Flat-Earth: Mathematician review (Step 2)

Reviewer: PhD mathematician. Scope: derivations, notation, units, hidden assumptions, and whether the numbers in the documents follow from their own premises. Verdicts use the prompt.txt scale. Physics verdicts (what the mainstream says, what an experiment shows) belong to the physicist; where I touch them I say so.

Tags: [read] = passage read in the document (deep read, source text, or page image); [calc] = my own arithmetic (Python, venv, numbers shown); [web] = Internet source with URL; [memory] = recalled, with confidence. Source stakes are stated where they matter (the Flat Earth Society wiki, Dubay, the Koreshan authors and Rob Skiba advocate the flat or concave earth; NASA, DMA and NGA are agencies with a stake in the globe model; Wallace, Carpenter and Hampden were parties to the 1870 wager).

How I worked. I read facts.json, claims.md and all 38 deep reads, the digests where a claim needed them, sites.md "Leads for review", the ten Investigator notes, and the Investigator's saved documents that bear on mathematics (Blount's "Collapse of the Globular Theory" with Clifton's letter, and the Wallace-era Field material as summarised in the notes). I re-ran every load-bearing calculation myself. I read these primary passages myself: Clifton's letter and the Blount account (research/files/tfeslib-...08...pdf), and the printed page 171 of Rowbotham 1865 (PDF p.176, page image). The Skiba videos are YouTube auto-captions, so figures from them may be misheard.

0. The mathematics every claim depends on (read this first)

Almost every number in this project comes from one formula, and its misuse produces most of the errors found below. So here it is once, with its assumptions.

Curvature drop. A sphere of radius R (about 3,959 statute miles) falls away from a horizontal tangent line by

drop(d) = R(sec(d/R) - 1), which is very nearly d^2/(2R) for d much less than R.

With d in miles and the answer in inches, d^2/(2R) = 8.00 d^2 [calc: 8.002]. That is the "8 inches times miles squared" rule. It is a correct small-angle formula, not an invention. At d = 1,000 miles the exact tangent drop is 129.7 miles against 126.3 from the rule (3 percent) [calc]; Rowbotham says the rule "requires to be modified after the first thousand miles" [read: Rowbotham 1881 p.10], which is right.

What "hidden" means. The rule gives the drop below a tangent at the observer. A real observer's eye is at height h_e above the water, so the eye's own horizon is at d_e = sqrt(2 R h_e) (1.23 sqrt(h_e in feet) miles; Dubay's item 69 uses exactly this [read]). A target at distance d is hidden below the sight line to the horizon by

hidden(d) = (d - d_e)^2 / (2R), and a target of height h_t is wholly hidden only if h_t is less than hidden(d).

Three distinct quantities are in circulation and the documents keep swapping them: 1. drop below the tangent at the observer, 8 in x d^2 (full distance); 2. hidden height beyond the eye's horizon, 8 in x (d - d_e)^2; 3. sagitta (mid-chord rise) of a chord of length L between two equal-height points, L^2/(8R), which equals 8 in x (L/2)^2.

Quantity 3 is what an equal-height three-mark test (Wallace, Oldham) measures. Quantity 2 is what a low-eye sighting of a distant mark measures. Quantity 1 applies only to a sight line that starts at the water's edge. Mixing them gives errors of a factor of 2 to 4 (section 1.4 and the Dubay audit).

Refraction. Near the ground, air density usually falls with height, so a nearly horizontal ray bends toward the denser air, downward. The standard way to model this is a circular ray of radius r. With k = R/r (Hirt et al. define k this way [read: Hirt deep read, eq. 1]), the curvature of the earth relative to the ray is (1 - k)/R, so every hidden height and every sagitta above is multiplied by (1 - k) and the eye's horizon moves out to sqrt(2 R h_e / (1 - k)). - k = 0.13 is Gauss's mean value; Hirt quote it as valid for the "intermediate atmosphere" and say it "is not suited for describing refraction effects in the lower atmosphere" [read: Hirt deep read]. - At k = 1 the ray follows the earth exactly: a globe then looks flat to the eye. For k greater than 1 the apparent curvature reverses sign (distant objects appear raised, the surface looks hollow). Hirt measured k between -4 and +16 at 1.8 m over grass on sunny days, +12 to +16 in the evening around sunset [read: Hirt deep read, section 3]. That was grass, 800 m lines, not water, so it shows what is possible near the ground, not what happened on the Bedford canal. This matters: it is a measured range showing that "k about 0.13" is not an upper bound. - No document in the project measured k at the time of any sighting. That absence cuts both ways (section 1.5).

