Simple infrared shader, how to lerp multiple colors in 4 'stages'?

Hello,

I am working on a simple infrared shader. The basic idea is that it should show a color based on the temperature. I have 4 colors, (black, red, blue and white) the coldest object should shade black, the hottest object should shade white. In between I wanna have a gradient. Each frame i pass the temperature range to the shader. This is the max temperature - min temperature (e.g: range= 500 - 100 = 400)

I made a setup, which shades the right color per 'stage':

this is what i want, but now i want the colors to blend into each other. Because I have 4 colors i decided i needed 3 lerps. That gave me the next result:

The first screen is based on 4 'stages', the second one is based on 3 'stages' because of the 3 lerps. How can i make a shader which blends the colors into each other from the first image?

This is my shader till now:

 Shader "Unlit/Infrared" //This shader displays a red gradient
 {
     Properties{
     //define some properties
         _Texture("Normal Texture",2D) = "white" {}
         _temperature("Temperature",float) = 0.0
 
         _infraOn("Infrared On/Off",int) = 0
 
         _whiteColor("White color",Color) = (1,1,1,1)
         _redColor("Red color",Color) = (1,0,0,1)
         _blueColor("Blue color",Color) = (0,0,1,1)
         _blackColor("Black color",Color) = (0,0,0,1)
 
         _temperatureRange("Temperature range",float) = 0
     }
 
     SubShader 
     {
         Tags { "Queue" = "Transparent" } //enable when making transparent texture
         Pass 
         {
             Blend SrcAlpha OneMinusSrcAlpha //enable when making transparent texture
             
             CGPROGRAM
             
             //tell CG what functions to use
             #pragma vertex vert
             #pragma fragment frag
             #include "UnityCG.cginc"
 
             //uniform so the texture stays in for multiple sessions
             uniform sampler2D _Texture;
             uniform sampler2D _TextureInfrared;
 
             uniform float4 _whiteColor;
             uniform float4 _redColor;
             uniform float4 _blueColor;
             uniform float4 _blackColor;
 
             uniform float _temperature;
             uniform float _maxTemperature;
             uniform float _minTemperature;
             uniform float _temperatureRange;
             uniform int _infraOn;
 
             //from unity to shader
             struct AppData {
                 float4 position : POSITION; //right semantics
                 float4 uv : TEXCOORD0; //
             };
 
             //from vertex to fragment
             struct VertexToFragment{
                 float4 position : SV_POSITION;
                 float4 uv : TEXCOORD0;
                 float4 objectOrigin : TEXCOORD2;
                 float3 localPos : TEXTCOORD1;
             };
 
             VertexToFragment vert(AppData input){
                 VertexToFragment output; // define output for fragment shader
                 output.localPos = input.position; // like position
                 output.position = mul(UNITY_MATRIX_MVP,input.position); // multiply the position by the matrix model view projection
                 output.uv = input.uv; // copy uv
 
                 //determine center:
                 output.objectOrigin = unity_ObjectToWorld[3];
 
                 return output;
             }
 
             //return the right color
             float4 frag(VertexToFragment v) : SV_Target{
                 float normalizedIntensity;
                 float normalizedTemperature;
                 float colorIntensity;
 
                 float4 color;
                 if (_infraOn==1) { // return the infrared texture
 
                     float4 center = v.objectOrigin;
 
                     float distanceToCenter = distance(v.localPos, center); // between 0.5 and 1.0
                     normalizedTemperature = _temperature / _temperatureRange; // between 0 and 1
 
                     colorIntensity = normalizedTemperature * distanceToCenter; // should be between 0 and 1, not sure though
                     
 
                     //debug for all intensity colors, the 4 'stages' according to the first screen: 
                     if (colorIntensity >= 0.0f && colorIntensity <= 0.25f) { /// coldest color
                         color = _blackColor;
                     }
                     else if (colorIntensity >= 0.25f && colorIntensity <= 0.5f) { // almost coldest
                         color = _blueColor;
                     }
                     else if (colorIntensity >= 0.5f && colorIntensity <= 0.75f) { // almost hottest
                         color = _redColor;
                     }
                     else if (colorIntensity >= 0.75f && colorIntensity <= 1.0f) { // hottest color
                         color = _whiteColor;
                     }
 
 
                     float normalizedIntensity;
 
 
                     //Lerp with 3 stages, according to the second screenshot: 
 
                     // 4 colors, 3 lerps but normalize intensity to make right gradient. 
                     //works with 1 lerp on each object...
                     //TODO:: make sure only 1 lerp happens per object
                     if (colorIntensity >= 0.0f && colorIntensity <= 0.33f) {
                         normalizedIntensity = colorIntensity / 0.33f;
                         color = lerp(_blackColor, _blueColor, normalizedIntensity); 
                     }
                     if (colorIntensity >= 0.33f && colorIntensity <= 0.66f) {
                         normalizedIntensity = colorIntensity / 0.66f;
                         color = lerp(_redColor, _blueColor, normalizedIntensity); 
                     }
                     if (colorIntensity >= 0.66f && colorIntensity <= 1.0f) {
                         normalizedIntensity = colorIntensity / 1.f;
                         color = lerp(_redColor, _whiteColor, normalizedIntensity); 
                     }
                 }
                 else { // just return the normal texture
                     color = tex2D(_Texture,v.uv);
                 }
 
                 return color;
             }
             ENDCG
         }
     }
 }
 

Thanks in advance! Thomas