# Apeiron AP1 – G31 balanced projective mode-chart reference

**Updated UTC:** 2026-09-02T21:42:32Z  
**Classification:** `M1_BALANCED_CHART_TRANSPORT_REFERENCE_PASS_STAGED_BACKGROUND_CONTINUATION_OPEN`  
**AP1 status:** ORANGE

The G29 coupled moving-split solver and the G30 first staged logarithmic
prefix checkpoint are hash-bound without changing any equation, physical
parameter, existing gate threshold, Pauli–Villars coefficient, or the fixed
boundary `q_cut = 0.6 Lambda`.

The same canonical mode equation is advanced in two dimensionless projective
charts,

- `R = v / (Lambda u)`, and
- `Q = Lambda u / v = 1 / R`.

The chart with the larger scaled denominator is selected after every
determinant-one transfer. This removes the coordinate pole of the original
single `v/u` chart without a vacuum reset or a change of state.

## Reference audit

- structural gates: `10/10 PASS`
- short single-chart ↔ balanced-chart source delta: `2.0122612347121938e-7`
- predeclared overlap cap: `1e-5`
- old single chart at relative `N = 0.13`: fail-closed at its projective pole
- balanced-chart dynamic pivots across the same interval: `5`
- vacuum resets: `0`
- coarse/fine mode-step source delta: `9.741665451760798e-11`
- predeclared mode-step cap: `5e-3`
- fine Wronskian relative error: `2.2447321779139907e-15`
- unchanged Wronskian cap: `1e-11`
- maximum fine active-chart ratio: `0.6666635715536942`
- maximum fine determinant defect before projection: `6.470386161084077e-7`
- predeclared determinant-defect cap: `1e-4`
- new regression tests: `9/9 PASS`
- combined M1 chain after G31: `93/93 PASS`
- registered project tests after G31: `211/211 PASS`
- local test discovery after G31: `186/186 PASS`

## Boundary

This is a projective transport and fixed-stored-source numerical reference,
not a coupled long-background candidate. No trajectory row, endpoint capsule,
new seed, production tolerance, present-anchor claim, physical response
kernel, observable, curve, fit, or significance is released. The next step is
to use the balanced chart in the next staged coupled logarithmic prefix while
retaining all existing physical and convergence gates.