.. _ls10-recommendations:

LS10 systematic-correction recommendations
==========================================

.. contents:: On this page
   :local:
   :depth: 2

.. seealso::

   :doc:`results_ls10` — full σ̂ and LRT tables for all nine samples and four resolutions.

Decision framework
------------------

**Method choice — MCMC-comb (WEIGHT\_COMB) for all samples.**

The Likelihood Ratio Test (LRT) rejects the additive-only null hypothesis at all four
map resolutions (NSIDE 32–256) and all nine stellar-mass thresholds, with
:math:`p < 10^{-9}` in every case.  This confirms a statistically significant
*multiplicative* component on top of the additive systematic.
``MCMC-comb`` (stored as ``WEIGHT_COMB = WEIGHT_SYS`` in the FITS files) is the only
method that models both components jointly.  The other methods are ranked as follows:

* ``OLS``, ``ISD-1``, ``MCMC-add``: nearly identical σ̂ values, miss the multiplicative term.
* ``ElasticNet``: equivalent to OLS when cross-validation selects zero amplitudes (some samples).
* ``ISD-3``: numerically unstable — σ̂ ≫ 1 at NSIDE 32 and poor at all other resolutions.
  **Never use ISD-3 for science.**

**Resolution choice — NSIDE 64 for all samples (primary).**

The selection criterion is the minimum :math:`\hat{\sigma}_{\rm comb}` among
valid NSIDEs (σ < 1, NSIDE ≥ 64).  Three additional constraints apply:

1. **NSIDE 32 is excluded** — its pixel scale (~108') exceeds the maximum
   angular bin in the w(θ) data (54'), producing zero measurable correction
   on the two-point function in the entire measured range.
2. **All 11 templates are at NSIDE 64** — analysing at finer resolutions
   upsamples the templates without adding systematic information.
3. **NSIDE 128/256 overfit** for sparse samples (M* ≥ 9.0, ≥ 11.25, ≥ 11.5):
   σ̂_comb exceeds 1.0 and the corrected w(θ) shows unphysically large corrections
   (up to 34%).

σ̂_comb is minimised at NSIDE 64 for all nine samples among resolutions where
w(θ) correction is non-zero.  NSIDE 128 is a viable *secondary* option for the five
densest samples (M* 9.5–11.25, σ̂_comb < 1), extending the correction scale to ~20'.

Recommended weight and resolution per sample
--------------------------------------------

.. csv-table::
   :header: "Sample (log M*)", "N_gal", "Method", "NSIDE", ":math:`\hat{\sigma}_{\rm comb}`", "θ corrected", "Key constraint"
   :widths: 12, 12, 12, 8, 10, 14, 50

   "≥ 9.0", "523,486", "MCMC-comb", "64", "0.751", "49–54'", "σ̂_comb N64=0.751 (minimum among valid NSIDEs); N128 overfits (σ=1.029); N256 overfits (σ=1.601); external |δw/w|=7.2% at 30'"
   "≥ 9.5", "1,432,502", "MCMC-comb", "64", "0.554", "49–54'", "σ̂_comb N64=0.554 (minimum among valid NSIDEs); N256 overfits (σ=1.052); external |δw/w|=4.8% at 30'"
   "≥ 10.0", "2,759,238", "MCMC-comb", "64", "0.382", "49–54'", "σ̂_comb N64=0.382 (minimum among valid NSIDEs); external |δw/w|=0.4% at 30'"
   "≥ 10.25", "3,308,841", "MCMC-comb", "64", "0.334", "49–54'", "σ̂_comb N64=0.334 (minimum among valid NSIDEs)"
   "≥ 10.5", "3,263,228", "MCMC-comb", "64", "0.310", "49–54'", "σ̂_comb N64=0.310 (minimum among valid NSIDEs)"
   "≥ 10.75", "2,802,710", "MCMC-comb", "64", "0.301", "49–54'", "σ̂_comb N64=0.301 (minimum among valid NSIDEs)"
   "≥ 11.0", "1,619,838", "MCMC-comb", "64", "0.305", "49–54'", "σ̂_comb N64=0.305 (minimum among valid NSIDEs); external |δw/w|=0.1% at 30'"
   "≥ 11.25", "541,855", "MCMC-comb", "64", "0.393", "49–54'", "σ̂_comb N64=0.393 (minimum among valid NSIDEs); N256 overfits (σ=1.212); external |δw/w|=2.2% at 30'"
   "≥ 11.5", "120,882", "MCMC-comb", "64", "0.710", "49–54'", "σ̂_comb N64=0.710 (minimum among valid NSIDEs); N128 overfits (σ=1.430); N256 overfits (σ=2.021); external |δw/w|=0.7% at 30'"