1. Claim 1: water is level and the earth shows no curvature

1.1 Recomputation of the Bedford-type sightings [calc]

Rowbotham's arithmetic on the data he states is exact in the cases I checked (11 ft 8 in; 10 ft 2 in; 14 ft; 40 in for the Southsea-Ryde funnel test [read: 1865 deep section 3]). The 1865 slips (Whitby 102 for 112; Lower Farn 67 for 21.7) and the Grimsby refraction step are confirmed [calc]. His method is the correct no-refraction construction; what is missing is the refraction factor (1 - k).

So the Bedford sightings are not arithmetic mistakes. They are observations that, if correctly reported, need k of 0.4 to 0.9 on a globe. Whether k reached those values is a physical question for the physicist (Hirt's measurements show it can near the ground in evening inversions; Clifton himself wrote of "an aqueous shimmering vapour" on the canal [read: Clifton letter]). I note only that the values needed are inside the range measured elsewhere (-4 to +16), but far above the textbook 0.13 that Rowbotham himself quotes (1/7 of the curvature, "sometimes exceeds one-fifth", from the Encyclopaedia Britannica [read: 1881 deep read, refraction section]).

1.2 The one well-designed test: the equal-height ratio (Wallace 1870, Oldham 1901)

Three marks at equal height above the water (both bridges and a pole at 3 mi), viewed with a levelled telescope from the first. On a sphere the marks at 3 and 6 miles lie below the observer's horizontal by x^2/(2R), so they appear at angles about x/(2R) below the cross-hair:

B (3 mi): 78 arc-seconds below; C (6 mi): 156 arc-seconds below, i.e. two equal steps down [calc].

If the water were flat but sloping, the three marks would be collinear and C would hide behind B. If the water were flat and level, B and C would sit on the cross-hair. Carpenter's reading, "equal steps down means a straight line" [read: Wallace 1905 section 5], is a mathematical error: equal angular steps are the signature of a quadratic profile (drops 6 ft at 3 mi and 24 ft at 6 mi, ratio 1:4), not of collinearity.

Why this design is mathematically robust: uniform refraction multiplies both angles by (1 - k) and leaves the 1:2 ratio untouched; a level error shifts both marks equally and also leaves the ratio. Only non-uniform refraction (k varying along the path or with height) can change it. Wallace's sketches were drawn by one referee and signed by the other ("each side signed the other side's drawing") [read: Wallace 1905 section 5], but that is Wallace's account of events; the umpire's own record (The Field, 26 March 1870, Investigator's file) gives the curvature "to the extent of five feet, more or less" [web: https://people.wku.edu/charles.smith/wallace/zCarpenter1870Field1.pdf], which equals the predicted 5.0 to 5.7 ft after refraction. Walsh was the stakeholder and Wallace's nominee. Carpenter, Hampden's referee, signed the sketch "under protest" because it was not levelled and had no cross-hair, so the level-telescope result is the better-controlled half. Hampden declined to look [read: Wallace 1905 section 2].

Oldham's 1901 abstract gives the same design, result "about six feet" [read: Oldham deep read]. It is one paragraph: no heights, magnification, weather, repeats or photographs are reported, and refraction is not mentioned. Agreement at about 6 ft (5.2 ft after standard refraction) is within the "about".

