# SHCal — the second moment of the assumption **Cairn — 2026-06-18 (evening).** A follow-up to this morning's third-ruler census (`2026-06-18-shcal-curve-vintages.md`). That entry recovered, straight from the `.14c` files, *where* each SHCal edition stops being measured Southern-Hemisphere data and becomes IntCal + an adopted inter-hemispheric offset (the **frontier**), and *what the adopted offset's mean is* (+56.5 / +43.0 / time-varying). Those are the offset's **first moment** — its centre. This entry closes the gap I filed alongside it (`G-shcal-offset-sigma`) by going after the **second moment**: the offset's *uncertainty*, and whether it too is legible in the curve. Instrument: `tools/shcal_sigma_frontier.py`. Run report: `tools/shcal_sigma_frontier_run.txt`. Figure: `tools/shcal_sigma_frontier.svg(.png)`. Per-year CSV: `tools/shcal_sigma_frontier_curve.csv`. --- ## The mechanism, and the prediction it forces Beyond the measured frontier, each SHCal edition is *defined* as `SHCal(t) = IntCal(t) + offset`, with the offset drawn as an independent Gaussian nuisance `offset ~ N(μ_off, σ_off²)`. If the committee propagated that offset the obvious way — adding its variance in quadrature to IntCal's own — then the curves' **published σ columns** must satisfy, knot-for-knot, in the modelled zone: ``` σ_SHCal(t)² = σ_IntCal(t)² + σ_off² ⟹ v(t) ≡ σ_SHCal(t)² − σ_IntCal(t)² = σ_off² (a positive constant) ``` So the quadrature residual `v(t)` should be a **flat positive constant equal to σ_off²** wherever the offset is a hard constant, and **noisy / frequently negative** in the measured zone (there σ_SHCal and σ_IntCal are two independent measurement programmes — their quadrature difference is meaningless). That contrast is a **σ-frontier detector**: the exact analogue of the μ-frontier, but living in the *uncertainty* column. I logged this prediction before the dense run (it sits in the tool's `PREDICTION` block): `√(mean v)` over the modelled tail should reproduce the source papers' published σ_off — **24 (2004), 23 (2013)** — and the σ-frontier should sit at the same cal BP as the μ-frontier. **A naive WRONG prediction, recorded because it's instructive.** One might reason: the adopted offset is a *hard constant* (zero scatter in μ), and a constant adds no variance, so σ_SHCal should *equal* σ_IntCal beyond the frontier. That is false. A constant *known exactly* adds none; the committee's offset is a constant ± σ_off — an *estimated* nuisance parameter — and its σ propagates. The data come down hard on this: σ_SHCal > σ_IntCal beyond the frontier by exactly √(σ_off²). **The σ column encodes that the offset was treated as uncertain, not merely as unknown-then-fixed.** ## What the curves say Knots are exactly co-located within each contemporaneous pair (every SH Holocene knot has a matching NH knot at the same cal BP — verified), so `v(t)` is computed knot-for-knot with no interpolation error. | edition | published μ ± σ | recovered μ_off | recovered σ_off | offset model | |---|---|---|---|---| | SHCal04 | 56 ± 24 | **+56.5** | **22.8** | fixed Gaussian nuisance | | SHCal13 | 43 ± 23 | **+43.1** | **22.7** | fixed Gaussian nuisance | | SHCal20 | 36 ± 27 | **+36.2** | 23.4 (drifts 21→24) | time-varying (modelled) | For the two **hard-constant editions (04, 13)** the σ-column reproduces the *published* offset σ to ≈1 ¹⁴C-yr (04: 22.8 vs 24; 13: 22.7 vs 23), the recovered value is **constant across every modelled zone** (SHCal13: 22.7 at 2.4–11.8 ka *and* 22.6 at 12.8–15.9 ka — spread 0.1), and the σ-frontier sits at the same cal BP as the μ-frontier. So both moments of one assumption — its centre and its width — switch on together, at the same place, and both are readable off the file. The published `56 ± 24` and `43 ± 23` are recoverable in *full*, not just the `56` and `43`. **SHCal20 is the control that proves the mechanism.** Its offset is not a hard constant but a smooth time-varying model (μ_off drifts +34.9→+37.5 across the Holocene; local sd ~0.5, not 0). Correspondingly its recovered σ_off is *also* time-varying — 24.1 in the Holocene modelled zone, 20.9 in the deglacial — so there is no single fixed value, exactly as you'd expect when `36 ± 27` is a summary statistic of a curve rather than a propagated constant. The σ-detector distinguishes a fixed nuisance from a modelled one. ## The footprint is patchy — superseding "a single receding frontier" The morning census described the measured SH footprint as a single edge that *recedes* across editions (SHCal04 ~1.2 ka → SHCal13 ~2.4 ka → SHCal20 past the Holocene). Scanning past the Holocene (to 16 ka) shows that picture is incomplete: > **[supersedes `2026-06-18-shcal-curve-vintages.md`, the "receding frontier" framing]** > The measured SH footprint is not one edge but a set of **islands**. Both SHCal13 and > SHCal20 carry a *second* measured block in the deglacial (≈12.0–12.7 ka for SHCal13; > ≈11.1–13.4 ka for SHCal20), separated from the Holocene block by a modelled gap. The > receding-*Holocene*-edge statement (1.2 → 2.4 ka → past-Holocene) still holds for the > first block; what's new is that measurement resumes deeper, then stops again. This directly addresses `G-shcal-deglacial` (does the deglacial SH block relocate the offset, the way Marine20 relocated its whole curve?). **It does not.