Essential mods: (Primed) [Continuity], [Brief Respite], [Rolling Guard], [Catalyzing Shields], and two Augur mods on warframe or weapons. Everything else is extra

With modding as Gauss your main priority goes Duration>Range>Strength>Efficiency

Duration:

• Affects energy Drain of holding Mach Rush, not very important but nice.
• Affects uptime of Kinetic Plating, since you want it up all the time it is very useful.
• Affects area duration and status duration of Thermal Sunder. It also increases the burst damage of the blast, as it consumes all damage you would have done if you kept it on the field.
• Affects uptime of Redline, and since you want it up all the time it is very useful. Also positively affects the Fire Rate Buff, Attack Speed Buff, Reload Speed Buff, and Casting Speed Buff given by Redline.
• Recommended to use all five Archon Shard slots for duration, that's how good it is.

Range:

• Affects the Knockdown Radius and Shockwave Radius of Mach Rush
• Affects range of Mach Crash which is incredibly useful if you're running that augment.
• Affects the Initial Radius and Final Radius of Thermal Sunder, incredibly good for the long range crowd control, long range nuke, and long range armor strip it provides.
• Does NOT affect the projectiles coming out of Gauss during Redline.

Strength:

• Efficiency stays at 100% for shield gating, you can put it lower using Blind Rage but strength doesn't do much for you compared to duration.
• Strength is last and not really needed as the damage and gauss' abilities aren't incredible and if you want a nuke frame there are much better options.
• Rolling Guard is in case you want to shield gate with this build and helps with survivability vs toxin or unexpected damage
• Catalyzing Shields is necessary for shield gating and allows one cast of Thermal Sunder to regain all your shields.
• Exilus: Honestly anything works in this slot, with the addition of more [Enemy Radar] on warframes you don't need to run [Enemy Sense], but you can if you want. [Vigilante Pursuit] is nice if you have other Vigilante mods in your build for the crit chance. [Rush] is also nice for the sprint speed which affects holding 1, which you won't do often but situationally okay. [Coaction Drift] allows you to run one less augur mod, and if your team is using an impactful aura, probably the most substantial difference found running this, but since solo you won't see much of a difference without removing Augur Reach, it's just okay.
• Arcanes: [Gauss] is so strong anything works, but Aegis synergizes so well with his kit that even a Rank 1 arcane would do wonders. Apart from that I'd recommend Energize or Molt Efficiency. Molt Efficiency becomes less useful when enemies start one shotting your shields, but it's pretty useful at lower levels when you aren't taking enough damage to rely on Arcane Aegis