.. note::

   NSIDE 128 is a valid secondary choice for the five densest samples
   (M* ≥ 9.5 through ≥ 11.25), extending the angular correction down to ~20 arcmin.
   It must **not** be used for M* ≥ 9.0 (σ̂_comb = 1.03) or M* ≥ 11.5 (σ̂_comb = 1.43),
   where it overfits shot noise.

   NSIDE 32 should never be used as the analysis resolution for angular clustering:
   its pixel scale exceeds the measured w(θ) range and delivers zero correction.

σ̂_comb goodness-of-fit across resolutions
-------------------------------------------

:math:`\hat{\sigma}_{\rm comb} < 1` indicates the model fits within the Poisson
expectation.  Values ≥ 1 (marked ✗) signal overfitting — the correction
is absorbing noise rather than real systematics.

.. csv-table::
   :header: "Sample (log M*)", "N_gal", "NSIDE 32", "NSIDE 64", "NSIDE 128", "NSIDE 256", "Recommended"
   :widths: 12, 12, 14, 14, 14, 14, 12

   "≥ 9.0", "523,486", "0.674", "0.751", "1.029 ✗", "1.601 ✗", "**N64**"
   "≥ 9.5", "1,432,502", "0.606", "0.554", "0.741", "1.052 ✗", "**N64**"
   "≥ 10.0", "2,759,238", "0.422", "0.382", "0.534", "0.786", "**N64**"
   "≥ 10.25", "3,308,841", "0.390", "0.334", "0.474", "0.696", "**N64**"
   "≥ 10.5", "3,263,228", "0.373", "0.310", "0.451", "0.659", "**N64**"
   "≥ 10.75", "2,802,710", "0.398", "0.301", "0.430", "0.630", "**N64**"
   "≥ 11.0", "1,619,838", "0.393", "0.305", "0.472", "0.726", "**N64**"
   "≥ 11.25", "541,855", "0.406", "0.393", "0.633", "1.212 ✗", "**N64**"
   "≥ 11.5", "120,882", "0.586", "0.710", "1.430 ✗", "2.021 ✗", "**N64**"

Systematic uncertainty budget
------------------------------

The table below quantifies the systematic correction applied and the residual
method uncertainty at the recommended NSIDE 64 + MCMC-comb.

* **Max |δw/w|_comb at 42–54'** — the largest fractional correction applied to w(θ)
  by the MCMC-comb weights relative to the uncorrected measurement.
  Observed only at the outermost bins because the NSIDE 64 pixel scale (~54')
  sets the lower angular limit of the weight-induced clustering modulation.
* **Method uncertainty >20'** — the standard deviation of the corrected w(θ)
  over the four well-behaved methods {OLS, ISD-1, MCMC-add, MCMC-comb},
  expressed as a fraction of w_comb.  This is the irreducible systematic floor
  from the choice of correction algorithm.
* Below ~40 arcmin at NSIDE 64, no measurable correction is applied in the
  wtheta data range.  The large-scale systematic bias (e.g. −7.2% at 30' for
  M* ≥ 9.0) operates primarily at degree scales where the template coherence
  length matches and the weights are most effective.