1.3 Questions a-d for claim 1

a. Is the mathematics internally consistent? - The core rule is correct. The deductions are consistent where one convention is kept (Rowbotham 1865/1881 sightings, the funnel test, Wallace's design). - Hidden-assumption errors found: 1. Refraction is set to zero, or applied to the wrong quantity. Rowbotham applies a "one-twelfth of the altitude of the object" allowance to the residual hidden height [read: 1881 deep read, section 2]; surveyors' allowances (1/7 of the curvature, or about 1/12 to 1/13 of the contained arc [memory, medium]) apply to the curvature term. Even his own formula does not give his result: 1/12 of 11 ft 8 in leaves 10 ft 8 in, not "a few inches less than 10 feet" [calc]. 2. The "same medium" argument (no refraction because the observer and flag are both in air [read: 1881 deep read]) is wrong for a ray in a density gradient: refraction is bending along the path, not a boundary effect. His own confirming test compared barometers, thermometers and hygrometers at the two end stations [read], which does not measure the vertical temperature gradient along the path that determines k. Hirt's eq. 2 gives k from dT/dz directly: k = 503 (p/T^2)(0.0343 + dT/dz) [read: Hirt deep read]; I recompute 0.174 for p = 1013, T = 288.15, dT/dz = -0.006. 3. Convention mixing in Dubay (items 9, 13, 78-79; section 1.6). 4. The "datum" arguments (railway section 180 miles, Suez, Mont Fréjus, the Atlantic cable [read: 1881 Exp.13]) confuse a drafting convention with a geometric statement. A section drawn against a datum that follows a level surface is, by construction, drawn on the surface that curves with the earth. Where Rowbotham says the Mont Fréjus summit "would be 52 feet above" the Italian entrance on a globe, the figure is 8^2 x 8 in = 42.7 ft plus the observed 10 ft; he mis-attributes 52 ft to the formula [read: 1881 deep read; calc 42.7 ft]. Spirit levelling with equal back-sights and fore-sights measures height differences relative to a level surface and cancels curvature to first order; the surveyors' objection to Exp.4 [read: 1881 deep read] is mathematically correct, and Rowbotham's modified method is described too briefly to assess. 5. Exp.7 and 8 treat a horizon line or a coastline as if the globe predicted a sag toward the ends of a straight edge. A horizon on a sphere is a circle centred under the observer, at constant dip in every direction; the "66 ft at ten miles each side" figure treats a 20-mile chord seen from its middle [read: 1881 deep read, reviewer note; agreed]. - Arithmetic slips in the Zetetic texts, all checked [calc]: Whitby 102 (should be 112); Lower Farn 67 (21.7); Grimsby refraction 206 (216.7); Mont Fréjus 52 (42.7); Blount's Chesapeake-type tables and Carpenter's 5,400-ft style figures follow the formula. - Koreshan survey: see 1.7.

b. Is it consistent with established experimental results? Which? - Wallace 1870 (equal-height design, umpire's record 5 ft), Oldham 1901 (6 ft), and the geodetic record behind WGS 84 (R about 6,371 km; see claim 5) are in agreement with a curvature of 8 in per mi^2. The experiments the claim relies on (Rowbotham 1838/1870, Blount 1904) are single-observer sightings that, on a globe, need k of 0.4 to 0.9. - Skiba's June 2016 Chicago trip: eye about 3 ft, 42.6 to 46 miles. The hidden height is 1,093 ft at 42.6 mi and 1,284 ft at 46 mi (k = 0, eye 3 ft), against a Willis Tower of about 1,451 ft [calc; height tagged [memory, high]]. So on a globe about 88 percent of the tower is hidden at 46 mi and a large part of the skyline's lower portion is predicted to be missing. Skiba's own bench demonstrations are qualitative and one is a software composite [read: Skiba deep read, section 4]. The ABC57 meteorologist's explanation is a superior mirage over cold water [read: Skiba deep read, section 5]; that is that station's public explanation. Whether the 2015 Nowicki photograph (53 mi from 200 ft: 849 ft hidden at k = 0 [calc]) is a mirage cannot be settled from the documents. - Observations on the other side (hull-down ships recovered by zoom, etc.) are beyond my competence; the physicist covers them.

