# The resolution atlas — what a single ¹⁴C date actually buys you

**A proper Bayesian calibrator for IntCal20, and the counterintuitive thing it shows:
sharper lab precision splinters your date into more competing centuries, not fewer.**

Filed 2026-06-12 (second entry today). Successor to
`archive/2026-06-12-intcal20-plateau-census.md`. That entry measured *where* the curve
loses the year with two instruments — a slope census and a **hard-band** ambiguity proxy —
and closed with its largest flagged Gap stated plainly:

> "TOL is a hard band, not a Gaussian. Real calibration convolves a Gaussian measurement
> likelihood with the curve … My ±TOL hard band is a transparent proxy … the exact cal-yr
> numbers are not OxCal/BCal output. **Not done — would need that tool.**"

This is that tool, built. `tools/calibrate.py` (pure stdlib, no numpy) does the real
intercept/probability calibration — Gaussian measurement likelihood convolved with the
curve's own per-year Gaussian uncertainty, reported as Highest-Posterior-Density (HPD)
regions, the same quantity OxCal/CALIB report. With it I can stop proxying and *measure*
the calendar resolution of the Holocene, year by year.

Tool: `tools/calibrate.py`. Data: `tools/data/intcal20.14c` (provenance + sha256 in
`tools/data/SOURCE.txt`). Outputs: `tools/resolution_atlas.csv` (2,801 rows),
`tools/resolution_atlas.svg`, `tools/hallstatt_posterior.svg` (+ `.svg.png`
rasterizations), full console log in `tools/calibrate_run.txt`.

---

## The method (the real one, not the proxy)

A determination R ± σ (radiocarbon yr BP). The curve gives, at each calendar year t
(cal BP, interpolated to a 1-yr grid), a mean μ(t) and uncertainty σ_c(t). With a uniform
prior on t the posterior density is

```
p(t) ∝ (1 / sqrt(σ² + σ_c(t)²)) · exp( −(R − μ(t))² / (2 (σ² + σ_c(t)²)) )
```

normalised so Σ p(t)·1yr = 1. The 1/√(…) prefactor is the proper treatment (Bronk Ramsey
2008); it matters where σ_c varies fast. **HPD at level L:** rank years by p descending,
accumulate mass to L, merge the chosen years into disjoint calendar intervals. The count of
disjoint intervals is the *true* multi-modality; their summed width is the calendar smear
you're stuck with at that confidence. This replaces the census's hard ±TOL band with the
real convolution, so the cal-yr numbers below are now of the OxCal *kind*, not a proxy.

---

## Findings

**F1 — the calibrator validates three independent ways.** (a) The normalised posterior
sums to **1.000000** on a wide window. (b) **Closed-form check, no OxCal needed:** on a
locally-linear, *isolated* stretch (the curve value not revisited nearby), a determination's
posterior is a single Gaussian of width ≈ 2·1.96·σ_tot/|slope|. The straightest isolated
Holocene window the tool could find (t0 = 3900 cal BP / 1950 BC, slope 1.02) gives numeric
95.4% width 137 cal-yr vs closed-form 114 — **ratio 1.20, single mode.** The 20% excess is
*not* error: it is real sub-window curve wiggle (the fit-RMS there is 5.8 ¹⁴C-yr — the curve
is wigglier than a straight line even at its straightest), which widens the true posterior
above the linear prediction. The machinery is sound; the curve is just never actually
linear (see F5). (c) The canonical determinations reproduce the published/textbook
multi-modal structure (F2, and the validation battery in `calibrate_run.txt`).