.. csv-table::
   :header: "Sample (log M*)", "N_gal", "Recommended NSIDE", "Max |δw/w|\ :sub:`comb` at 42–54'", "θ of first correction", "Method uncertainty >20'", "Regime"
   :widths: 12, 12, 14, 18, 16, 18, 22

   "≥ 9.0", "523,486", "64", "8.75%", "49–54'", "1.64%", "Strongly contaminated"
   "≥ 9.5", "1,432,502", "64", "6.82%", "49–54'", "0.80%", "Strongly contaminated"
   "≥ 10.0", "2,759,238", "64", "3.40%", "49–54'", "0.12%", "Weakly contaminated"
   "≥ 10.25", "3,308,841", "64", "5.09%", "49–54'", "0.07%", "Low contamination"
   "≥ 10.5", "3,263,228", "64", "8.10%", "49–54'", "0.14%", "Low contamination"
   "≥ 10.75", "2,802,710", "64", "12.98%", "49–54'", "0.11%", "Low contamination"
   "≥ 11.0", "1,619,838", "64", "19.52%", "49–54'", "0.15%", "Weakly contaminated"
   "≥ 11.25", "541,855", "64", "18.93%", "49–54'", "0.24%", "Moderately contaminated"
   "≥ 11.5", "120,882", "64", "29.29%", "49–54'", "0.29%", "Weakly contaminated"

Per-sample angular correction profiles
---------------------------------------

The following tables show, for each sample at the recommended NSIDE 64,
the observed w(θ), the MCMC-comb corrected w(θ), the fractional correction
applied, and the method-to-method spread as a proxy for the systematic floor.
All corrections are expressed as percentages of w_obs.

.. _rec-9p0:

M* ≥ 9.0 (z < 0.08, N = 523,486)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ -7.2% at 30', max |δw/w| = 8.4% at 23'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.751).  No finer valid NSIDE available (σ̂ ≥ 1 at NSIDE 128+).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "3.71714", "3.71714", "+0.00", "0.00"
   "0.69", "3.41581", "3.41581", "+0.00", "0.00"
   "0.86", "3.06270", "3.06270", "+0.00", "0.00"
   "1.06", "2.70489", "2.70489", "+0.00", "0.00"
   "1.31", "2.39533", "2.39533", "+0.00", "0.00"
   "1.63", "2.21736", "2.21736", "+0.00", "0.00"
   "2.01", "1.95239", "1.95239", "+0.00", "0.00"
   "2.49", "1.74937", "1.74937", "+0.00", "0.00"
   "3.08", "1.55497", "1.55497", "+0.00", "0.00"
   "3.82", "1.39614", "1.39614", "+0.00", "0.00"
   "4.72", "1.23611", "1.23611", "+0.00", "0.00"
   "5.85", "1.08708", "1.08708", "+0.00", "0.00"
   "7.23", "0.94986", "0.94986", "+0.00", "0.00"
   "8.96", "0.81964", "0.81964", "+0.00", "0.00"
   "11.08", "0.69699", "0.69699", "+0.00", "0.00"
   "13.72", "0.58956", "0.58956", "+0.00", "0.00"
   "16.99", "0.49512", "0.49512", "+0.00", "0.00"
   "21.03", "0.40304", "0.40304", "+0.00", "0.00"
   "26.03", "0.32596", "0.32596", "+0.00", "0.00"
   "32.22", "0.27140", "0.27140", "+0.00", "0.00"
   "39.89", "0.22417", "0.22417", "+0.00", "0.00"
   "49.38", "0.18634", "0.17003", "+8.75", "1.64"
   "61.12", "0.15643", "0.14727", "+5.85", "1.11"
   "75.66", "0.13447", "0.12775", "+5.00", "0.95"
   "93.64", "0.11412", "0.10478", "+8.18", "1.16"
   "115.92", "0.09485", "0.08971", "+5.42", "0.84"
   "143.50", "0.08309", "0.07744", "+6.80", "0.86"
   "177.62", "0.06393", "0.05976", "+6.52", "0.78"
   "219.84", "0.05130", "0.04763", "+7.16", "0.73"
   "272.13", "0.04051", "0.03717", "+8.24", "0.70"

.. _rec-9p5:

