--------------
Euler Identity
--------------

This directory contains a single Notation3 file that expresses the 
Euler identity in the ARC style: it states the Answer, gives a short 
Reason, and carries several independent Checks that verify the claim 
each time the file is reasoned over. The file uses only the official N3 
math built-ins in rule bodies and models complex numbers with small 
backward rules, which act as lightweight, user-defined built-ins for 
complex sum, product, and the mapping e^{iθ} → (cos θ, sin θ).

The Answer is asserted when tolerant comparisons show that cos(π) is 
essentially −1 and sin(π) is essentially 0. The Reason is given as 
plain text that points to Euler’s formula viewed as “rotation on 
the unit circle.” The Checks verify the claim from different angles: 
- a direct trigonometric evaluation
- a computation of e^{iπ} via the  user-defined complex exponentiation
- a unit-magnitude check |e^{iπ}| = 1
- an angle-doubling identity e^{iπ}·e^{iπ} = e^{i·2π}.
All comparisons use a small tolerance of 1e−12 so harmless floating-point 
residues do not mask success.

To run it, load the N3 file into an N3 reasoner that supports the math 
built-ins and backward rules, such as EYE, and materialize the inferred 
triples. Any reasoner that executes the built-ins for cosine, sine, 
arithmetic, absolute value, and order will do. After reasoning you 
should see the concise summary alongside the checks, for example:

@prefix : <http://example.org/arc/euler#>.
@prefix math: <http://www.w3.org/2000/10/swap/math#>.

:answer :text "e^{iπ} + 1 = 0".
:answer :pair (0 0).
:reason :says "Use e^{iθ} = (cos θ, sin θ); with θ = π we get (-1,0), so e^{iπ}+1=0.".
:reason :uses math:cos.
:reason :uses math:sin.
:reason :at 3.141592653589793 .
:reason :cos -1.0 .
:reason :sin 1.2246467991473532e-16 .
:check1 :ok true.
:check2 :ok true.
:check3 :ok true.
:check4 :ok true.
:check5 :ok true.

If you wish to adapt the file, you can change the angle, tighten or 
loosen the tolerance, or add more checks, so long as rule bodies remain 
triple patterns that call only the documented N3 built-ins. In this way 
the ARC artifact stays self-contained, readable, and repeatable: it 
says the answer, explains the reason, and proves itself every time it 
runs.

