docs: UCNS-G prime-cylinder metric geometry v3 (mirror)

docs/handoffs/2026-05-22-ucns-g-prime-cylinder-v3.md

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+# Session Handoff v3 — UCNS-G Prime-Cylinder Metric Geometry
+
+**Status:** Supersedes UCNS metric geometry handoff v2 where v2 describes UCNS-G as a single `(r, θ, z)` point or treats twist as θ/double-cover behavior.
+**Date pinned:** 2026-05-22
+**Scope:** UCNS-G / EDCM metric geometry only. Does not modify UCNS-A theorem scope.
+**Firewall:** UCNS-A Theorem N remains scoped to UCNS-A factorization algebra. Nothing here inherits proof status from UCNS-A.
+
+## 0. Carry the UCNS-A / UCNS-G firewall
+
+UCNS-A and UCNS-G remain distinct unless an explicit bridge is proven.
+
+UCNS-A is the recursive factorization algebra in `The-Interdependency/ucns`.
+
+UCNS-G is the EDCM metric geometry described here.
+
+Do not allow UCNS-A proof claims to validate UCNS-G metric claims.
+
+## 1. v2 correction
+
+v2 described UCNS-G as:
+
+```text
+(r, θ, z)
+```
+
+That is too small.
+
+A single `(r, θ, z)` point describes one projected metric-disk state. It does not describe the full EDCM metric geometry.
+
+Corrected:
+
+```text
+UCNS-G = prime-indexed tensor of non-closing Möbius-cylinder metric disks.
+```
+
+Each metric receives its own disk/cylinder trace.
+
+Each turn is a tensor sample across metric disks.
+
+Each round is an ordered tensor of turn tensors.
+
+Each session is an ordered tensor of round tensors.
+
+A “disk of disks” may be used for visualization, but the canonical structure is an ordered tensor/bundle of disks.
+
+## 2. Metric disks
+
+For every metric axis `A`, define a metric disk:
+
+```text
+D_A
+```
+
+A metric point is:
+
+```text
+MetricPoint_A = {
+  axis: A,
+  prime_axis: p_A,
+  grain: token | turn | round | session | archive,
+  twist_ordinal: τ,
+  phase: φ,
+  face: +1 | -1,
+  sign: -1 | 0 | 1,
+  magnitude: m,
+  gauge: radius | circumference | area | depth,
+  confidence: c
+}
+```
+
+## 3. Signed ternary axes
+
+Every EDCM metric axis has state:
+
+```text
+state_A(W) = (s_A, m_A)
+```
+
+where:
+
+```text
+s_A ∈ {-1, 0, 1}
+m_A ∈ [0, 1]
+```
+
+`0` is not scalar absence. It is neutral / unresolved / no measurable directional commitment.
+
+## 4. Phase/percent remains a projection
+
+Percent/phase is still valid for:
+
+```text
+operator density
+area-percent
+metric magnitude
+completion fraction
+window-relative proportions
+```
+
+But it is not canonical identity.
+
+```text
+phase/percent view = local projection
+ordinal twist view = traversal identity
+axis/sign view = directional EDCM measurement
+```
+
+## 5. Twist is ordinal zero-boundary
+
+Twist is not an angle value.
+
+Twist is the ordinal seam / zero-boundary.
+
+```text
+zero = twist_0
+360° = phase zero at twist_1
+720° = phase zero at twist_2
+1080° = phase zero at twist_3
+```
+
+Therefore:
+
+```text
+same phase ≠ same state
+same zero-looking point ≠ same ordinal zero
+```
+
+Canonical state requires:
+
+```text
+(twist_ordinal, phase, face/orientation)
+```
+
+not only:
+
+```text
+θ mod 360°
+```
+
+or:
+
+```text
+θ mod 720°
+```
+
+## 6. Non-closing Möbius-cylinder graph
+
+UCNS-G graphs do not close.