M* ≥ 9.5 (z < 0.12, N = 1,432,502)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ -4.8% at 30', max |δw/w| = 4.9% at 15'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.554).  NSIDE 128 is a valid secondary option (σ̂=0.741).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "3.02828", "3.02828", "+0.00", "0.00"
   "0.69", "2.68379", "2.68379", "+0.00", "0.00"
   "0.86", "2.34430", "2.34430", "+0.00", "0.00"
   "1.06", "2.12172", "2.12172", "+0.00", "0.00"
   "1.31", "1.85966", "1.85966", "+0.00", "0.00"
   "1.63", "1.64785", "1.64785", "+0.00", "0.00"
   "2.01", "1.46712", "1.46712", "+0.00", "0.00"
   "2.49", "1.29353", "1.29353", "+0.00", "0.00"
   "3.08", "1.13963", "1.13963", "+0.00", "0.00"
   "3.82", "0.99403", "0.99403", "+0.00", "0.00"
   "4.72", "0.86649", "0.86649", "+0.00", "0.00"
   "5.85", "0.74101", "0.74101", "+0.00", "0.00"
   "7.24", "0.63716", "0.63716", "+0.00", "0.00"
   "8.96", "0.53442", "0.53442", "+0.00", "0.00"
   "11.09", "0.44365", "0.44365", "+0.00", "0.00"
   "13.73", "0.36584", "0.36584", "+0.00", "0.00"
   "17.00", "0.30018", "0.30018", "+0.00", "0.00"
   "21.04", "0.24555", "0.24555", "+0.00", "0.00"
   "26.04", "0.20359", "0.20359", "+0.00", "0.00"
   "32.24", "0.17215", "0.17215", "+0.00", "0.00"
   "39.91", "0.14930", "0.14930", "+0.00", "0.00"
   "49.40", "0.12794", "0.11999", "+6.21", "0.80"
   "61.14", "0.10998", "0.10521", "+4.33", "0.54"
   "75.68", "0.09423", "0.09068", "+3.77", "0.44"
   "93.67", "0.07730", "0.07227", "+6.51", "0.61"
   "115.95", "0.06843", "0.06557", "+4.18", "0.40"
   "143.53", "0.05427", "0.05115", "+5.74", "0.44"
   "177.64", "0.04707", "0.04464", "+5.17", "0.38"
   "219.89", "0.03238", "0.03025", "+6.59", "0.38"
   "272.18", "0.02931", "0.02731", "+6.82", "0.36"

.. _rec-10p0:

M* ≥ 10.0 (z < 0.18, N = 2,759,238)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ -0.4% at 30', max |δw/w| = 2.0% at 120'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.382).  NSIDE 256 is a valid secondary option (σ̂=0.786).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "2.32873", "2.32873", "+0.00", "0.00"
   "0.69", "2.02841", "2.02841", "+0.00", "0.00"
   "0.86", "1.76627", "1.76627", "+0.00", "0.00"
   "1.06", "1.54218", "1.54218", "+0.00", "0.00"
   "1.31", "1.33191", "1.33191", "+0.00", "0.00"
   "1.63", "1.15594", "1.15594", "+0.00", "0.00"
   "2.01", "0.99820", "0.99820", "+0.00", "0.00"
   "2.49", "0.85471", "0.85471", "+0.00", "0.00"
   "3.08", "0.72759", "0.72759", "+0.00", "0.00"
   "3.82", "0.61737", "0.61737", "+0.00", "0.00"
   "4.72", "0.51672", "0.51672", "+0.00", "0.00"
   "5.85", "0.43088", "0.43088", "+0.00", "0.00"
   "7.24", "0.35531", "0.35531", "+0.00", "0.00"
   "8.96", "0.29081", "0.29081", "+0.00", "0.00"
   "11.09", "0.23767", "0.23767", "+0.00", "0.00"
   "13.73", "0.19574", "0.19574", "+0.00", "0.00"
   "17.00", "0.16276", "0.16276", "+0.00", "0.00"
   "21.05", "0.13598", "0.13598", "+0.00", "0.00"
   "26.05", "0.11506", "0.11506", "+0.00", "0.00"
   "32.24", "0.09877", "0.09877", "+0.00", "0.00"
   "39.91", "0.08372", "0.08372", "+0.00", "0.00"
   "49.41", "0.07394", "0.07271", "+1.67", "0.12"
   "61.15", "0.05939", "0.05854", "+1.42", "0.10"
   "75.69", "0.05103", "0.05035", "+1.33", "0.08"
   "93.70", "0.04113", "0.04029", "+2.06", "0.09"
   "115.97", "0.03283", "0.03229", "+1.67", "0.07"
   "143.55", "0.02782", "0.02724", "+2.06", "0.07"
   "177.67", "0.02007", "0.01959", "+2.38", "0.07"
   "219.92", "0.01382", "0.01339", "+3.11", "0.09"
   "272.23", "0.01186", "0.01146", "+3.40", "0.10"

