From 9ae74d5fcd9936c064df816b32515dada477b219 Mon Sep 17 00:00:00 2001
From: pt2710 <132405221+pt2710@users.noreply.github.com>
Date: Fri, 22 May 2026 00:29:34 +0200
Subject: [PATCH 1/2] Add triadic UEO axis domain model

---
 triadic_domains.py | 177 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 177 insertions(+)
 create mode 100644 triadic_domains.py

diff --git a/triadic_domains.py b/triadic_domains.py
new file mode 100644
index 0000000..7b587cb
--- /dev/null
+++ b/triadic_domains.py
@@ -0,0 +1,177 @@
+"""
+Triadic Completeness domain model for McCrackn's Prime Law.
+
+This module keeps the TC / UEO layer separate from the gap-motif scheduler.
+It models the paper-level bridge:
+
+    U -> E -> O
+    M_j = <{2} union P_j>
+    p_{j+1} = min(N>=1 \ M_j)
+
+The implementation is deliberately finite-prefix and audit oriented.  It does
+not perform trial division or primality testing.  Composite placement is derived
+from already realized axes.
+"""
+from __future__ import annotations
+
+from dataclasses import dataclass
+from heapq import heappop, heappush
+from typing import Iterable, Iterator, Literal
+
+
+AxisKind = Literal["unity", "even", "odd-prime", "odd-composite"]
+
+
+@dataclass(frozen=True)
+class AxisPlacement:
+    """Canonical U/E/O placement for a positive integer in a realized prefix."""
+
+    value: int
+    domain: str
+    axis: str
+    kind: AxisKind
+    least_prime_factor: int | None = None
+    cofactor: int | None = None
+
+
+class TriadicDomains:
+    """
+    Finite-prefix TC axis machine.
+
+    The odd domain is split into:
+    - O1: the free / prime odd axis in the user's notation.
+    - O{k}: least-prime-factor composite strata, where O2 is lpf=3,
+      O3 is lpf=5, O4 is lpf=7, and so on.
+
+    The paper notation uses O_0 for the prime/free layer and O_i^(min)
+    for least-prime-factor composite strata.  This class exposes both via
+    explicit placement metadata.
+    """
+
+    UNITY = 1
+
+    def __init__(self, realized_primes: Iterable[int]):
+        primes = tuple(int(p) for p in realized_primes)
+        if not primes or primes[0] != 2:
+            raise ValueError("realized_primes must start with 2")
+        if any(p <= 0 for p in primes):
+            raise ValueError("realized_primes must be positive")
+        self.realized_primes = primes
+        self.odd_primes = tuple(p for p in primes if p != 2)
+        self._odd_axis_index = {p: i + 2 for i, p in enumerate(self.odd_primes)}
+
+    @staticmethod
+    def odd_axis_value(position: int) -> int:
+        """Return the primitive odd-axis value at zero-based position."""
+        if position < 0:
+            raise ValueError("position must be non-negative")
+        return 2 * position + 1
+
+    def even_face(self, n: int) -> tuple[int, int]:
+        """Return (v2(n), odd_face) for a positive integer."""
+        if n <= 0:
+            raise ValueError("n must be positive")
+        depth = 0
+        value = n
+        while value % 2 == 0:
+            depth += 1
+            value //= 2
+        return depth, value
+
+    def emitted_smooth_numbers(self, limit: int) -> list[int]:
+        """
+        Emit the finite prefix of <{2} union P_j> up to limit.
+
+        This uses merge/multiply stream operations over already realized axes.
+        It is a finite-prefix witness for Axis Law coverage, not a primality
+        test over arbitrary candidates.
+        """
+        if limit < 1:
+            return []
+        seen = {1}
+        heap = [1]
+        out: list[int] = []
+        while heap:
+            value = heappop(heap)
+            if value > limit:
+                continue
+            out.append(value)
+            for axis in self.realized_primes:
+                nxt = value * axis
+                if nxt <= limit and nxt not in seen:
+                    seen.add(nxt)
+                    heappush(heap, nxt)
+        return out
+
+    def first_hole_after(self, current_prime: int, limit: int | None = None) -> int:
+        """
+        Return the first positive integer after current_prime not emitted by
+        the realized-axis monoid within a finite bound.
+        """
+        if current_prime < 1:
+            raise ValueError("current_prime must be positive")
+        bound = limit if limit is not None else max(current_prime * current_prime, current_prime + 32)
+        covered = set(self.emitted_smooth_numbers(bound))
+        for n in range(current_prime + 1, bound + 1):
+            if n not in covered:
+                return n
+        raise ValueError("finite bound did not expose a first hole")
+
+    def axis_law_successor_gap(self, current_prime: int, limit: int | None = None) -> tuple[int, int]:
+        """Return (next_prime, gap) from the finite first-hole law."""
+        nxt = self.first_hole_after(current_prime, limit=limit)
+        return nxt, nxt - current_prime
+
+    def odd_composite_stream(self, limit: int) -> Iterator[AxisPlacement]:
+        """Yield canonical odd composite placements up to limit by lpf strata."""
+        if limit < 3:
+            return
+        for p in self.odd_primes:
+            for odd in range(p, limit // p + 1, 2):
+                value = p * odd
+                if value > limit:
+                    break
+                if self._least_realized_odd_prime_factor(value) == p and value != p:
+                    yield AxisPlacement(
+                        value=value,
+                        domain="O",
+                        axis=f"O{self._odd_axis_index[p]}",
+                        kind="odd-composite",
+                        least_prime_factor=p,
+                        cofactor=odd,
+                    )
+
+    def place(self, n: int) -> AxisPlacement:
+        """Return the canonical finite-prefix U/E/O placement for n."""
+        if n <= 0:
+            raise ValueError("n must be positive")
+        if n == 1:
+            return AxisPlacement(n, "U", "U1", "unity")
+
+        depth, odd_face = self.even_face(n)
+        if depth > 0:
+            return AxisPlacement(n, "E", f"E{depth}", "even", cofactor=odd_face)
+
+        if n in self.odd_primes:
+            return AxisPlacement(n, "O", "O1", "odd-prime")
+
+        lpf = self._least_realized_odd_prime_factor(n)
+        if lpf is None:
+            return AxisPlacement(n, "O", "O1", "odd-prime")
+
+        return AxisPlacement(
+            value=n,
+            domain="O",
+            axis=f"O{self._odd_axis_index[lpf]}",
+            kind="odd-composite",
+            least_prime_factor=lpf,
+            cofactor=n // lpf,
+        )
+
+    def _least_realized_odd_prime_factor(self, n: int) -> int | None:
+        for p in self.odd_primes:
+            if p * p > n:
+                break
+            if n % p == 0:
+                return p
+        return None

