research: integrate stack-local UCNS/EPAC/PCEA worktree research

Preserved from the pre-sync dirty working tree: UCNS standing-wave candidates,
EPAC boundary audits, PCEA gonol successor/transition candidates, and
research/ucns recursive-scale experiments. README conflicts with newer main
resolved by keeping the newer EPAC/PCEA standing and adding the new content.

Author: erinepshovel-code

libs/ucns/README.md

@@ -8,6 +8,11 @@ Its active scope is limited to constructions that directly represent or test geo
 - the directed lifted carrier and its 360° visible / 720° complete return;
 - exact framed Möbius motion;
 - Möbius vesica and Seed-of-Life constructions and certificates;
+- the native two-turn Möbius root-loop standing-wave candidate, including its
+  operator, domain, boundary conditions, admissible modes, and derived seam
+  trace;
+- the canonical Möbius band coordinate standing-wave candidate, including its
+  physical-edge boundary conditions and sampled continuous-boundary trace rank;
 - prime-indexed ribbon, link, interval, Milnor, Alexander, and related topological geometry;
 - numerical machinery used to certify those constructions.
 
@@ -33,6 +38,22 @@ One visible turn preserves phase and reverses the local frame. Two visible turns
 
 The active package also retains the Möbius vesica/seed family and the `prime_*` topological geometry family. Generated geometry certificates remain evidence; semantic receipts do not.
 
+`src/ucns/mobius_standing_wave.py` is the smallest explicit continuum
+standing-wave model currently admitted by UCNS canon: a scalar second-order
+wave equation on the native two-turn Möbius root loop. Its boundary trace is
+the derived value/derivative trace at the quotient seam. That trace has rank at
+most two through every finite mode cutoff, so this model falsifies direct
+induction of PCEA-style boundary-measure capacity channels.
+
+`src/ucns/mobius_band_standing_wave.py` extends that construction to the
+canonical Möbius band coordinate domain `[0,2] x [-w,w]` with the native
+quotient `(t+1,u)~(t,-u)`. Its coordinate-domain Laplacian, Neumann physical
+edge condition, admissible parity-matched modes, and sampled continuous-boundary
+trace are executable. Finite capacity channels survive only when they fit the
+derived boundary trace rank. Bulk-only counting is falsified when interior
+transverse modes exceed the independent boundary traces. The embedded
+Laplace-Beltrami operator and exact PCEA channel encoding remain `hmmm`.
+
 ## Usage
 
 ```python
@@ -59,4 +80,4 @@ python -m build
 python -m twine check dist/*
 ```
 
-`hmmm`: the complete higher-dimensional UCNS construction and the exact geometric operation of every Public Gonol function position remain unresolved. Unresolved geometry stays unresolved; semantic machinery is not used to fill it.
+`hmmm`: the embedded continuum surface operator, exact continuum boundary-measure capacity theorem, derived PCEA channel encoding, complete higher-dimensional UCNS construction, and exact geometric operation of every Public Gonol function position remain unresolved. Unresolved geometry stays unresolved; semantic machinery is not used to fill it.