.. _rec-10p25:

M* ≥ 10.25 (z < 0.22, N = 3,308,841)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: not available for this sample.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.334).  NSIDE 256 is a valid secondary option (σ̂=0.696).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "2.04201", "2.04201", "+0.00", "0.00"
   "0.69", "1.74577", "1.74577", "+0.00", "0.00"
   "0.86", "1.50298", "1.50298", "+0.00", "0.00"
   "1.06", "1.28995", "1.28995", "+0.00", "0.00"
   "1.31", "1.10343", "1.10343", "+0.00", "0.00"
   "1.63", "0.93653", "0.93653", "+0.00", "0.00"
   "2.01", "0.79371", "0.79371", "+0.00", "0.00"
   "2.49", "0.67131", "0.67131", "+0.00", "0.00"
   "3.08", "0.55949", "0.55949", "+0.00", "0.00"
   "3.82", "0.46909", "0.46909", "+0.00", "0.00"
   "4.73", "0.38466", "0.38466", "+0.00", "0.00"
   "5.85", "0.31790", "0.31790", "+0.00", "0.00"
   "7.24", "0.25879", "0.25879", "+0.00", "0.00"
   "8.96", "0.21090", "0.21090", "+0.00", "0.00"
   "11.10", "0.17245", "0.17245", "+0.00", "0.00"
   "13.74", "0.14305", "0.14305", "+0.00", "0.00"
   "17.00", "0.12102", "0.12102", "+0.00", "0.00"
   "21.05", "0.10244", "0.10244", "+0.00", "0.00"
   "26.05", "0.08609", "0.08609", "+0.00", "0.00"
   "32.25", "0.07519", "0.07519", "+0.00", "0.00"
   "39.92", "0.06292", "0.06292", "+0.00", "0.00"
   "49.41", "0.05613", "0.05524", "+1.60", "0.07"
   "61.16", "0.04478", "0.04414", "+1.42", "0.06"
   "75.70", "0.03708", "0.03654", "+1.47", "0.05"
   "93.70", "0.03161", "0.03097", "+2.04", "0.05"
   "115.98", "0.02248", "0.02202", "+2.03", "0.04"
   "143.55", "0.02054", "0.02006", "+2.33", "0.04"
   "177.68", "0.01295", "0.01253", "+3.20", "0.03"
   "219.94", "0.00851", "0.00812", "+4.54", "0.03"
   "272.24", "0.00724", "0.00687", "+5.09", "0.04"

.. _rec-10p5:

M* ≥ 10.5 (z < 0.26, N = 3,263,228)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: not available for this sample.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.310).  NSIDE 256 is a valid secondary option (σ̂=0.659).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "1.83274", "1.83274", "+0.00", "0.00"
   "0.69", "1.54648", "1.54648", "+0.00", "0.00"
   "0.86", "1.29930", "1.29930", "+0.00", "0.00"
   "1.06", "1.10542", "1.10542", "+0.00", "0.00"
   "1.31", "0.92508", "0.92508", "+0.00", "0.00"
   "1.63", "0.77624", "0.77624", "+0.00", "0.00"
   "2.01", "0.64651", "0.64651", "+0.00", "0.00"
   "2.49", "0.53988", "0.53988", "+0.00", "0.00"
   "3.08", "0.44303", "0.44303", "+0.00", "0.00"
   "3.82", "0.36563", "0.36563", "+0.00", "0.00"
   "4.73", "0.29859", "0.29859", "+0.00", "0.00"
   "5.85", "0.24366", "0.24366", "+0.00", "0.00"
   "7.24", "0.19921", "0.19921", "+0.00", "0.00"
   "8.97", "0.16323", "0.16323", "+0.00", "0.00"
   "11.10", "0.13584", "0.13584", "+0.00", "0.00"
   "13.74", "0.11425", "0.11425", "+0.00", "0.00"
   "17.01", "0.09842", "0.09842", "+0.00", "0.00"
   "21.05", "0.08323", "0.08323", "+0.00", "0.00"
   "26.05", "0.07059", "0.07059", "+0.00", "0.00"
   "32.25", "0.06188", "0.06188", "+0.00", "0.00"
   "39.92", "0.05086", "0.05086", "+0.00", "0.00"
   "49.41", "0.04559", "0.04464", "+2.08", "0.05"
   "61.16", "0.03581", "0.03512", "+1.94", "0.04"
   "75.70", "0.02985", "0.02923", "+2.08", "0.05"
   "93.70", "0.02554", "0.02483", "+2.78", "0.04"
   "115.98", "0.01760", "0.01706", "+3.08", "0.06"
   "143.55", "0.01645", "0.01589", "+3.42", "0.05"
   "177.69", "0.00970", "0.00920", "+5.17", "0.09"
   "219.95", "0.00661", "0.00614", "+7.21", "0.12"
   "272.25", "0.00566", "0.00520", "+8.10", "0.14"