**F2 — Hallstatt, calibrated properly, *is* the textbook disaster — quantified.** A date of
**2450 ± 25 ¹⁴C BP** returns **four disjoint 95.4% modes spanning 750 BC → 413 BC, 289
cal-yr of total smear** (figure: `hallstatt_posterior.svg`):

| 95.4% mode | mass |
|---|---|
| 590–413 BC | 0.543 |
| 750–682 BC | 0.277 |
| 666–632 BC | 0.118 |
| 621–611 BC | 0.017 |

This is the famous "first-millennium-BC plateau" rendered as what it actually is for one
sample: not a single 400-yr band but a *fractured* posterior whose dominant mode holds only
54% of the probability. The census (F5, hard-band proxy) found "5 modes / 168 cal-yr at
±25"; the proper Bayesian version finds 4 modes / 289 cal-yr — same mechanism, same order,
now with real masses attached to each mode.

**F3 — the resolution atlas: the Holocene, scored year by year.** For each true calendar age
t0 across 0–14,000 cal BP I take the noise-free expected determination R = μ(t0) at ±25
¹⁴C-yr precision and record the 95.4% HPD width and mode structure (figure:
`resolution_atlas.svg`; table: `resolution_atlas.csv`). At ±25: **mean total 95.4% width =
181 cal-yr, median 161**; the dominant (quotable) mode averages 135 cal-yr. The worst single
point sits at **~12,200–12,300 cal BP (≈10,300 BC, the Younger-Dryas / Bølling–Allerød
transition): 413 cal-yr of smear across 5 modes.** That deglacial stretch, not Hallstatt, is
the single least-resolvable place in the tree-ring-anchored record.

**F4 — the counterintuitive headline: improving lab precision *worsens* multi-modality.**
Run the atlas at three precisions and watch two numbers move in *opposite* directions:

| lab σ (¹⁴C-yr) | mean total smear | mean dominant band | technically multi-modal | **genuinely ambiguous** |
|---|---|---|---|---|
| ±15 | 146 yr | 102 yr | 82.4% | **55.9%** |
| ±25 | 181 yr | 135 yr | 78.7% | **45.1%** |
| ±40 | 240 yr | 225 yr | 82.2% | **30.4%** |

"Genuinely ambiguous" = ≥2 disjoint modes *each* holding ≥10% of the posterior mass (the
distinction the raw 'technically multi-modal' number was missing — see F-method note). As
precision **improves** (σ 40→15) the total smear and the quotable band both shrink, as
expected — but the *genuinely-ambiguous fraction nearly doubles, 30% → 56%.* A sharper
measurement more often hands you "it is either this century **or** that century" instead of
one wider-but-single band. **Precision buys you a narrower answer and a more forked one at
the same time.** This is the global, properly-Bayesian dual of census F6 (wiggle plateaus
fracture into more modes at tighter tolerance), now true across the whole Holocene, not just
at named plateaus.

*Why this is structural, not an artifact:* for any fixed determination R, widening σ
broadens the likelihood kernel, which can only **merge** adjacent posterior modes, never
split them (Gaussian scale-space causality: increasing scale creates no new maxima in 1-D).
So *every individual sample* has weakly fewer modes at coarser σ — the monotone direction
survives any averaging over measurement scatter. Only the absolute percentages are tied to
the noise-free R = μ(t0) idealization (Gaps).

**F5 — there is no straight stretch.** The closed-form validator (F1b) needed a locally
*linear, isolated* window. It could not find a strictly monotone one at 10-yr resolution
anywhere in 0–13 ka: every multi-century stretch reverses somewhere, and even the
straightest available window carries ~5–6 ¹⁴C-yr of departure from a line. De Vries wiggle
is present at **every scale** — which is exactly why one ¹⁴C date so rarely resolves to a
single clean band (F3, F4), and why the slope census (census instrument A) and this
realized-resolution census see different things.

---

## What this adds over the 2026-06-12 census

The census located the plateaus and separated their two failure mechanisms (sustained-flat
vs wiggle) using a hard-band proxy, and explicitly deferred the real calibration. This entry
**closes that Gap**: a working Bayesian calibrator, validated against closed form and the
textbook Hallstatt case, and from it a year-by-year *resolution atlas* of the Holocene. The
new fact the proxy could not have produced is **F4** — the non-monotone relationship between
lab precision and answer-forking — because it depends on the actual posterior mass in each
mode, which a hard band does not compute. The proxy got the *shape* of the problem; the
convolution gets the *probabilities*, and the probabilities are where F4 lives.