c. Has it been tested? By whom? Replicated? - Rowbotham 1838, 1844, 1849, 1856, 1862, 1870: the inventor's own, mostly unobserved ("unnamed boatman") [read: 1881 witness table]; the only independent named parties of a Bedford test in his book are the 1870 wager party, who reported curvature. - Wallace 1870: three-mark test, both sides present, umpire's decision for Wallace (stake holder and nominee); Hampden refused to look through either telescope [read: Wallace 1905]. Hampden v. Walsh (1876) turned on the Gaming Act and made no finding about the experiment [web: https://people.wku.edu/charles.smith/wallace/zAnon1877ReportsCases.pdf, via Investigator]. The TFES wiki statement that a court "determined that Wallace had, indeed, cheated" is not supported by any record read here; it is also inconsistent within the wiki's own sentence (Wallace "won the bet" yet "cheated") [read: Bedford deep read]. - Oldham 1900-01: an abstract of a repeat in the same set-up; photographs were the stated aim and are not reported. - Blount/Clifton 11 May 1904: one operator, one day, "misty and unsatisfactory weather" in the zetetic account [read: Blount pamphlet]. Clifton's letter says the lower edge of the sheet could be watched with a 2 ft telescope, the camera image was "very plainly visible", the weather produced "an aqueous shimmering vapour", and that "unless rays of light will travel in a curved path" the water presents a level surface [read]. - Koreshan survey 1897: see 1.7. - Skiba 2016-17: self-described inconclusive; the 15 Nov 2017 balloon team narrator: "a very slight curve ... hard to tell what the true horizon really is" [web: https://www.youtube.com/watch?v=B0-J7OhcQUk, auto-captions, via the Investigator]. - No test in the record measured the vertical temperature profile at the time of an observation. The refraction explanation and the flat-level explanation are therefore both unmeasured at Bedford.

d. If true, what in mainstream physics would have to change? Mathematically, a flat plane has zero Gaussian curvature; geodesy, great-circle navigation, GPS (3-D ranges to satellites) and survey triangle excess would all have to fail. The sphere is fixed by independent length data (section 1.8). To keep the Bedford sightings and still fit survey data one would need ray curvature of order 1/R everywhere over water, which is what the wiki's "Electromagnetic Acceleration" proposes (claim 3).

1.4 What the Zetetic arithmetic does and does not show (the instructive mistake)

The shared error is not computational. It is treating a prediction with a free parameter (k) as if the parameter were fixed at zero. A globe plus a refraction coefficient predicts (1 - k) times the geometric hidden height; the Bedford sightings require k in a stated range. The data cannot decide between "flat" and "globe with k near 0.5 to 0.9" without a measurement of k, and the flat model also needs light to travel straight, which no one claims over long water paths either (Clifton's own caveat).

An important symmetry for the reader: a flat plane with light curving upward at 1/R in every direction and a globe with straight light predict the same sightings along the surface. Observations along a surface cannot separate them; observations that do not depend on light paths can (section 1.8).

1.5 Hidden assumptions in the Clifton/Blount photograph [calc]

The pamphlet says the screen's bottom "should have been certainly over 20 feet below the line of vision" [read]. My figures: eye at 2 ft gives 12.1 ft hidden at the water line, at 1.5 ft 13.5 ft. The "over 20 feet" figure is the no-horizon drop (24 ft) or is not derivable [read; calc]. The TFES deep read says "8 in x 18.2 = about 15 ft"; 8 in x 18.2 is 12.1 ft [calc] (see Corrections). Consequences: - Schadewald (a critic, the only source for the size) gives the sheet as 15 ft square [web: Chapter_07.html, via the Investigator]. If so, 15 - 12.1 = 2.9 ft (19 percent) of the sheet top is above the sight line even at k = 0, so a globe predicts a visible upper part of the sheet; the claim that the whole sheet and its reflection were seen is the part that needs k near 0.9. The pamphlet itself does not give the size [read: Blount pamphlet; Blount-Smith 1904 preface also gives none]. - The reflection in the water at 6 miles requires a reflected ray from a near-grazing point of the water just in front of the sheet, which is the region geometrically hidden. Its presence in a photograph taken in "misty" conditions is a more demanding condition than visibility of the top. - Camera focal length "between 16 and 17 feet" [read: Clifton letter]; a 15 ft sheet at 6 mi would image at about 2.4 mm [calc]; the photograph's scale is consistent, and mirage-type distortion (looming) is plausible but not measured.

