Galactic Key emblem Galactic Key · Interface Calculus Studio Live Compiler · 4 Regimes

Galactic Key · Interface Calculus Compiler & Synthesis Studio

A browser compiler that turns proportional-font user interfaces into a branchless math circuit to differentiate, prove, and auto-fix layout laws across every screen width.

Compiled Circuit vs Oracle Measured
1,427 ops → 0.00 px Δ

Matches imperative layout across 1,500 random trials with zero mismatches.

Interval Proof [320, 1920] px Measured
1/5 → 5/5 laws proven

Taylor intervals and root-splitting certify every pixel width in 106 ms.

Inverse Design (40 Steps) Measured
2.68 → 0.00 cut-offs

Sigmoid annealing drops design loss 1,170.2 → 217.3 in 345 ms.

Who benefits: Designers and engineers get mathematical proof of zero clipped text, 44 px touch targets, and zero sideways scroll across every device width.
Second-order effect: Layout bugs become differentiable loss terms repaired automatically by gradient descent before code ships.
Risk and limit: Proportional word-wrap thresholds depend on the active font metric table, so custom web fonts must be measured before circuit compilation.

Recommendation & decision needed: Nothing needed from Berkeley; press Auto-Fix Design in the studio below to watch gradient descent repair all five laws live.

Compiled Circuit Blueprint Stage

1/5 Laws Proven 3 Cut-Off Cards Oracle Δ: 0.00 px
Interactive Studio, Mathematical Engines & Receipts

Circuit Controls

1,427 ops · 0.00 px oracle error

Every control below feeds directly into the compiled arithmetic circuit; tap any question mark or toggle Help to read what each parameter changes.

Selects which declarative UI specification to compile into a straight-line arithmetic circuit.

Chooses which design variable drives the purple forward-AD sensitivity arrows on the blueprint stage.

Inverse solver

Runs 40 steps of sigmoid-annealed gradient descent and interval certification to repair all five layout laws.

Reset state

Restores the initial broken design parameters so you can inspect the counterexample intervals again.

390px

Sweeps viewport width W from 320 px phone to 1,920 px desktop through the compiled circuit.

36px

Adjusts horizontal and vertical outer page padding p in pixels across all viewports.

26px

Adjusts grid gap g between adjacent cards and rows in pixels across the layout.

210px

Sets the minimum card width threshold m that triggers responsive column count transitions.

1.18×

Scales proportional typography σ, shifting greedy word-wrap step thresholds across all cards.

34px

Controls action button height b; must be at least 44 px to satisfy the touch target law.

0.00 (Exact)

Replaces hard Heaviside steps with smooth sigmoids σ(x/τ) so discrete jumps become differentiable.

Four-Regime Mathematical Inspector

Continuous Viewport Interval Proof

How Proportional Typography Compiles to Step Gates

Unlike monospace character counting, proportional fonts assign distinct pixel widths to every word. For any text string with measured word widths w₁, …, wₘ and space width s at base font size, greedy line wrapping is monotonically non-increasing in normalized line width u = textW / fontScale. The compiler binary-searches the exact critical thresholds U₂, U₃, …, Uₖ where the line count jumps from n - 1 to n, emitting one Heaviside step gate per line break: lines(textW, σ) = 1 + ∑ 1[textW < σ · Uₙ].

UI Specification Active Circuit Ops Oracle Error (500 Trials) Broken Laws Proven Solved Laws Proven Expected Height (Before → After)
Fleet Incident Cockpit 1,427 (860 +, 284 ×, 80 ReLU, 203 step) 0.00 px (0 mismatches) 1/5 (39.9% no-trunc) 5/5 (100.0% proven) 1,714.6 px → 1,521.7 px
SaaS Pricing & Checkout 925 (556 +, 192 ×, 54 ReLU, 123 step) 0.00 px (0 mismatches) 1/5 (5.4% no-trunc) 5/5 (100.0% proven) 1,188.5 px → 1,017.2 px
Executive Decision Brief 1,118 (680 +, 216 ×, 70 ReLU, 152 step) 0.00 px (0 mismatches) 1/5 (60.6% no-trunc) 5/5 (100.0% proven) 1,622.6 px → 1,424.9 px