthonormal presentation in which the action is rotation in a two-dimensional representation plane.
4. Readout: The manifest progression of the form is represented by orthogonal component readout along one fixed axis.
5. Amplitude: The primitive amplitude $Q$ is fixed and positive.

In the minimal representation, the rotation can be written:

$R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.$

For a starting point with radius $Q$ and phase $\varphi$,

$\mathbf u_0=Q(\cos\varphi,\sin\varphi),$

the progression becomes:

$\mathbf u(\theta)=R_\theta\mathbf u_0=Q\bigl(\cos(\theta+\varphi),\sin(\theta+\varphi)\bigr).$

The component readout thus gives:

$\mathcal F(\theta)=\pi_2\bigl(\mathbf u(\theta)\bigr)=Q\sin(\theta+\varphi).$

The sinusoidal form is necessary under these structural and readout premises. The orthogonal component readout of the phase-homogeneous cycle's minimal rotational representation is a sine. When the premises are present, the form is not an additional free choice.

The choice $\varphi=0$ places the beginning at an upward zero crossing. A global sign reversal switches the orientation of the displacements, but does not change the primitive form type.

Continuity, closure and replicability alone are not sufficient to derive a sine. Phase homogeneity, the representation condition and the component readout must be stated explicitly.

The principle of absolute continuity establishes that the reading is not interrupted. It does not replace the other premises that determine the impulse's form.

Replica identity establishes that subsequent primitive SENSATIONS have the same form and amplitude. It is not alone the justification for the first form being a sine.

MECH — the basic reading

1. Continuous reading
FOCUS remains fixed. The manifest impulse unfolds in the current field of view.

In the mathematical presentation, the reading progression is defined over the entire interval:

$[0,2\pi]_\theta.$

None of the subintervals between the qualitative boundaries is omitted. The reading therefore does not consist only of the registrations at $E_0,\ldots,E_4$.

FOCUS reads one and the same $\mathcal F_0$ through four connected manifest progressions:

$[0,\tfrac{\pi}{2}],\qquad[\tfrac{\pi}{2},\pi],\qquad[\pi,\tfrac{3\pi}{2}],\qquad[\tfrac{3\pi}{2},2\pi].$

The division describes the progression. It does not divide the reading into independent, separate acts.

2. Qualitative boundary events
Here $E_k$ is used as shorthand for the manifest boundary events $E_k^M$. $E_0$ is the starting boundary. The four subsequent boundaries $E_1,\ldots,E_4$ provide the checkpoints for assessing the reading result that has emerged. Five boundary events therefore do not imply five AE checks.

Boundary	Phase $\theta_k$	Readout value	Manifest status
$E_0$	$0$	$0$	First registration at the starting boundary
$E_1$	$\pi/2$	$Q$	Positive extremum; the first part has been read
$E_2$	$\pi$	$0$	First return to zero; one half-form has been read
$E_3$	$3\pi/2$	$-Q$	Negative extremum; three parts have been read
$E_4$	$2\pi$	$0$	The entire sinusoidal progression has been read

The reading order is:

$E_0\prec_{\mathrm{read}}E_1\prec_{\mathrm{read}}E_2\prec_{\mathrm{read}}E_3\prec_{\mathrm{read}}E_4.$

$\prec_{\mathrm{read}}$ denotes the order in the manifest reading. It does not denote an internal process in KNOWING.

Let

$\mathcal G(\mathcal F_0):=\{E_0,E_1,E_2,E_3,E_4\}$

denote the five qualitative boundary events. «Boundary» here means a boundary in the division of the reading progression, not the mathematical boundary of the sine graph.

Each boundary event belongs to the same SENSATION. None of them alone denotes a new, independent impulse.

3. CHANGE activates examination, not restarting
At a qualitative CHANGE, the examination of whether the form that has emerged so far can be understood as a concept is activated.

This is different from a reading point arising. The reading point is present continuously; the examination is activated at the CHANGE.

The whole of the current THIS is already registered in FOCUS. The examination does not produce this whole, but concerns the understanding of what is now present.

The examination does not introduce a break in the reading. If the form has not yet been completed and understood, the reading continues through the further manifest progression.

The O-principle, the AE checks and their positions in relation to each other are elaborated in the following points. Here only the distinction between continuous reading and activated examination is established.

4. Zero value and closure
The zero values must be distinguished from the status of the events:

$\mathcal F_0(0)=\mathcal F_0(\pi)=\mathcal F_0(2\pi)=0,$

but $E_0$, $E_2$ and $E_4$ are three different boundary events.

An equal readout value therefore does not imply an equal reading status. At $E_0$ the reading begins, at $E_2$ the half-form has been read, and at $E_4$ the whole form has been read.

The zero value denotes that the impulse's current displacement is zero. It does not imply that IS-NESS or the complete THIS ceases. The relationship between the zero axis and VOID is explained in point 5.5.

At a full cycle:

$\mathbf u(2\pi)=\mathbf u(0).$

This is geometrical phase closure. It does not imply that the complete THIS at the end is identical to THIS at the beginning.