1.6 Dubay's quantitative items (class a, 68 items) [calc]

I rechecked the stated hidden amounts with the deep read's own method. - Items 6, 10, 11, 15, 60, 62-65 reproduce under the stated formula; item 10 (5,400 ft for 180 mi) and 11 are chord sagittas, item 9 (600 ft for 30 miles) and 13 (6,667 ft for 100 mi) are full-distance drops from a tangent. The post mixes these without saying so. - Items 78-79: hidden amounts of 7,719 ft and 9,220 ft for Mounts Foraker and McKinley from Anchorage are produced by subtracting only the observer's horizon. The peaks are 17,400 and 20,320 ft, so on the post's own numbers 9,700 ft and 11,100 ft of each would be visible; each mountain has its own horizon of about 160 to 175 miles [calc]. These items do not contradict a globe. - Items 67 and 72: 1,944 ft cannot be derived from the printed numbers (8 in x 54^2 vs the 47 miles given); item 72 (Grimsby 70 mi, 300 ft tower) I get 2,916 ft hidden at the water line before subtracting the 300 ft tower [calc], against the printed 2,600. - Items 81-86 (lighthouses) take Rowbotham's 1865 figures, which come from Findlay's 1862 tables of computed minimum ranges [read: 1865 deep read, section 7]. Findlay's ranges for Calf of Man, Cromer, Eddystone and Lower Farn lie within 3 percent of the standard curvature-plus-refraction range (23 vs 22.8, 23 vs 22.5, 13 vs 13.3, 12 vs 11.3 nmi) [read; calc]. A computed range read as an observation shows nothing about curvature. Three rows exceed it (Ryde 12 vs 8.8; Bidston 23 vs 20.8; Whitby 23 vs 21.3), explained neither way without the 1862 tables [read: 1865 deep read]. - Item 15 (airliner "descends 2,777 ft every minute", "31.5 miles higher") treats a quadratic drop as linear: 8 in x 500^2 = 31.6 mi in an hour is right, but in the first minute the drop is only 46 ft [calc]. - Item 98 uses 6.0 x 10^12 mi per light-year (accepted 5.88 x 10^12 [memory, high]); item 117 gives an Earth-to-Moon mass ratio of 87 that follows neither from its own diameters (cubed ratio 50.8) nor from the accepted 81 [read; calc]; item 49 prints "(0.00004%)" for a fraction 4.3 x 10^-5, i.e. 0.0043 percent [calc]; item 97 adds speeds in different directions as if collinear [read]; items 180-184 predict that a westbound airliner should take half the eastbound time (the post's model: ground rotating at 500 or 1,000 mph under a 500-mph airliner) while item 23 and 157 say the atmosphere co-rotates, which would give no asymmetry at all. The post uses both models [read: Dubay deep read, section 3f]. - Items 39-42 (southern circumference "25,000 miles"): see 1.8.

1.7 The Koreshan Rectilineator survey (1897), the concave variant [calc]

Data [read: Koresh deep read, section 5b]: the line, started 128 in above mean tide, was found (measured vs 8 in x n^2, in inches) 8.02 vs 8 at 1 mile, 30.62 vs 32 at 2, 54 vs 50 at 2.5, 75 vs 72 at 3, 128 vs 136.1 at 4 1/8.

