Paper 3: the psi field and where leakage comes from

Paper 3 gives the boundary between branches a microphysics. Not a metaphor, not an information-theoretic sketch: a field, with a Lagrangian, an instanton calculation, and a saturation function. It is also the paper whose early drafts contained two of the worst errors in the programme, both since corrected, and this post covers both the physics and the scars.

The field

Psi is the boundary permeability field: a scalar whose value measures stress on the wall between two branches. Its source is specific. In the framework’s bulk, a four-dimensional Walker-Wang substrate carries a dilaton field sourced by trace anomalies, the same quantum effect that makes the energy-momentum tensor’s trace nonzero in curved or stressed media. Where the bulk is stressed, the boundary value of the dilaton, which is Psi, rises. The equation of motion is a wave equation with a mass term, driven by the trace of the bulk stress-energy. Nothing in this construction is exotic: dilatons, anomalies and domain walls are all standard. The exotic claim is narrowly that the boundary value of such a field modulates permeability between sectors.

Instantons, twice

The suppression of leakage is instantonic, and the paper’s history holds both the failure and the repair of this calculation. The failure: a naive thin-wall estimate, multiplying a wall tension by a wall thickness cubed, gave an instanton action of exactly 1 for any wall, meaning a leakage probability of about 0.37. A wall that leaks a third of the time on every crossing is not a boundary; the estimate failed by two orders of magnitude in the exponent, and it failed independently of every parameter, which is the signature of a broken estimate rather than a tuned one.

The first repair was the area-quantised version: in a surface-code architecture, leakage requires a defect string to traverse the code distance, 64, and the action is the distance times a logarithmic factor, 147, giving a suppression of order ten to the minus 64. That repair is now itself demoted: the logarithmic ratio inside it was reverse-engineered from a pre-committed target, so the 147 is a proxy with a target’s fingerprints, and the papers keep it only as an illustration of the scaling shape.

The surviving version replaces the proxy with the standard Yang-Mills exponent: leakage is theta-vacuum mixing, its action is 8 pi squared over g squared with the coupling pinned by the boundary category’s central charge, and the action is 4 pi squared times 7, which is 276.35. That matches the logarithm of the independently known vacuum suppression to five hundredths of a per cent, and with the Z6 correction, to four thousandths. The correction banner on the paper states the ordering honestly: proxy retained, physical parent identified, mechanism pending a boundary calculation.

Saturation, and the two scars

The probability function took its own repair. Written naively as proportional to e to the Psi, leakage diverged: at moderate stress it exceeded one, reaching absurd values like ten to the 27, which is not a probability. The fix is the logistic saturation, p equals delta naught e to the Psi over one plus delta naught e to the Psi, bounded in the unit interval for every Psi, agreeing with the naive form at small stress where all applications live. Both scars, the broken action and the broken probability, are already in the corrections series of this archive with their arithmetic; this post’s summary is here because Paper 3 is where they live in the physics.

What the paper contributes now is a leakage operator that is linear, saturated, exponentially suppressed, and sourced by standard bulk physics, with kill criteria stated: a boundary computation that returns a different coupling, or a blinded experiment that finds unstructured correlations, and the hypothesis loses its edge. The next paper builds the bulk that sources it.