How does the 2/5 come from the Hexarchon?
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Three Tests: A derivation counts as a derivation rather than a fit if it passes all three of the following:
Test 1 — Independence from the muon. The derivation must not use the muon mass, the muon Compton wavelength, or the value (2φ/5) mod 1, at any step. If the derivation is written down and the reader can see that the muon never appears, Test 1 passes. If the factor 2/5 appears only after the muon has been used to select which geometric features to consider, Test 1 fails.
Test 2 — Necessity. The derivation must show that 2/5 is forced by the geometry, not available in it. Many rational factors appear in the cross-sectional geometry of any polyhedron; 1/3, 1/4, 1/5, 2/3, 3/4, 2/5, 3/5, 3/8, and others will all be constructible in some sense. A derivation that shows 2/5 is one of the factors the geometry permits is a fit. A derivation that shows 2/5 is the factor that a specific geometric construction produces is a derivation.
Test 3 — Specificity. The derivation must identify which feature of the Hexarchon produces the factor, and must show that a nearby but different feature produces a different factor. If the argument is “the geometry contains φ, and 2/5 is a natural rational multiple of φ,” that is not specific enough — it would equally support 1/5, 3/5, 2/3, or any other small rational multiple. The derivation must say: this ratio, from this construction, and not another.
A successful derivation has the following shape:
- A named construction. Start from a specific, named feature of the Hexarchon — a ratio of edge lengths, a ratio of face diagonals, an angle between symmetry axes, a ratio of volumes of inscribed solids, a cross-sectional area ratio, or a ratio of the radii of inscribed and circumscribed spheres. The construction must be specific enough that a reader can reproduce it without ambiguity.
- A computation. Show that the named construction yields the rational factor 2/5 (or a quantity that reduces to 2/5 under a stated operation — for example, a ratio whose fractional part is 2/5, or a ratio multiplied by a stated constant). The computation should be short enough to check by hand.
- A uniqueness statement. Show that a nearby but different construction yields a different factor. For example: “the ratio of the inscribed sphere radius to the circumscribed sphere radius of the octahedron is 1/√3, whose continued-fraction expansion begins [0; 1, 1, 2, …] and does not produce 2/5; the ratio of the face diagonal to the edge produces √2, which likewise does not. The construction that does produce 2/5 is [X].” This is what makes the derivation specific rather than available.
- A statement of what would have counted as failure. “If the construction had yielded 1/3 instead of 2/5, the muon match would have been to a different target and the pre-registration would have failed.” Stating this in advance is what distinguishes a derivation from a post-hoc fit.
What the derivation would look like if it fails
A failed derivation has one of these shapes, and it is worth naming them so you can recognize them if you find yourself producing one:
- Reverse-engineering. The construction is chosen because it produces 2/5. The choice of which ratio to compute is made after noticing that 2/5 is the target. This is a fit, not a derivation.
- Under-determination. The construction produces a family of factors, of which 2/5 is one. The derivation does not say why 2/5 rather than another member of the family.
- Circularity. The derivation uses φ, and φ is related to 2/5 by a construction that has already assumed the target. For example, “2/5 = (φ − 1)/(φ + something)” where the “something” is chosen to make the equation hold.
- Over-generation. The derivation produces so many rational factors that 2/5 is bound to be among them. A geometry that produces every rational factor up to some denominator is not a derivation of any particular one.
If you find yourself producing one of these, the honest move is to say so, and to report a negative result on H1. That is a perfectly respectable outcome, and it is what the pre-registration is designed to make visible.
The specific question to ask of the Hexarchon
Given the above, the concrete question to put to the geometry is:
Does the Hexarchon contain a construction whose ratio, or fractional part of a ratio, is exactly 2/5 — where the construction is named in advance and the computation does not use the value 2/5 to select the construction?
If the answer is yes, and if the construction is specific and unique in the sense of Test 3, then H1 is satisfied and the pre-registration can proceed to inventory construction. If the answer is no, H1 fails, and the muon near-coincidence remains a coincidence candidate.
The question is deliberately narrow. It does not ask whether the Hexarchon “contains” the golden ratio, or whether 2/5 is “natural” in some broader sense. It asks whether a specific construction yields 2/5 specifically, without the answer being used to select the construction. That is the smallest question that can be answered yes or no, and it is the one the pre-registration turns on.
A note on what “the Hexarchon” needs to be for this to work
For the derivation to be checkable, the Hexarchon needs a precise enough definition that a reader can reproduce the construction. If the Hexarchon is defined as “the cross-sectional geometry of the octahedron” in a way that is already standard in the 81018 papers, the derivation can cite that definition and proceed. If the Hexarchon is defined more loosely — as a family of related constructions — then the derivation should specify which member of the family is being used, and why that member rather than another.