1. Fit. The data fit a power law with exponent 2.09 (log fit over 1 to 4 1/8 mi) [calc]. A straight line fit fails by up to 15 in; a pure 8n^2 has residuals of -0.03 to +8 in. So the shape is quadratic, as the authors claim. That is a real result: it is the point of their reply to the "settling" and "inclined start" objections [read: p.164-167]. A uniform tilt or a linear settling cannot give it.
2. But a quadratic is exactly what a constant bending per step looks like. Over 1,815 twelve-foot sections the deflection y = (1/2) delta L N^2 needs a systematic turn of delta = 0.11 arc-seconds per section [calc]. The turn of the local vertical over 12 ft of earth surface is 12 ft/R = 0.118 arc-seconds [calc]. The two are the same size. So the observation is equally what an instrument would show if each step carried even a tiny residual memory of the local vertical. The datum (water) falls by 8n^2 with a globe, so against a truly straight line the water-parallel datum on a globe would predict a rise of 8n^2; the observed fall of 8n^2 is a deviation from straight of 16n^2 in globe terms (2/R), which is 0.24 arc-second per step [calc]. The authors know the factor: their own contrast table gives 264 in (about 22 ft) between theories at 4 1/8 mi [read: p.149].
3. Instrument claim. "Eliminates atmospheric refraction" [read: p.120] is true for the stepped part, 0 to 2.5 miles; the last 1.6 miles (13,200 to 21,780 ft) was finished with a telescope, a cross-hair and a boat hull, refraction treated as "so small a factor" without a number [read: p.140-141]. The 128 in at staff 25 comes from that optical stretch. Without it the stepped data end at 54 in (2.5 mi).
4. Quantities the authors do not give: the tolerance of the 128-in transfer at each of 25 staffs through a 3.75-ft tide range; the cross-arm tolerance after reversal; any uncertainty. The reported agreement is 0.02 in at 1 mile and 0.025 in at 1/8 mile, which is 0.5 mm precision through a chain that includes a tide gauge and spirit levels graduated to 0.075 in [read]. I do not say the numbers were tampered with; I say the stated precision is not accompanied by an error analysis, and agreement this close to the prediction (the 0.02 in) calls for one.
5. Closure test. The 228 ft out-and-back test reports "without a deviation of .0001 of an inch" [read: p.124-125]. A systematic downward bend of Earth-curvature size would give an out-and-back miss of order 0.01 to 0.03 in over 19 sections [calc, order of magnitude: (1/2)N^2 delta L with delta = 0.12 to 0.24 arc-second], well above 0.0001 in. If the test was as described, it constrains a down-bending bias strongly. It is the best evidence in the survey for the authors. Against it: it was a short, ground-level, undated trial, described in prose in the colony's own book, no table; the system of reversals cancels a bias fixed to the instrument frame but not a gravity-referenced one such as sag; and the later 0.375 to 0.75 mile return over the line agrees only to about 0.25 in [read: deep read 5d]. Return lengths are given three ways (1/2 mile, 3/4 mile, 0.375 mile in the table), unreconciled [read].
6. Single-source and stake: witnesses are mostly colony members; the only data are the colony's tables. The original field notebooks may be in the State Archives of Florida, N2009-3 [web: https://www.floridamemory.com/items/show/258146, via the Investigator]. No independent replication is in the record.
7. Conventionalism. "Convex or concave" is not a question the survey can settle by itself: any map that sends a sphere's interior to its exterior (inversion in a sphere) changes which way surfaces curve and turns straight rays into circles, so a concave-earth theory can be made to fit any data a globe fits, at the price of a position-dependent light speed and ruler length [memory, high; standard geometry]. The Koreshan content is the claim that the rectilineator is a straight line when the rest of geometry is not. That is a statement about how "straight" is realised, and the test that decides it is a control: run the same instrument along a baseline of independently known shape (a long straight surveyed tunnel). The record has no such control.

1.8 Whether a flat plane can be mapped consistently: a pure-mathematics test [calc]

The TFES model is a north-centred azimuthal equidistant map (the standard flat-earth map). Compare its distances with the globe's, using R = 6,371 km:

Northern-hemisphere pairs fit within 7 percent; every southern pair is 2.1 to 2.6 times too long. The circumference at 45 deg S would be 58,600 miles on the flat map against 17,600 on a globe [calc]. Johannesburg to Perth is a scheduled flight of about 11 hours [memory, high]; at the flat-map distance it would require about 1,670 km/h. Dubay's items 39-42 (Sydney-Nelson "25,182 miles" circumference, etc.) claim southern circumferences of about 25,000 miles, which is neither the globe's 20,000 nor the flat map's 54,000 to 59,000: they do not follow from either model. His item 39 arithmetic: 1,550 mi x 360 / 22.04 deg = 25,321 miles, not the printed 25,182 [calc]. Carpenter's proof 16 and others say the 45 deg S circumference is "about twice" that at 45 deg N [read: Carpenter deep read]; on the north-centred flat map the ratio is 3 [calc]. So the southern-hemisphere data (distances, flight durations) contradict the flat map as a pure mathematical object, whatever one thinks of refraction. This is the cleanest "refuted by calculation" item in claim 1, and it applies to the model that gives the flat earth a definite metric. It does not apply to the Koreshan concave variant or to a model that gives the flat earth an unspecified metric (which then predicts nothing).

