What changed, when, and why

Part IV's alphabet, (1,−1,t,t⁻¹), is the second member of a family: the full set of t-th roots of unity together with one free reciprocal pair. arXiv:2608.09619 evaluates s_λ(μ_t, z, z⁻¹) for every t ≥ 2 and every λ, with no hypothesis on the shape: the value is a signed product of exactly three factors over a fixed denominator, or zero, and the three arguments are read off the t-quotient of λ. At t = 2 the alphabet is Part IV's. (The letter t changes job between the two papers: in Part IV it is the free variable, there the order of the root of unity.)

Part IV grades its closed form an Observation: checked against the bialternant in exact arithmetic on all 3060 partitions with four parts ≤ 14 and all 4845 with parts ≤ 16, with no mismatch, and not proved. The general statement carries a proof — a Laplace expansion along the t frozen rows of the bialternant with one cancellation lemma in the symmetric group, which also delivers the sign, the one already in Littlewood's evaluation at the roots of unity.

No number in Part IV moves and its record does not need a new version. What moves is a status word: at t = 2 the closed form stops being an Observation.

The rank-two case Part IV stopped at, and said so, is now conjectured to be unrepairable rather than merely unrepaired: Conjecture 10.4 there says the value is a product of characters for every λ if and only if the free part is a single reciprocal pair. Four deformations of the alphabet are measured against it and each destroys the product. That is evidence, not a proof, and it is labelled a conjecture.

The closed form for the η-dependence of the Higgs mass matrix — ΔH₁₁ = −2(2π)²L₁·M₂/8, ΔH₂₂ = −2(2π)²L₂·M₂/8, ΔH₁₂ = 0 — was published against five worked cases. The instrument runs it against the direct winding sum on all 119 multiplets of the catalogue: worst disagreement 0.0162 %. On the 16 multiplets the catalogue declares blind, the closed form predicts exactly zero and the winding sum measures exactly zero.

Five cases cannot distinguish "the formula is right" from "the formula is right on the cases it was built on". 119 can, and the blind ones are the sharpest of the 119 because a prediction of exactly zero has nowhere to hide.

The record is unaffected — no number in Parts IV or V moves. What changes is the strength of the evidence behind one of them.

A trap paid for here, and it is the reason the blind cases are stated separately: 16 multiplets have no Part IV box, and the first sweep silently printed M₂ = 0 for them and concluded "invisible to η". That was the right answer for the wrong reason — those 16 happen to be exactly the 16 the catalogue declares blind. On another group the same code would have lied. A missing datum is now an explicit unknown, never a zero.

Part VI says, in its own words, that the potential modified by the escape is not recomputed there. With the Part VII kernel present in the same instrument it is: the modified potential is evaluated, the vacuum is re-minimised on it, and the result is displayed labelled measured, carrying the Part VI §7 anchor band and marked explicitly as going beyond what the paper claimed.

Part VI reached a sign, deliberately, and stopped there. The kernel that turns the sign into a number was written for Part VII. Once both live in one instrument, refusing to run the second on the first's output would be a presentation choice, not an honesty one.

The record is unaffected: Part VI's statement that it does not recompute remains true of Part VI. The site is where the recomputation lives, and it is labelled as this page's work, not the paper's. Every absolute number in it inherits the open question about the published α column — see the tool's own "what this cannot tell you" panel.

Our α disagrees with the α published in Table 1 of arXiv:2503.04090 on every one of the five rows, and not by a constant: the ratio runs 1.03× to 2.08× with no pattern in 8D, in A₄ or in the content. A constant would be a convention and could be absorbed; this cannot. Row (2) is the only row that agrees, and the only one whose Higgs mass lands inside the 125–127 GeV window — and it is the row this tool opens on, which it inherited rather than chose.

Everything upstream of the α column reproduces. Their §3 was rebuilt from their own eqs. (11)–(13) and (54) — structure constants, KK spectrum, all four branchings — and agrees; their eq. (67) is proved rather than sampled; their eq. (68) is rederived; their Table 1 is internally consistent with their own eq. (82). No rescaling reconciles the two columns: neither a per-representation nor a per-channel factor, and a single common coefficient on the antiperiodic sector fails by 18 % at best while also breaking the Higgs-mass column.

No absolute number on this site (TeV, GeV) is citable until this closes. The three things that escape it entirely — the mass ratio, the bill in eighths, and the two arithmetic laws — are marked as such, because no normalisation enters any of them.

Two things have moved since. A discriminating sixth row is registered in advance in Part VI, so whoever computes such a row next can read our answer off in print rather than by asking us. And Part VII adds a second, independent published anchor — von Gersdorff, Irges and Quirós, arXiv:hep-th/0204223, whose five-dimensional single-parity model on S¹/Z₂ is exactly the Δ = 0 corner of our formula — and it reproduces our criterion, its three critical flavour numbers and one of its two published minima exactly. That does not close the gap. It relocates it: the residual is no longer free to sit in the potential itself.

Where the rule forbids halving the domain, halving anyway moves the minimum of the Wilson-line potential in 40 of the 60 representations tested — the worst by 45.1 % of the potential's own depth |V|. The other 20 are cases where the two domains happen to share their minimum.

A selection rule that never changed an answer would be a bookkeeping convention rather than a constraint. Nothing in Part III measured how often it matters, because the paper's question was which domain is legal, not what it costs to use the wrong one. The instrument can ask the second question cheaply, so it does.

The record is unaffected. This is the missing sentence about why the rule is worth stating, and it is now a panel rather than a claim: the numbers are recomputed in the page, over the same potential the calculator uses, so a reader can watch the minimum move.

One trap paid for here, recorded so nobody repeats it: the first version scanned the half-domain on its own grid, which gave it twice the resolution of the full-domain scan and manufactured a "loss" of −5.3 × 10⁻⁴ that was mesh, not physics. Two regions must be compared from one set of evaluations.

Part III's admissibility test — a + 2b + 3c odd, or the representation degenerate — is a single parity: the half-domain is legal if and only if a + c is even. Since a + 2b + 3c ≡ a + c (mod 2), and both branches of the degenerate case independently force a + c even, the second disjunct can never be the one that decides. Checked on all 3375 triples with a, b, c ≤ 14, of which 399 are degenerate: zero exceptions.

The rule was stated as a disjunction because that is how it was derived — one clause from the centre charge, one from the degenerate orbits. Written that way it hides that the two clauses never disagree. The instrument evaluates both and compares them, which is why the coincidence showed up at all.

The record is unaffected: the printed rule is correct, and this is a reduction of it, not a correction. Nothing in Part III depends on the disjunction being irredundant. The one-bit form is what the tool displays, with the disjunctive form beside it.