.. _rec-10p75:

M* ≥ 10.75 (z < 0.31, N = 2,802,710)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: not available for this sample.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.301).  NSIDE 256 is a valid secondary option (σ̂=0.630).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "1.67913", "1.67913", "+0.00", "0.00"
   "0.69", "1.39828", "1.39828", "+0.00", "0.00"
   "0.86", "1.15311", "1.15311", "+0.00", "0.00"
   "1.06", "0.96659", "0.96659", "+0.00", "0.00"
   "1.31", "0.78652", "0.78652", "+0.00", "0.00"
   "1.63", "0.64253", "0.64253", "+0.00", "0.00"
   "2.01", "0.53047", "0.53047", "+0.00", "0.00"
   "2.49", "0.43243", "0.43243", "+0.00", "0.00"
   "3.08", "0.34959", "0.34959", "+0.00", "0.00"
   "3.82", "0.28335", "0.28335", "+0.00", "0.00"
   "4.73", "0.23140", "0.23140", "+0.00", "0.00"
   "5.85", "0.18662", "0.18662", "+0.00", "0.00"
   "7.24", "0.15564", "0.15564", "+0.00", "0.00"
   "8.97", "0.12849", "0.12849", "+0.00", "0.00"
   "11.10", "0.10935", "0.10935", "+0.00", "0.00"
   "13.74", "0.09324", "0.09324", "+0.00", "0.00"
   "17.00", "0.08058", "0.08058", "+0.00", "0.00"
   "21.05", "0.06795", "0.06795", "+0.00", "0.00"
   "26.05", "0.05777", "0.05777", "+0.00", "0.00"
   "32.25", "0.04913", "0.04913", "+0.00", "0.00"
   "39.92", "0.04046", "0.04046", "+0.00", "0.00"
   "49.41", "0.03593", "0.03471", "+3.37", "0.09"
   "61.16", "0.02669", "0.02587", "+3.07", "0.11"
   "75.70", "0.02368", "0.02296", "+3.04", "0.11"
   "93.71", "0.01827", "0.01741", "+4.70", "0.09"
   "115.98", "0.01312", "0.01250", "+4.74", "0.10"
   "143.56", "0.01200", "0.01133", "+5.52", "0.09"
   "177.70", "0.00658", "0.00601", "+8.64", "0.09"
   "219.95", "0.00423", "0.00369", "+12.78", "0.09"
   "272.26", "0.00403", "0.00350", "+12.98", "0.09"

.. _rec-11p0:

M* ≥ 11.0 (z < 0.35, N = 1,619,838)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ -0.1% at 30', max |δw/w| = 11.5% at 181'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.305).  NSIDE 256 is a valid secondary option (σ̂=0.726).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "1.95256", "1.95256", "+0.00", "0.00"
   "0.69", "1.59664", "1.59664", "+0.00", "0.00"
   "0.86", "1.32136", "1.32136", "+0.00", "0.00"
   "1.06", "1.07241", "1.07241", "+0.00", "0.00"
   "1.31", "0.85831", "0.85831", "+0.00", "0.00"
   "1.63", "0.68550", "0.68550", "+0.00", "0.00"
   "2.01", "0.55324", "0.55324", "+0.00", "0.00"
   "2.49", "0.43709", "0.43709", "+0.00", "0.00"
   "3.08", "0.34639", "0.34639", "+0.00", "0.00"
   "3.82", "0.27836", "0.27836", "+0.00", "0.00"
   "4.73", "0.22881", "0.22881", "+0.00", "0.00"
   "5.85", "0.18603", "0.18603", "+0.00", "0.00"
   "7.24", "0.15573", "0.15573", "+0.00", "0.00"
   "8.96", "0.13200", "0.13200", "+0.00", "0.00"
   "11.10", "0.11279", "0.11279", "+0.00", "0.00"
   "13.74", "0.09547", "0.09547", "+0.00", "0.00"
   "17.00", "0.08340", "0.08340", "+0.00", "0.00"
   "21.05", "0.07102", "0.07102", "+0.00", "0.00"
   "26.05", "0.06002", "0.06002", "+0.00", "0.00"
   "32.24", "0.05002", "0.05002", "+0.00", "0.00"
   "39.91", "0.04213", "0.04213", "+0.00", "0.00"
   "49.40", "0.03462", "0.03324", "+3.99", "0.05"
   "61.15", "0.02839", "0.02753", "+3.03", "0.05"
   "75.70", "0.02487", "0.02413", "+2.96", "0.05"
   "93.69", "0.01556", "0.01460", "+6.19", "0.05"
   "115.96", "0.01629", "0.01563", "+4.05", "0.04"
   "143.55", "0.00966", "0.00894", "+7.50", "0.05"
   "177.68", "0.00802", "0.00740", "+7.67", "0.06"
   "219.94", "0.00301", "0.00242", "+19.52", "0.15"
   "272.24", "0.00431", "0.00374", "+13.30", "0.09"

.. _rec-11p25:

M* ≥ 11.25 (z < 0.35, N = 541,855)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ +2.2% at 30', max |δw/w| = 17.4% at 181'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.393).  NSIDE 128 is a valid secondary option (σ̂=0.633).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "2.88285", "2.88285", "+0.00", "0.00"
   "0.69", "2.46602", "2.46602", "+0.00", "0.00"
   "0.86", "1.95053", "1.95053", "+0.00", "0.00"
   "1.06", "1.53776", "1.53776", "+0.00", "0.00"
   "1.31", "1.25039", "1.25039", "+0.00", "0.00"
   "1.62", "0.97024", "0.97024", "+0.00", "0.00"
   "2.01", "0.80113", "0.80113", "+0.00", "0.00"
   "2.49", "0.59543", "0.59543", "+0.00", "0.00"
   "3.08", "0.46615", "0.46615", "+0.00", "0.00"
   "3.82", "0.36119", "0.36119", "+0.00", "0.00"
   "4.72", "0.29551", "0.29551", "+0.00", "0.00"
   "5.85", "0.24319", "0.24319", "+0.00", "0.00"
   "7.24", "0.19466", "0.19466", "+0.00", "0.00"
   "8.96", "0.17246", "0.17246", "+0.00", "0.00"
   "11.09", "0.14584", "0.14584", "+0.00", "0.00"
   "13.73", "0.12429", "0.12429", "+0.00", "0.00"
   "17.00", "0.10843", "0.10843", "+0.00", "0.00"
   "21.04", "0.09061", "0.09061", "+0.00", "0.00"
   "26.04", "0.07547", "0.07547", "+0.00", "0.00"
   "32.24", "0.06300", "0.06300", "+0.00", "0.00"
   "39.90", "0.05287", "0.05287", "+0.00", "0.00"
   "49.39", "0.04264", "0.04085", "+4.19", "0.09"
   "61.13", "0.03611", "0.03510", "+2.81", "0.08"
   "75.67", "0.02624", "0.02543", "+3.07", "0.07"
   "93.67", "0.02222", "0.02104", "+5.31", "0.08"
   "115.94", "0.01504", "0.01433", "+4.74", "0.07"
   "143.51", "0.01258", "0.01176", "+6.53", "0.08"
   "177.66", "0.00800", "0.00734", "+8.25", "0.09"
   "219.89", "0.00332", "0.00269", "+18.93", "0.24"
   "272.19", "0.00490", "0.00428", "+12.55", "0.14"

.. _rec-11p5:

M* ≥ 11.5 (z < 0.35, N = 120,882)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

External two-point measurement: δw/w ≈ +0.7% at 30', max |δw/w| = 9.7% at 97'.