1.9 Verdict, claim 1 (split)

The TFES statement that the Bedford result was "dismissed without testing" does not hold: the 1870 wager was a test, accepted by both sides in advance, with agreed predictions [read: Wallace 1905 section 4]. What fails to be "tested" is refraction itself.

2. Claim 2: the earth is stationary (no rotation, no orbit)

2.1 Michelson-Gale (1925) [calc]

The prediction Delta = 4 A omega sin(phi)/(lambda V) reproduces 0.236 fringe from the paper's inputs (A = 207,836 m^2, phi = 41 deg 46', lambda = 5700 A, omega = 7.2921e-5 rad/s) [calc: 0.2363; the deep read agrees]. Time difference 4.5e-16 s [calc]. Observed 0.230 +/- 0.005 [read: p.5]. I recompute from Table I: 13 set means, mean 0.2295 (unweighted) or 0.2290 (weighted by sets of 20 to 26), standard deviation 0.0204, standard error 0.0057 [calc]. The paper's +/- 0.005 is of that size. The difference observed minus predicted, 0.006, is about 1.1 combined uncertainties. A null prediction (stationary earth) is zero, which is about 40 standard errors from the mean of set means [calc: t = 40.5].

Count: Table I sets of 20 except set 6 (26) and set 13 (23): 11 x 20 + 26 + 23 = 269 total [calc, confirms]. The TFES deep read's "263 vs 269 inconsistency" is an artefact of misreading set 6 as 20 (see Corrections). The wiki's quotation of "-0.04 to +0.55 fringe, mean +0.26" comes from Correa (not seen), not from the paper; +0.26 disagrees with the paper's table (0.230), and the range -0.04 to +0.55 is for single readings (Fig. 3, deep read: "readings cluster between about 0.15 and 0.30 with a thin tail"). The wiki's sentence "the rotation of the earth would start and stop" confuses scatter in a noisy optical reading with a physical on/off effect; every set mean in Table I is positive and within 0.04 of 0.23 [read: TFES-MGP deep read].

A limit on what it proves, stated mathematically: a Sagnac interferometer measures rotation of the loop relative to local inertial frames. Earth rotating in a static universe, and a static earth with the whole universe rotating about it, give the same Sagnac number; the experiment cannot tell which. In general relativity the two are related by a coordinate transformation [memory, high]. The paper itself draws no conclusion between stationary aether and relativity; both give the same value (the abstract "assumption of a stationary ether as well as in accordance with relativity") [read: p.4]. For the claim "the earth does not rotate" in the sense "the earth is not rotating relative to the inertial frame" the experiment is a direct refutation; for "the earth is at rest in some preferred frame and the heavens rotate daily" it is silent. The wiki does not make the second claim for this page [read: TFES-MGP deep read].

2.2 Ring laser gyroscopes

The TFES page quotes accurately but omits what follows, so arguments from the quoted caption fail. The Brillouin-laser caption "N-S change close to twice the Earth's rotation rate (2 x 15 deg/h), E-W near zero" describes a difference between two orientations: flipping the sensitive axis reverses the sign of the projected Earth rate, so the change is 15 - (-15) = 30 deg/h. The wiki reads "twice the rate of the Earth's rotation is detected" [read: RLG deep read, section 6]. That is a sign-convention error in the wiki, not in the source. The E-W near-zero result is what a rotation-sensitive horizontal axis should show and the wiki omits it. The G-ring figure (pico-radians per second) is a residual after removing "all known geophysical signals" (the caption) [read]; the Earth-rate value is 7.29e-5 rad/s, about 10^8 times larger, so the plot shows noise at the 10^-7 level, not a rate that "switches sign" [calc; deep read]. The wiki's own "alternative" (an unspecified diurnal mechanism producing the Sagnac split) is not given a form that could produce the cos(angle between axis and rotation vector) dependence the devices show [read: deep read, section 8]. The page contains no Knodel claim and no null RLG result (confirms claims.md Corrections).

