problem(selection-principle): record what the inverse-problem axis now establishes

The selection-principle question splits cleanly and the two halves now answer
differently:

- Topological rarity (Q-A). (b2, b3) = (21, 77) is reached by no twisted
  connected sum construction; sits in a genuine gap of the realized G2 Betti
  census; reached only via the JK / Donaldson K3-fibration route. But rarity
  is representation-dependent: the CY3 shadow (h11, h21) = (21, 27) is
  realized by 668,607 distinct reflexive 4-polytopes in the Kreuzer-Skarke
  database. The G2 rarity argument does not transport to the CY3
  representation of the same numbers.

- Inverse-problem specialness (Q-B). A blind mirage-proof sweep (alphabet
  size held constant; only the Betti leaf values change) over the full KS
  Hodge range (b2, b3 each in [0, 491], 242,064 grid points), via the Sieve
  methodology (D33), re-runs the search per pair and places (21, 77) BELOW
  THE MEDIAN on every discriminating axis: ~64% of grid pairs strictly beat
  it on Betti-sensitive coverage at the discriminating tolerance, expensive
  quartile at tight tau. The CY3 shadow (21, 27) is even less special. The
  verdict grows as the haystack grows.

Reading. The two halves are independent and both honest. Q-B closes one
candidate channel for the selection principle: the Betti pair is not
arithmetically privileged at fitting the freeze. Q-A keeps the
topological-distinguishedness channel open, in the K3-lattice direction the
constraints already name. Charter-mandated honest outcome ("publish either
way").

Next step revised: continue mapping K3-lattice classifications that single
out (15, 7, 1) and (21, 77) simultaneously; do not look for selection in the
inverse-problem statistic — that channel is closed by the landscape sweep.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

Author: gift-framework

problems/selection-principle.md

@@ -3,7 +3,7 @@ title: The selection principle
 axis: selection
 status: active
 opened: 2026-06-11
-updated: 2026-06-11
+updated: 2026-06-18
 ---
 
 **Statement.** Why this geometry? Identify the principle that selects K7,
@@ -24,6 +24,45 @@ standard template. None of this yet amounts to a selection principle; it
 amounts to the geometry being a distinguished point in several classifications
 at once.
 
-**Next step.** Map which classifications single out (15, 7, 1) and (21, 77)
-simultaneously; test whether distinguishedness propagates from the K3 lattice
-to the G2 level.
+**What is now established (2026-06-18).** The question splits cleanly and the
+two halves answer differently.
+
+- *Topological rarity (Q-A).* (b2, b3) = (21, 77) is reached by no twisted
+  connected sum construction (exhaustive 3852-config no-go, CHNP b2_max
+  about 18); it sits in a genuine gap of the realized G2 Betti census
+  (nearest realized (19, 65)); reached only via the
+  Joyce-Karigiannis / Donaldson K3-fibration route. So (21, 77) IS rare as
+  a G2 manifold. But rarity is representation-dependent: its CY3 shadow
+  via the Kunneth decomposition CY3 x S^1, namely (h11, h21) = (21, 27),
+  is realized by 668,607 distinct reflexive 4-polytopes in the
+  Kreuzer-Skarke database. The G2 rarity argument does not transport to
+  the CY3 representation of the same numbers.
+
+- *Inverse-problem specialness (Q-B).* In a blind, mirage-proof sweep
+  (alphabet size held constant across the grid; only the Betti leaf
+  values change) over the full Kreuzer-Skarke Hodge range
+  (b2, b3 each in [0, 491], 242,064 grid points), the
+  Sieve methodology
+  ([arithmon/sieve](https://github.com/arithmon/sieve), D33) re-runs
+  the search per pair and asks how cheaply / how widely each pair fits
+  the frozen Standard Model set. (21, 77) sinks BELOW THE MEDIAN on
+  every discriminating axis: ~64% of grid pairs strictly beat it on
+  Betti-sensitive coverage at the discriminating tolerance, and it sits
+  in the expensive quartile at tight tau. The CY3 shadow (21, 27) is
+  even less special. The verdict GROWS as the haystack grows: smaller
+  prior grids were the conservative reading, in (21, 77)'s favour.
+  Specialness, in the inverse-problem sense, is not where the geometry
+  lives.
+
+Reading. The two halves are independent and both honest. Q-B closes one
+candidate channel for the selection principle: the Betti pair is not
+arithmetically privileged at fitting the freeze, so the principle, if there
+is one, is NOT "(21, 77) is the cheapest set of leaves to reach the
+Standard Model." Q-A keeps the topological-distinguishedness channel
+open, in the K3-lattice direction the constraints already named; it does
+not amount to a principle on its own.
+
+**Next step.** Continue mapping which classifications single out (15, 7, 1)
+and (21, 77) simultaneously; test whether distinguishedness propagates from
+the K3 lattice to the G2 level. Do not look for selection in the inverse-
+problem statistic — that channel is closed by the landscape sweep.