∀x [(P(x) → Q(x))]4 nodes / depth 3closed formulaA small studio treating symbolic problems atypically
Don't look a gift horse in the mouth! We did...
Across generations
What they built matters.
So do their choices.
A lifetime's work carries more than a price. When family wishes, records and assumptions become tangled, we bring mathematical reasoning and patient investigation to the question: what is known, what follows, and what still needs checking.
The person whose future is being discussed should be heard, understood and free to decide.
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01 / symbolic engine
Make the inference undeniable.
Choose a worked proof. Inspect the premises, name the next rule, and watch an argument become a trace you can check.
proof consoleREADY FOR INPUT
Select a theorem
Loaded theoremThe Arkansas Derby (9 furlongs)A small, suspiciously well-behaved proof.
GoalHorse(comet)
derive the consequent
without hand-waving
without hand-waving
1given
∀x (GiftHorse(x) → Horse(x))premise2given
GiftHorse(comet)premise3next
GiftHorse(comet) → Horse(comet)choose rule4next
· · ·lockedRace distanceStarting gate
00 / 02 derived
Choose the next rule, or play the worked proof.
02 / evolutionary grammar
Let the syntax evolve.
Start with formula trees, exchange branches, mutate symbols, and inspect the statements that emerge. The grammar holds; meaning can change.
TTerminalatomic formula: P(x), Q(y)
NNon-terminalopens a branch: ∀, ¬, →
∴Phenotypethe parsed statement we can read
population sandbox generation 00
Structure first. Meaning is yours to inspect. Choose a node in one tree, or matching nodes in two.
∀x [ P(x) → Q(x) ]∃y [(R(y) ∧ S(y))]4 nodes / depth 3closed formula¬ [(A(x) ∨ B(x))]4 nodes / depth 3free: x∀y [(F(y) → G(y))]4 nodes / depth 3closed formulaselection nonewaiting for a branch
Crossover exchanges equal-arity branches. Mutation changes one symbol or grows a replacement branch.
offspring bayNo offspring staged
Select branches in the population, then choose an operator. The next parseable statement will appear here.
Population seeded. Select a tree or branch to begin.
Every leaf is an atomic formula. Quantifiers bind x or y; free variables are shown on each card. Mutation and crossover can change variable binding and truth. No fitness score or equivalence claim is attached.
01Subtree crossover
Two parents exchange marked branches, making two new trees without touching the originals.
02Point mutation
One node label changes while its arity stays fixed, so the tree remains well formed.
03Undo / redo
Adopted generations are reversible. Exploration should not require a sacrificial keyboard.
03 / maps into pictures
Dessin d'enfant / a map leaves a trace.
A Belyi map sends a surface to the sphere. Its inverse image of the interval [0, 1] leaves a graph you can inspect. Choose an example and inspect the vertices, edges and faces.
β⁻¹([0, 1])2 sheets / 1 face
preimage of 0 preimage of 1
04 / follow the root
Homotopy continuation / take the long way.
Start with roots you can name. Deform one system into another, predict the next position, correct it, and keep the path alive until the endpoint becomes legible.
known / t = 0G(x, y) = (x² − 1, y² − 1)four easy seed roots
target / t = 1F(x, y) = (x² + y − 1, y² + x − 1)the roots we want to reach
H = (1 − t)G + tFt = 0.00
reference branch tracked predictor corrected root
| Root | x | y | Residual ‖H‖∞ |
|---|---|---|---|
| α₁ | -1.000000 | -1.000000 | 0.00e+0 |
| α₂ | -1.000000 | 1.000000 | 0.00e+0 |
| α₃ | 1.000000 | -1.000000 | 0.00e+0 |
| α₄ | 1.000000 | 1.000000 | 0.00e+0 |
Across generations
A lifetime's work.
A question worth care.
There is a person behind the papers, and a history behind what they have built. Make time for their stories. Ask what matters to them now.
When the records, numbers or assumptions need closer inspection, bring us the question. We help make the reasoning clear enough to discuss, test and take further.
Our work is research and logical investigation. Legal advice and representation are outside our services.
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