libs/ucns/docs/STANDING_WAVE_MOBIUS_BAND.md

@@ -0,0 +1,118 @@
+# UCNS Mobius Band Standing-Wave Continuum
+
+Status: SURVIVED as an explicit UCNS continuum standing-wave construction on
+the canonical Mobius band coordinate domain; SURVIVED for finite capacity
+requests that fit the derived sampled boundary trace rank; FALSIFIED for
+bulk-only capacity counts that exceed that trace rank; UNRESOLVED for PCEA
+promotion.
+
+## Domain Claim
+
+Surface form: standing-wave continuous-boundary trace.
+
+Term id: `ucns.mobius_band_standing_wave.boundary_trace`.
+
+Claiming domain: UCNS geometry.
+
+Claimed sense: finite value samples, with optional normal derivative slots, on
+the single continuous physical boundary induced by a scalar coordinate-domain
+standing-wave operator on the canonical Mobius band.
+
+Scope: `src/ucns/mobius_band_standing_wave.py` and tests that consume that
+exact coordinate-domain band model.
+
+Excluded uses: embedded Laplace-Beltrami theorem, physical field theory,
+spectral/zeta theorem, PCEA runtime capacity, FFT/discrete spectral tooling,
+and arbitrary boundary-measure promotion.
+
+Status: provisional candidate.
+
+## Construction
+
+The canonical Mobius ribbon supplies the coordinate domain:
+
+```text
+(t, u) in [0, 2] x [-w, w]
+(t + 1, u) ~ (t, -u)
+```
+
+The spatial operator is:
+
+```text
+L X = -(d2 X / dt2 + d2 X / du2)
+```
+
+The standing-wave equation is:
+
+```text
+d2 psi / d tau2 = c^2 * (d2 psi / d t2 + d2 psi / d u2)
+```
+
+The boundary conditions are:
+
+```text
+X(t+1,u) = X(t,-u)
+dX/dt(t+1,u) = dX/dt(t,-u)
+dX/du(t+1,u) = -dX/du(t,-u)
+dX/du(t,+w) = 0
+dX/du(t,-w) = 0
+```
+
+The admissible basis is separable. The one-turn longitudinal character must
+match the transverse parity:
+
+```text
+n even: longitudinal cos/sin(n*pi*t) or constant, transverse cos(k*pi*u/w)
+n odd:  longitudinal cos/sin(n*pi*t), transverse sin((k+1/2)*pi*u/w)
+```
+
+## Boundary Trace
+
+The Mobius strip has one continuous physical boundary. The executable trace
+uses `u=+w` over the complete two-turn boundary traversal:
+
+```text
+X(2*i/sample_count, w), 0 <= i < sample_count
+```
+
+Optional normal-derivative trace slots are present, but the declared Neumann
+condition makes them zero for admissible modes. They therefore do not increase
+rank.
+
+For longitudinal cutoff `N` and transverse cutoff `K`, the interior basis
+dimension is:
+
+```text
+(K + 1) * (1 + 2*N)
+```
+
+The sampled boundary value trace rank is bounded by the boundary harmonics and
+the sample count:
+
+```text
+rank <= min(sample_count, 1 + 2*N)
+```
+
+Increasing `K` adds transverse bulk modes but does not add independent boundary
+trace channels at the same longitudinal cutoff.
+
+## Boundary-Capacity Result
+
+The boundary-capacity principle survives in this finite tested form:
+
+- finite channel requests that fit the derived sampled boundary trace rank are
+  `SURVIVED`;
+- channel requests that fit the interior basis dimension but exceed the
+  boundary trace rank are `FALSIFIED`;
+- arbitrary promotion from this sampled trace to PCEA runtime capacity remains
+  `UNRESOLVED`.
+
+No capacity channel is assigned by fiat. Discrete FFT/epicycle tooling is not
+used as a PDE derivation.
+
+## hmmm
+
+- embedded Laplace-Beltrami operator on the realized three-dimensional ribbon;
+- exact continuum boundary-measure capacity theorem rather than finite sampled
+  rank;
+- derived encoding from continuum trace functions into PCEA runtime channels.

libs/ucns/docs/STANDING_WAVE_ROOT_LOOP.md

@@ -0,0 +1,97 @@
+# UCNS Mobius Standing-Wave Root Loop
+
+Status: SURVIVED as the smallest explicit UCNS continuum standing-wave
+construction; FALSIFIED as a direct source of PCEA-style boundary-measure
+capacity channels.
+
+## Domain Claim
+
+Surface form: standing-wave boundary trace.
+
+Term id: `ucns.mobius_standing_wave.boundary_trace`.
+
+Claiming domain: UCNS geometry.
+
+Claimed sense: endpoint value/derivative trace induced by the scalar
+second-order standing-wave operator on the native two-turn Mobius root loop.
+
+Scope: `src/ucns/mobius_standing_wave.py` and tests that consume that exact
+root-loop model.
+
+Excluded uses: physical field theory, spectral/zeta theorem, PCEA runtime
+capacity, FFT/discrete spectral tooling, and Möbius band or higher-dimensional
+domains.
+
+Status: provisional candidate.
+
+## Construction
+
+The native UCNS Mobius root loop supplies the continuum coordinate:
+
+```text
+t in [0, 2]
+```
+
+The spatial operator is:
+
+```text
+L X = -d2 X / dt2
+```
+
+The standing-wave equation is:
+
+```text
+d2 psi / d tau2 = c^2 d2 psi / d t2
+```
+
+The complete-return boundary conditions are:
+
+```text
+X(0) = X(2)
+X'(0) = X'(2)
+```
+
+The admissible spatial basis is:
+
+```text
+1
+cos(n*pi*t), n >= 1
+sin(n*pi*t), n >= 1
+```
+
+Each positive mode changes sign under one visible turn when `n` is odd and
+returns after two visible turns.
+
+## Boundary Trace
+
+For each basis function, the trace is derived from endpoints:
+
+```text
+(X(0), X(2), X'(0), X'(2))
+```
+
+The trace image has rank `1` for only the constant mode and rank `2` once any
+positive sine/cosine mode is admitted. Increasing the interior mode cutoff adds
+more interior basis functions, but it does not add independent trace channels in
+this one-dimensional root-loop model.
+
+## PCEA Capacity Induction
+
+This model does not induce the capacity channels required by PCEA's
+boundary-capacity hypothesis. The derived trace rank is at most `2`, while
+PCEA-style boundary capacity asks for a boundary-measure-indexed family of
+independent coupling channels.
+
+No channels are assigned by fiat. Discrete FFT/epicycle tooling is not used as a
+PDE derivation.
+
+The next larger construction is `src/ucns/mobius_band_standing_wave.py`, which
+adds the canonical Mobius band coordinate domain and its continuous physical
+boundary. That construction does not retroactively change this root-loop
+falsifier.
+
+## hmmm
+
+- embedded Laplace-Beltrami operator on the realized ribbon;
+- exact continuum boundary-measure capacity theorem;
+- derived map from continuum trace functions into PCEA capacity channels.