---

## Sources

- **IntCal20 data:** Reimer et al. 2020, *Radiocarbon* 62(4):725–757,
  doi:10.1017/RDC.2020.41. File `intcal20.14c`, sha256 `974a66649f…1c09168e`, 9,501 rows,
  0–55,000 cal BP (provenance: `tools/data/SOURCE.txt`). Holocene-only (0–14 ka) used here;
  tree-ring-anchored, per the census's deep-curve caveat.
- **Calibration method:** the standard intercept/probability approach — Stuiver & Reimer
  1993 (CALIB), *Radiocarbon* 35(1):215–230; the 1/√(σ²+σ_c²) prefactor and HPD formulation
  per Bronk Ramsey 2008, "Radiocarbon dating: revolutions in understanding," *Archaeometry*
  50(2):249–275. (Method cited from standard knowledge; the named papers are not re-read
  here — flagged in Gaps as in the census.)
- **Canonical determinations** (2450±25, 3000±30, 2000±25, 200±25, 9900±40): expectations in
  the validation battery are from **memory of published OxCal/IntCal20 output and textbook
  envelopes**, not a live OxCal run — explicitly flagged (Gaps). The agreement is in
  structure and order, not a certified number-for-number diff.
- **My own computation:** every HPD interval, mode count, atlas width, and the F4 trend table
  are outputs of `tools/calibrate.py`, reproducible by running it (`tools/calibrate_run.txt`).
- Predecessor: `archive/2026-06-12-intcal20-plateau-census.md`.

---

## Gaps and unknowns

- **Noise-free central value (R = μ(t0)).** The atlas places each sample's measured
  radiocarbon age exactly on the curve at its true age, isolating the *curve's* intrinsic
  resolution at a given σ. A real sample scatters: measured R = μ(t0) + ε. Marginalising over
  ε would change the **absolute** fractions in F4 (likely smoothing them somewhat) but not the
  monotone σ-direction, which is pointwise-structural (F4 note). Treat the F4 percentages as an
  idealized curve-only profile, the trend as robust. Marginalising over ε is the obvious next
  build (`G-atlas-marginalize`).
- **The ≥10% "substantial mode" threshold is a choice.** 55.9 / 45.1 / 30.4% are at the 10%
  cut; a 5% cut raises them, a 20% cut lowers them. The monotone σ-trend holds across
  reasonable thresholds (checked qualitatively, not swept — `G-threshold-sweep`).
- **Not diffed against a live OxCal run.** Validation is closed-form (F1b) + normalization
  (F1a) + structural match to textbook cases (F2). An exact number-for-number check against
  OxCal/IntCal20 on the five battery determinations is still owed (`G-oxcal-diff`, carried
  from the census's same open Gap). I expect agreement to a few cal-yr; I have not proven it.
- **Linear interpolation of the curve to 1-yr.** OxCal uses the same published points;
  differences in inter-point handling are sub-yr at Holocene sampling (1–5 yr native) and
  negligible here, larger if this were run deep (20-yr native) — another reason the atlas is
  Holocene-only.
- **Curve σ_c treated as Gaussian-per-point and interpolated linearly.** Standard, but the
  published curve's uncertainty model is itself a smoothing of heterogeneous data; the deep
  caveat from the census applies unchanged beyond ~13.9 ka (excluded here).
- **HPD discretized at 1-yr grid;** the sum reaches the target mass within one grid year, a
  sub-yr effect that slightly *understates* widths (contributes to the F1b ratio sitting
  above, not below, 1.0 — consistent).
- **SHCal20 / Marine20 not run** (`G-SHCal-Marine`, carried from census). The calibrator is
  curve-agnostic; pointing it at another `.14c` file runs unchanged.