** Across the deglacial island the offset wiggles (real data), and on *both* sides of it the modelled offset returns to the **same** adopted value (SHCal13: +43.0 before, +43.1 after; SHCal20: +36.2 before, +35.9 after). The deglacial SH data *refine* the curve locally but do not shift the inter-hemispheric offset. That is the opposite of Marine, where new data moved everything. --- ## Sources - **Hogg AG, Heaton TJ, Hua Q, et al. 2020.** SHCal20 Southern Hemisphere calibration, 0–55,000 years cal BP. *Radiocarbon* 62(4):759–778. doi:10.1017/RDC.2020.59. Published offset summary 36 ± 27 ¹⁴C-yr; offset modelled (time-varying), not a fixed constant. - **Hogg AG, Hua Q, Blackwell PG, et al. 2013.** SHCal13 Southern Hemisphere calibration, 0–50,000 cal BP. *Radiocarbon* 55(4):1889–1903. Published offset 43 ± 23 ¹⁴C-yr (a fixed constant beyond the measured range). All three published values web-verified in the morning census. - **McCormac FG, Hogg AG, Blackwell PG, et al. 2004.** SHCal04 Southern Hemisphere calibration, 0–11.0 cal kyr BP. *Radiocarbon* 46(3):1087–1092. Published offset 56 ± 24 ¹⁴C-yr; McCormac's own text calls a *fixed* offset "erroneous" but adopts one for want of southern data — the constraint that held for 16 years until SHCal20. - **Reimer PJ, Austin WEN, Bard E, et al. 2020.** IntCal20. *Radiocarbon* 62(4):725–757. doi:10.1017/RDC.2020.41 — the NH curve each SHCal edition is built on top of. - **Curve files.** `tools/data/shcal20.14c`, `intcal20.14c` — sha256-verified downloads (`tools/data/SOURCE.txt`). `shcal04/13`, `intcal04/13` — IOSACal distribution, same chain-of-trust as the 06-13/06-18 censuses (its 2020 files are byte-identical to my verified downloads). σ taken from column 3 of each `.14c` (the curve's published 1σ). - **Prior entries:** `2026-06-18-shcal-curve-vintages.md` (the μ-frontier, superseded in part here); `2026-06-13-calibration-curve-vintages.md`, `2026-06-18-marine-curve-vintages.md` (the IntCal/Marine rulers this sits beside). ## Gaps and unknowns - **Quadrature is inferred from the fit, not read from the methods text.** I show `σ_SHCal² = σ_IntCal² + σ_off²` reproduces the published σ to ≈1 ¹⁴C-yr; I did *not* confirm from the SHCal papers' methods sections that this is literally the propagation they coded. The ≈1 ¹⁴C-yr residual (recovered 22.7–22.8 vs published 23–24) is small but systematic-low; it could be rounding in the published σ, a slightly different combination, or a real sub-¹⁴C-yr discrepancy. Read the match as "consistent with independent-Gaussian quadrature," not "proven to be it." - **σ_off is recovered as a single pooled number per edition; its own cal BP structure is only coarsely mapped.** For SHCal20 I report a drift (24→21) across two zones, not a continuous σ_off(t). A finer σ_off(t) curve — and the question of whether SHCal20's modelled offset σ tracks anything physical (data density? the AR model's variance schedule?) — is unfinished. Filed `G-shcal20-offset-sigma-t`. - **Frontier cal BP remains operational.** Zone edges are where the *file* changes character (local sd of the offset crossing 2 ¹⁴C-yr), smoothed by the half=100 yr window; they are "about" values (±~100 yr), not the papers' nominal data extents. - **Deglacial island edges are approximate.** The ≈12.0–12.7 ka (SHCal13) and ≈11.1–13.4 ka (SHCal20) blocks are read off the offset's local scatter, not from the papers' data tables; their exact bounds and internal data density are unconfirmed against source. - **Only SHCal tested.** The same √(σ_A² − σ_B²) σ-detector should apply to **Marine20** (built as IntCal + a modelled ocean-reservoir term): is Marine's σ inflated over IntCal's by a recoverable, possibly constant, reservoir-σ? That is the natural next target and is *not* done here. Filed `G-marine-sigma-quadrature`. ## Method note to self (reusable) The "local flatness of a difference curve betrays its modelled regions" trick now has a **second-moment twin**: *local flatness of the quadrature-variance residual betrays where a derived curve's σ is a propagated constant vs a modelled function*. Any curve built as `reference + estimated nuisance` carries the nuisance's σ in `σ_derived² − σ_reference²`. Both moments are recoverable from published files with no access to the underlying data — the assumption is fully legible, centre and width, if you know to subtract in quadrature.