THE MATHEMATICAL CITYOBSERVATORY / 01
YOUR EXPLORATION, ON THIS DEVICE
A LIVING WORLD OF IDEAS 01 / EXPLORE

Complex ideas.
Made playable.

Follow your curiosity through a city of signals, learning, poetry, and proof. Every discovery has another way in.

THE FINITE CORE125→47Explore the mathematics.
Keep its boundaries visible.
CONSTELLATION MAPChoose a district · drag to orbit
A shared home for your existing worlds.EXPLORE / EXPERIMENT / VERIFY
YOUR FIRST DISCOVERY

A signal finds its shape.

Change a signal. Make a prediction. Watch what remains. Then inspect the evidence.

  1. Explore
  2. Predict
  3. Experiment
  4. Inspect
TOUCH THE MATHEMATICS 02 / SIGNAL GARDEN

Watch a signal
find what stays.

Seven spectral sectors. Two retained. Change the initial overlap and see how the filter reshapes the signal.

SPECTRAL LANDSCAPESTEP 0
Conditional fidelity—
Success probability—
Complement norm—
Unnormalized sector population after filtering
SectorDimensionPopulationSelection
FROM YOUR ATTACHED QGNN CODE EXACT COMPRESSED SIMULATION

Evolve inside the retained carrier.

Four graph-derived quantum gates change a fixed retained input while preserving its norm. Move an angle and watch the observable respond. These controls are separate from the spectral filter above.

47 RETAINED COORDINATESLOADING
Observable expectation—
State norm—
Norm error—
Inspect the 47 populations
Coordinates in the supplied retained eigenbasis, not torus-node addresses
CoordinatePopulation

WHAT YOU ARE SEEING

Retention and contraction.

The filter leaves the J = 2 and J = 5 sectors unchanged and contracts the other sectors. The complement norm is bounded by (15/17)n times its initial norm. Conditional fidelity is undefined when the selected overlap is zero.

LEARN BY CHANGING THINGS 03 / NEURAL LAB

Teach a boundary.
See learning happen.

A small neural network learns from synthetic points. Predict, train, compare—and keep neural accuracy separate from spectral projection.

DECISION FIELDREADY
Epoch0
Training accuracy—
Mean squared error—

A real trainer, a clear boundary.

This browser lab runs deterministic gradient training on synthetic data. Its training accuracy does not measure performance on unseen real-world data. The existing ENTRAIN instrument includes an additional feature encoding and E47 projection monitor; the Signal Garden here is a separate experiment.

Inspect every training point and prediction
All 128 synthetic points: inputs, labels, and current predictions
PointXYLabelProbabilityPrediction
MANY WORLDS, ONE WAY IN 04 / ATLAS

Your instruments.
Connected.

Explore the existing surfaces. Each destination keeps its own purpose and evidence boundary.

CURIOSITY WITH A TRACE 05 / EVIDENCE

See how
we know.

Inputs, assumptions, sources, and limits stay attached to each experiment.

FINITE MATHEMATICS

The spectral core

H = V₂ ⊗ V₂ ⊗ V₂, dimension 125.
K = (C − 6I)(C − 30I).
ker K has dimension 47.
Γ = I − K²/99144.

The browser Signal Garden computes seven-sector norm observables. It does not reconstruct phases, compile quantum gates, or run on quantum hardware.

Inspect the matrix proof ↗Open the research workbench ↗
SYNTHETIC EXPERIMENT

The learning model

A two-input, one-hidden-layer neural network with a scalar probability output. Deterministic datasets and initialization make experiments replayable.

Training uses synthetic labels. Accuracy and loss are training metrics. A low loss is not evidence of quantum advantage or a physical theory.

Read the executable model ↗Inspect numerical checks ↗
REPRODUCIBILITY

Take the experiment with you

Replay files contain validated inputs. Imported results are recomputed. Saved progress stays on this browser unless you export it.

MAKE TIME FOR CURIOSITY

Plan an exploration.

Choose your own date and save a 30-minute session with your current replay link. Calendar changes happen only when you confirm in your calendar app.

Published evidence

A curated snapshot of explicitly public registry records. Registration is distinct from independent verification.

Published snapshot · 13 October 2026

MAKE IT YOURS

Your way into the City.

All experiments are available through controls and text. Nothing requires dragging, color recognition, or a timer.

PASS THE DISCOVERY ON

Share an idea.
Keep its context.

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