libs/ucns/src/ucns/__init__.py

@@ -4,17 +4,17 @@
 #   module_kind: facade
 #   summary: geometry-only UCNS public surface
 #   owner: Erin Spencer
-#   public_surface: carrier geometry, framed Mobius root loop, exact Public Gonol carrier, Mobius vesica and seed geometry
+#   public_surface: carrier geometry, framed Mobius root loop, exact Public Gonol carrier, Mobius vesica and seed geometry, native root-loop and band standing-wave candidates
 #   internal_surface: none
 #   auth_boundary: none
 #   storage_boundary: none
 #   network_boundary: none
 #   user_data_boundary: none
 #   admin_only: false
-#   tests: tests.test_public_gonol, tests.test_geometry_public_surface, tests.test_carrier
+#   tests: tests.test_public_gonol, tests.test_geometry_public_surface, tests.test_carrier, tests.test_mobius_standing_wave, tests.test_mobius_band_standing_wave
 #   rollout: active geometry-only package facade
 #   rollback: restore prior facade from Git history
-#   requires: directed_carrier_floor, ucns_native_mobius_geometry, ucns_public_gonol_geometry, ucns_mobius_vesica_candidate, ucns_mobius_seed_of_life_candidate
+#   requires: directed_carrier_floor, ucns_native_mobius_geometry, ucns_public_gonol_geometry, ucns_mobius_vesica_candidate, ucns_mobius_seed_of_life_candidate, ucns_mobius_standing_wave_root_loop, ucns_mobius_band_standing_wave
 #   since: 2026-08-20
 #   unresolved: canonical completion of the full UCNS geometric construction
 # === END MODULE_BUILD ===
@@ -30,7 +30,8 @@
 """UCNS geometry.
 
 The active package surface is deliberately geometric: carriers, exact motion,
-Möbius constructions, geometric certificates, and topological/prime geometry.
+Möbius constructions, root-loop and band standing-wave candidates, geometric
+certificates, and topological/prime geometry.
 Lexical semantics, corpora, morphology, definition recursion, evaluator
 frameworks, PTCNA state, and cross-stack adapters are not UCNS package content.
 """
@@ -69,6 +70,10 @@
 from .mobius_vesica import *  # noqa: F401,F403 - geometric public module
 from .mobius_seed import __all__ as _mobius_seed_all
 from .mobius_seed import *  # noqa: F401,F403 - geometric public module
+from .mobius_standing_wave import __all__ as _mobius_standing_wave_all
+from .mobius_standing_wave import *  # noqa: F401,F403 - geometric public module
+from .mobius_band_standing_wave import __all__ as _mobius_band_standing_wave_all
+from .mobius_band_standing_wave import *  # noqa: F401,F403 - geometric public module
 
 __all__ = list(dict.fromkeys([
     "DirectMobiusError",
@@ -95,4 +100,4 @@
     "radius_from_breadth",
     "same_lifted_position",
     "same_visible_position",
-] + list(_mobius_vesica_all) + list(_mobius_seed_all)))
+] + list(_mobius_vesica_all) + list(_mobius_seed_all) + list(_mobius_standing_wave_all) + list(_mobius_band_standing_wave_all)))