+
+Local disk projection:
+
+```text
+phase 1 → phase 0
+```
+
+Canonical ordinal graph:
+
+```text
+(twist_n, phase=1) → (twist_{n+1}, phase=0)
+```
+
+Möbius face rule:
+
+```text
+face_{n+1} = -face_n
+```
+
+or bit form:
+
+```text
+face_{n+1} = face_n XOR 1
+```
+
+Spec rule:
+
+```text
+The disk closes only from the local perspective.
+The graph does not close from the ordinal perspective.
+```
+
+## 7. Sine-wave projection
+
+A circle projected onto one axis is a sine wave:
+
+```text
+y = r · sin(2πφ)
+```
+
+The Möbius/ordinal lift is:
+
+```text
+y_n = (-1)^n · r · sin(2πφ)
+```
+
+where:
+
+```text
+n = twist_ordinal
+```
+
+If `n` is collapsed, the twist is lost.
+
+## 8. Unit gauge
+
+A metric disk is unit-normalized but not unit-identical.
+
+Typed unit ones:
+
+```text
+1_R = full radius
+1_C = full circumferential traversal
+1_A = full area coverage
+1_Z = one ordinal recurrence / depth layer
+```
+
+These may all display as `1`, but they are not the same unit.
+
+Spec line:
+
+```text
+The twist hides zero.
+The gauge hides one.
+```
+
+## 9. Perspective
+
+Perspective is load-bearing.
+
+Inside a metric disk:
+
+```text
+the disk is centered
+zero is local
+the circle appears closed
+```
+
+From the parent/auditor frame:
+
+```text
+the disk is displaced
+zero is ordinal seam/contact
+the graph does not close
+```
+
+## 10. Epicycles
+
+An epicycle is:
+
+```text
+locally centered
+globally uncentered
+```
+
+Formal:
+
+```text
+An epicycle is a locally centered child disk whose center is externally
+located at an anchor of a parent disk.
+```
+
+Classical:
+
+```text
+center(child) ∈ circumference(parent)
+```
+
+General UCNS-G:
+
+```text
+center(child) = anchor(parent)
+```
+
+The parent anchor may be a circumference point, twist point, prime carrier point, metric-axis point, or grain boundary.
+
+## 11. Prime-cylinder audit projection
+
+Prime-cylinder visualization is the canonical audit projection for UCNS-G across granularity.
+
+It renders each metric disk as a prime-indexed non-closing Möbius-cylinder trace.
+
+Suggested primitive prime assignment:
+
+```text
+P → 2
+K → 3
+Q → 5
+T → 7
+S → 11
+
+C → 13
+R → 17
+D → 19
+N → 23
+L → 29
+O → 31
+F → 37
+E → 41
+I → 43
+```
+
+Primitive metrics sit on prime anchors.
+
+Composite positions may represent interactions/couplings.
+
+Spec line:
+
+```text
+Prime cylinder shows where the disk closes locally but not canonically.
+```
+
+## 12. Grain tensor
+
+Canonical grain structure:
+
+```text
+MetricDisk_A = non-closing prime-indexed Möbius-cylinder trace for axis A
+
+Turn_i = tensor sample across metric disks
+
+Round_j = ordered tensor of Turn_i states
+
+Session_k = ordered tensor of Round_j states
+
+Archive = ordered tensor of Session_k states
+```
+
+Parent disks are projections, not identity.
+
+```text
+parent_disk = projection(child_tensor)
+```
+
+not:
+
+```text
+parent_disk = child_tensor
+```
+
+## 13. hmmm
+
+Open items:
+
+* Formal exchange rules between typed units `1_R`, `1_C`, `1_A`, `1_Z`.
+* Exact prime assignment for all EDCM and Operator axes.
+* Whether prime interactions should be represented by multiplication, graph edges, epicycle attachment, or separate coupling tensors.
+* How DRIFT/DVG/DA map onto area/count/length readouts in the new prime-cylinder graph.
+* Whether UCNS-A can provide a formal bridge to this UCNS-G structure, or whether UCNS-G remains a parallel construction.
+* Whether current scalar EDCM code should be migrated through schema first or code first.
+* How to encode signed ternary axis states in frontend/API types without breaking current scalar projections.
+
+## hmmm