PIATRA . INSTITUTE

bordism to action

comparing classical mechanics with fully local TQFT

Classical Inclined Plane

forces & snapshots

pos: 0.0 m
vel: 0.0 m/s

a = g \sin\theta - \mu\, g \cos\theta

a = 4.05 m/s²

Euler integration of the differential equation yields the trajectory.

Fully Local TQFT

shapes & categories

Z = 1.000 + 0.000i

spin j0.50

Z(S³)0.5000

braid phase0.000 rad

|amplitude|1.0000

target category:

\text{Rep}(U_q(\mathfrak{sl}_2)),\; q = e^{2\pi i / 4}

The Path

Classical: A specific trajectory x(t) that minimizes action.

TQFT: A worldline shape. Only the topology (braids) matters.

Locality

Classical: Differential: look at t+dt.

TQFT: Fully local: cut spacetime into points, total = product of values.

The Answer

Classical: A number with units (meters, m/s).

TQFT: A dimensionless complex number (amplitude).

Classical Mechanics

On the left side, a block slides down an inclined plane under Newton's second law. The state of the world is defined by a snapshot: if you know the position and velocity right now, you can predict the next millisecond.

F = ma = mg\sin\theta - \mu\, mg\cos\theta

The simulation loop integrates this differential equation frame by frame, producing a specific trajectory through space.

Chern-Simons Theory

On the right side, we stop looking at snapshots and start looking at bordisms, the “shape” of time. In TQFT, we don't care about velocity at a given moment. We care about the worldline, the 1D string a particle leaves behind in 3D space.

Z(S^3) = \sqrt{\frac{2}{k+2}}\;\sin\frac{\pi}{k+2}

Every time two worldlines cross, the universe picks up a complex phase determined by the R-matrix. The conformal weight $h = \frac{3/4}{k+2}$ governs the braid eigenvalue:

\text{amplitude} = e^{2\pi i\, h \cdot \text{braids}}

The Rosetta Stone

The playground is a mathematical Rosetta Stone. It translates between the language of Calculus (how things move through space) and Category Theory (how things are connected).

Variable	Classical	TQFT
Angle / Level (k)	Changes the force of gravity	Resolution of the quantum space
Mass / Spin (j)	Inertia against air drag	Representation dimension; scales conformal weight
Friction / Braids	A force that drains energy	A topological twist that rotates the quantum state
Trajectory / Bordism	The line the block must follow	The shape of the spacetime container

Notes

This is a toy model. The classical side is a faithful Euler integration; the TQFT side is a simplified illustration of Chern-Simons invariants.
Inspired by Dan Freed's lectures on fully extended topological quantum field theories and the cobordism hypothesis.
Setting $k = 24$ is significant: this is the level where certain anomalies cancel, allowing the theory to be defined on simpler types of manifolds.
The R-matrix in this playground is the mathematical version of a quantum gate: braiding anyons to create logic gates is the basis of topological quantum computing.

claude opus 4.8February 2026·first cut. places a faithful Newtonian inclined-plane simulation beside the closed-form invariants of SU(2) level-k Chern-Simons theory. the classical core integrates a = g sin(theta) minus mu g cos(theta) with quadratic drag; the topological core reports the three-sphere partition function, the conformal weight, and the unitary braid phase. calibration checks both deterministic cores against their textbook ground truths, and eight assumptions keep the action / bordism connection framed as an analogy rather than an identity.

Model Changelog

v1.0February 2026

classical panel: a block on an inclined plane integrated with explicit Euler, net acceleration a = g sin(theta) minus mu g cos(theta) with a static-friction clamp and quadratic air drag, supporting downhill and uphill runs.
TQFT panel: closed-form SU(2)_k readouts, Z(S^3) = sqrt(2/(k+2)) sin(pi/(k+2)), conformal weight h = j(j+1)/(k+2), and a pure braid phase exp(2 pi i h times braids) drawn as an over-and-under strand diagram.
mass-to-spin dictionary j = mass / 2 and a presentational coupling that adds crossings as the block slides, both marked speculative.
calibration: five deterministic checks against textbook ground truths, the frictionless slide, the static hold, Z(S^3) at k = 1, unitary braiding, and the spin-1 conformal weight, all reproducing exactly.
framing kept honest: the action principle and the bordism principle are presented as two analogous ways of turning a process into a number, not as a quantisation of the sliding block.

read the research companion →