Upgrading The Orientation Rule (\kappa_u > 0) From Operational To Derived

The programme currently uses the condition \kappa_u > 0 as an operational rule to fix the physical forward/readout orientation (the remaining global \mathbf Z_2). This is compatible with both brackets for the selector u (kinematic or dynamical).

Separately, the programme keeps a derivation target: explain why the physical readout must land on the constructive/persistent branch, i.e. why the forward/readout choice should coincide with \kappa_u > 0.

This note lists the clean upgrade routes and what they would have to assume.

After the recent static cleanup, these routes now matter in a more specific way than before. The even-sector selector in the current one-generation static closure is already the observable/readout projector P_obs, so any successful D1-D3 upgrade no longer settles only an orientation convention. It also controls whether the present static closure remains merely conditional or becomes part of a more unified parent-level derivation.

See also:

  • kernels/dynamics.md (orientation rule and its geometric content)
  • kernels/epistemics.md (operational readout rule and the remaining burden)
  • kernels/discrete-symmetries.md (disambiguates the different Z2 flips)
  • kernels/u-selector-bracketing.md (the [K]/[D] fork map and no-go statements)
  • kernels/orientation-d1-bulk-stability.md (a focused D1 attempt and its minimal gate)

Route D1: Bulk Stability / Attractor Principle

Assume there exists a bulk functional (energy, Lyapunov, entropy production, or equivalent) whose extremum/monotonicity selects the long-lived readout channel.

Target form:

  • show that for the readout-relevant family of bulk states, the long-time attractor/fixed point must satisfy Re_u(AB)|_* > 0
  • conclude \kappa_u > 0 for the physically realized direct readout branch

What this would buy:

  • turns \kappa_u > 0 into a bulk stability consequence rather than a named convention

What it must overcome:

  • symmetry and linearized branch-space stability do not fix the sign (see u-selector-bracketing.md)

One concrete candidate for the missing sign-sensitive stability functional is the conjugate-sum readout intensity |A+\bar B|^2, whose interference term is 2 Re_u(AB); see kernels/orientation-d1-bulk-stability.md.

If EM/readout coupling is taken to be axis-aligned (defined only after u is selected, e.g. gauge-fix u=e1) and to sample the conjugate-sum channel, then the D1 gate reduces to Re_u(AB)|_*>0 and the orientation rule \kappa_u>0 follows immediately from the fixed-point identity in kernels/dynamics.md.

This means D1 is now conditionally closed at the algebraic level: once the conjugate-sum readout channel is granted, the remaining implication to \kappa_u > 0 is immediate. What remains open is the physical justification for that readout channel.

Route D2: Ambient Scale/Readout Selector Descent

Assume there is an ambient selector (scale-flow / readout generator) upstream of the reduced slice (for example in SO(2,4)), whose induced reduced image fixes the forward/readout orientation.

Target form:

  • identify D_amb (or a flow field)
  • show that its induced orientation agrees with the constructive/persistent branch convention
  • read \kappa_u > 0 as the sign alignment between the ambient forward arrow and the odd bulk scalar

What this would buy:

  • makes the forward/readout arrow an induced geometric object rather than an epistemic postulate

What it must specify:

  • what data selects D_amb in the first place, and why it is physical rather than gauge

Route D3: Observer Coupling / Readout Arrow As A Physical Input

Assume that the observer/readout mechanism is itself a physical coupling that breaks the Z2 degeneracy (for example, a semigroup arrow t>=0 plus a coupling that requires persistence rather than inversion).

Target form:

  • model the readout coupling so that the only stable, consistently readable branch is the constructive/persistent one
  • show that this physical readout coupling forces \kappa_u > 0 in the phase-normalized gauge

What this would buy:

  • admits that “observability” is not purely kinematic, while still turning the rule into a consequence of the readout mechanism

What it concedes:

  • the sign is not fixed by the bare transport algebra; it is fixed at the interface of dynamics and epistemics

Minimal Writing Rule

Until one of D1–D3 is completed, statements should be written in one of two explicit forms:

  • Operational rule: “we adopt \kappa_u > 0 to fix the forward/readout orientation.”
  • Derivation target: “we aim to derive \kappa_u > 0 from {bulk stability / ambient selector / readout coupling}.”

Current Priority

The routes are not all equally valuable at the current stage.

  • D1 is the sharpest route for the sign itself.
  • D2 is the highest-leverage route overall, because it may explain both the observable projector and the static branch selection in one parent language.
  • D3 remains a valid physical-coupling route, but unless it also explains the auxiliary rule it will likely leave the static conditional closure only partially upgraded.

So the current default next move should be: pursue D2 first, use D1 as a side route for orientation-sign closure, and keep D3 available if the readout physics becomes concrete sooner than the parent selector.