From c1a72802812b5100e5083266ce51ae07986c7e49 Mon Sep 17 00:00:00 2001
From: pt2710 <132405221+pt2710@users.noreply.github.com>
Date: Fri, 22 May 2026 00:30:19 +0200
Subject: [PATCH 2/2] Add TC axis consistency tests

---
 tests/test_triadic_domains.py | 62 +++++++++++++++++++++++++++++++++++
 1 file changed, 62 insertions(+)
 create mode 100644 tests/test_triadic_domains.py

diff --git a/tests/test_triadic_domains.py b/tests/test_triadic_domains.py
new file mode 100644
index 0000000..35631a4
--- /dev/null
+++ b/tests/test_triadic_domains.py
@@ -0,0 +1,62 @@
+from triadic_domains import TriadicDomains
+
+
+def test_unity_even_odd_placements():
+    domains = TriadicDomains([2, 3, 5, 7])
+
+    assert domains.place(1).domain == "U"
+    assert domains.place(1).axis == "U1"
+
+    placed_even = domains.place(40)
+    assert placed_even.domain == "E"
+    assert placed_even.axis == "E3"
+    assert placed_even.cofactor == 5
+
+    placed_prime = domains.place(31)
+    assert placed_prime.domain == "O"
+    assert placed_prime.axis == "O1"
+    assert placed_prime.kind == "odd-prime"
+
+
+def test_odd_composites_are_lpf_axis_placed():
+    domains = TriadicDomains([2, 3, 5, 7])
+
+    expected = {
+        9: ("O2", 3, 3),
+        15: ("O2", 3, 5),
+        21: ("O2", 3, 7),
+        25: ("O3", 5, 5),
+        27: ("O2", 3, 9),
+        35: ("O3", 5, 7),
+        49: ("O4", 7, 7),
+    }
+
+    for value, (axis, lpf, cofactor) in expected.items():
+        placement = domains.place(value)
+        assert placement.domain == "O"
+        assert placement.kind == "odd-composite"
+        assert placement.axis == axis
+        assert placement.least_prime_factor == lpf
+        assert placement.cofactor == cofactor
+
+
+def test_axis_law_first_hole_examples():
+    domains_after_3 = TriadicDomains([2, 3])
+    assert domains_after_3.axis_law_successor_gap(3, limit=16) == (5, 2)
+    assert domains_after_3.axis_law_successor_gap(5, limit=16) == (7, 2)
+    assert domains_after_3.axis_law_successor_gap(7, limit=16) == (11, 4)
+
+    domains_after_7 = TriadicDomains([2, 3, 5, 7])
+    assert domains_after_7.axis_law_successor_gap(29, limit=64) == (31, 2)
+    assert domains_after_7.place(31).axis == "O1"
+
+
+def test_composite_stream_has_canonical_lpf_entries_without_duplicate_axes():
+    domains = TriadicDomains([2, 3, 5, 7])
+    placements = {p.value: p for p in domains.odd_composite_stream(50)}
+
+    assert placements[15].axis == "O2"
+    assert placements[15].least_prime_factor == 3
+    assert placements[25].axis == "O3"
+    assert placements[35].axis == "O3"
+    assert placements[49].axis == "O4"