Recommended: ``WEIGHT_COMB`` from NSIDE 64 analysis (σ̂_comb = 0.710).  No finer valid NSIDE available (σ̂ ≥ 1 at NSIDE 128+).

.. csv-table::
   :header: "θ (arcmin)", "w_obs", "w_comb (MCMC-comb)", "δw/w (%)", "Method spread (%)"
   :widths: 12, 14, 18, 12, 16

   "0.56", "5.94974", "5.94974", "+0.00", "0.00"
   "0.69", "4.79663", "4.79663", "+0.00", "0.00"
   "0.86", "3.38167", "3.38167", "+0.00", "0.00"
   "1.06", "2.83741", "2.83741", "+0.00", "0.00"
   "1.32", "2.09777", "2.09777", "+0.00", "0.00"
   "1.62", "1.51437", "1.51437", "+0.00", "0.00"
   "2.01", "1.41378", "1.41378", "+0.00", "0.00"
   "2.48", "1.01514", "1.01514", "+0.00", "0.00"
   "3.08", "0.73878", "0.73878", "+0.00", "0.00"
   "3.81", "0.58870", "0.58870", "+0.00", "0.00"
   "4.71", "0.49227", "0.49227", "+0.00", "0.00"
   "5.84", "0.38506", "0.38506", "+0.00", "0.00"
   "7.23", "0.33236", "0.33236", "+0.00", "0.00"
   "8.96", "0.25731", "0.25731", "+0.00", "0.00"
   "11.09", "0.22303", "0.22303", "+0.00", "0.00"
   "13.73", "0.18840", "0.18840", "+0.00", "0.00"
   "16.98", "0.16781", "0.16781", "+0.00", "0.00"
   "21.03", "0.13840", "0.13840", "+0.00", "0.00"
   "26.03", "0.12239", "0.12239", "+0.00", "0.00"
   "32.23", "0.08921", "0.08921", "+0.00", "0.00"
   "39.88", "0.07715", "0.07715", "+0.00", "0.00"
   "49.38", "0.06260", "0.05872", "+6.19", "0.20"
   "61.12", "0.05113", "0.04871", "+4.73", "0.28"
   "75.64", "0.03836", "0.03642", "+5.06", "0.29"
   "93.63", "0.02901", "0.02644", "+8.86", "0.21"
   "115.88", "0.02264", "0.02096", "+7.44", "0.27"
   "143.47", "0.01488", "0.01306", "+12.27", "0.23"
   "177.60", "0.01079", "0.00928", "+14.03", "0.25"
   "219.81", "0.00557", "0.00414", "+25.70", "0.24"
   "272.11", "0.00477", "0.00337", "+29.29", "0.23"

Caveats and limitations
-----------------------

* **Angular range**: the wtheta_data.json files cover θ = 0.57–54 arcmin only.
  Systematic contamination at degree scales (θ > 1°) is not captured in these tables.
  The external δw/w measurements at 30–180' (listed in the per-sample rows above)
  come from a separate analysis that covers a broader angular range.

* **Zero correction at θ < 40' for NSIDE 64**: the correction from NSIDE 64 weights
  is zero below ~40 arcmin in the measured range because the NSIDE 64 pixel scale
  (~54') sets the minimum angular scale over which the weight field varies.
  This does not mean the systematic is zero at small scales — it means that
  sub-degree corrections require a finer weight map (NSIDE 128, see secondary option).

* **NSIDE 128 secondary option**: for the five densest samples (M* 9.5–11.25),
  σ̂_comb < 1 at NSIDE 128, and corrections extend to θ ~ 20'.
  The method uncertainty remains below 0.8% in all cases.
  However, since all templates are at NSIDE 64, the NSIDE 128 weight field
  is derived from upsampled templates with no additional angular information.

* **ISD-3 must not be used**: σ̂ >> 1 at NSIDE 32 for all samples; the corrected
  weights are numerically unstable.  See :doc:`results_ls10` for full ISD-3 σ̂ values.

* **FITS column names**: the recommended weights are stored as ``WEIGHT_COMB``
  (equivalent to ``WEIGHT_SYS``) in the ``*_NSIDE0064_WEIGHTS.fits`` files.