5. Completed form and necessary conceptualisation
At each reading point, FOCUS registers the whole of the current THIS. This completeness must be distinguished from the completed reading of the whole impulse. At $E_4$, the entire sinusoidal progression has been read. The terminal O-perspective makes the fully read FORM available as a whole for the examination addressed further in 5.4–5.5.

Having the form fully read and seeing the form as SEPARATION are different determinations.

At the completed sinusoidal form that has returned to zero, conceptualisation must occur. This point establishes this transition as part of the author's description of the mechanism, but does not yet derive why the result is necessarily SEPARATION.

The status of the AE result in KNOWING and the conceptualising observation are addressed later. That the explanation here stops at a completely registered form does not imply any pause in the EC.

PLACEMENT
Point 5.3 builds on point 5.2:

$\begin{aligned}&I_0\text{ is present without confirmed KNOWING of its own IS-NESS}\\&\quad\text{under the necessity principle: IS-NESS must KNOW}\\&\longrightarrow\mathcal F_0\text{ felt through AWARENESS}.\end{aligned}$

This point describes the sinusoidal form and the continuous reading up to a completely registered form.

The conceptualising observation, in which the form is seen as SEPARATION, is addressed separately. The further connection to the next unresolved relation is described when the entire EC is to be explained.

Point 5.4 explains space and time as aspects of the same reading process. The order of explanation does not imply that the reading takes place first and then has space and time added to it.

X

• One primitive SENSATION has one whole sinusoidal form.
• The justification for the sine uses phase homogeneity, minimal rotational representation and orthogonal component readout.
• All subsequent primitive SENSATIONS are replicas of the same primitive form and amplitude.
• FOCUS remains fixed.
• A current reading point is present throughout the entire reading, including between the qualitative boundaries.
• The reading of FOCUS is uninterrupted.
• CHANGES activate examinations of comprehensibility without restarting the reading.
• The five qualitative boundaries belong to one and the same impulse.
• THE FOCAL VIEW continuously follows the impulse's development. THE PANORAMIC VIEW encompasses the current THIS and registers the CHANGES. At the qualitative checkpoints, the examination of the reading result through the AE is activated.
• At completed reading, a complete reading result is present; the terminal O-perspective makes the FORM available as a whole.
• Complete form in the field of view and the form seen as SEPARATION are different determinations in the same uninterrupted progression.

NOT-X

• The sine is not a physical wave in a pre-given space.
• The rotational representation does not mean that FOCUS itself moves or rotates.
• A marker moving along a drawn curve is not FOCUS moving through a space.
• The component readout is not a domain transfer.
• The representation's circular rotation is not in itself the O-principle's full all-round registration or the later conceptualising quarter-turn.
• The reading point is not activated only at qualitative CHANGES.
• An examination against KNOWING does not imply a pause in the reading or an internal temporal process in KNOWING.
• $\theta$ does not presuppose an external clock.
• Fixed amplitude does not mean a constant readout value.
• The five boundary events are not five independent impulses.
• The negative half is not a separate counter-SENSATION.
• A zero readout value does not mean absolute non-IS-NESS.
• A completed sine reading is not the entire EC.

BOUNDARY-X
A distinction must be made between:

• the ontological SENSATION and its mathematical representation;
• the cycle's phase homogeneity and the readout's different values;
• invariant amplitude and varying displacement;
• stationary FOCUS and phase progression in the reading progression;
• a continuous reading point and a qualitative boundary event;
• continuous reading and an activated examination against KNOWING;
• the complete current THIS and a fully read impulse;
• completed form in the field of view of FOCUS and conceptualisation as SEPARATION.

This point introduces absolute continuity as an ontological principle. It does not alone provide a full mathematical description of continuity through all EC phases and between different impulses.

Nor does this point go through the AE checks, the thirteenth step or the entire EC. The significance of the O-principle for space and the significance of the reading progression for time are addressed in point 5.4.

The sinusoidal form is necessary under the listed structural and readout premises. The ontological justification must make these premises explicit; they must not be hidden in the word «minimal». When they are present, the sinusoidal form follows as the mathematical result.

FAIL_COND
The following claims would conflict with this point's definitions or justification:

• that continuity, closure and replicability alone uniquely determine the sinusoidal form;
• that the representation or readout premise can be omitted without affecting the sine argument;
• that THE SENSATION changes its primitive form or amplitude from one primitive SENSATION to the next;
• that FOCUS itself must move spatially in order to read the impulse;
• that reading points are present only at the five qualitative boundaries;
• that the reading stops during the examinations against KNOWING;
• that the end of one impulse necessarily introduces a break before further reading;
• that a partial sine is already a fully read form;
• that one of the five boundary events is in itself a new, independent SENSATION;
• that the same readout value necessarily implies the same THIS;
• that complete registration of the current THIS implies that the rest of the impulse has already been read;
• that the representation presupposes an established IDENTITY, a pre-given physical space, external clock time or physical movement as a startup premise.

These are criteria for internal consistency and for valid use of the sine argument. They do not replace the justification for the ontological premises.

Close FORMAL PRESENTATION »