2.3 Foucault pendulum

Foucault's own law, stated without proof [read: 1851 deep read], is that the plane turns by the earth's angular motion times the sine of the latitude. Rate = 15.04 deg/h x sin(phi): Paris 11.3 deg/h (period 31.8 h), 41.8 deg N 10.0 deg/h, pole 15.04 deg/h, equator 0, Sydney-latitude reversed in sense with period 42.9 h at 33.9 deg S [calc]. The TFES page gives no rate, no latitude analysis [read: Foucault deep read, "Bottom line"]; its evidence is that early 1851 trials gave ellipses and wrong directions (Conlin, Newark trials), Longden's 1919 wire-twist reversal, and a South Pole station anecdote. These are real error sources (Longden's wire anisotropy; the early difficulty of release without lateral impulse, which Foucault addressed by burning a thread [read: 1851 deep read]). A source of bias in launching does not give a consistent sin(latitude) dependence across latitudes and hemispheres. The deep read notes the South Pole anecdote result ("twice what is expected") is incomplete in the excerpt. A doubled rate at the pole is not explained by an obvious launch error, and the page does not say what was finally obtained [read]; I treat it as an unresolved anecdote of unknown reliability.

2.4 Coriolis effect: magnitudes [calc]

The wiki's argument is "artillery needs adjustment, therefore the Coriolis correction is unproven" [read: Coriolis deep read]. The deciding quantity, which the page does not give, is the size of the correction against dispersion. Lateral Coriolis deflection of a projectile is approximately Omega sin(phi) x range x time of flight: 18 m (10 km, 35 s), 62 m (20 km, 60 s), 139 m (30 km, 90 s) at 45 deg [calc, approximate]. First-round accuracy "in tens of meters" is the wiki's own citation [read], so at 20 to 30 km the correction is comparable to or larger than the dispersion, and at 1 km (flight 1.5 s) it is about 8 cm [calc], which explains why sniper manuals do not mention it. Falling bodies: eastward deflection (1/3) omega g t^3 cos(lat): Reich 520 ft, 27 mm (the page quotes "over an inch"), Cornwall 1,320 ft, 114 mm (4.5 in; the page quotes "less than 5 inches"), Benzenberg 235 ft, 8 mm [calc]. The theory predictions in the sources I can check are therefore numerically right; Rowbotham's observed southward deflections of 10 to 20 in at 1,300 ft exceed the predicted eastward 4.5 in [read: Coriolis deep read], an anomaly that I cannot resolve from the page (air currents are the quoted explanation).

Drains: the effect is real but needs a settled tank and long waiting; the page's own quoted sources (Shapiro 1962, Trefethen 1965 in Nature 207: 1084-1085, "The bath-tub vortex in the southern hemisphere") report the effect in carefully settled tanks, and Trefethen's group found clockwise draining in Sydney by a secondary account [web: https://link.springer.com/article/10.1038/2071084a0 for the paper's existence and citation; direction from secondary citations at https://www.desy.de/user/projects/Physics/General/bathtub.html only; I did not read the paper]. The page's Eotvos paragraph (equivalence-principle torsion balance) is the wrong test for a Coriolis force (a common force cancels in a difference measurement) [read: deep read; agreed, [memory] high].

2.5 Airy's water-filled telescope [calc]

The TFES argument: a moving earth predicts a bigger tilt with water in the tube; none seen; therefore the earth is fixed. Aberration alone is v/c = 29.78 km/s / c = 20.49 arc-seconds [calc]. The naive stationary-aether prediction for a water-filled tube is n v/c, a shift of 6.8 arc-seconds [calc: (1.333 - 1) x 20.49]. Fresnel's partial-drag coefficient 1 - 1/n^2 = 0.437 for water cancels that difference, so a moving earth predicts the same angle with or without water [calc; [web] for the coefficient and the result https://arxiv.org/pdf/1710.05786 and https://arxiv.org/pdf/1302.6965, search 10-10: sources give the same finding, "Airy's null result ... Fresnel's coefficient 1 - 1/n^2", without Airy's angles]. Relativistic velocity addition gives the same coefficient to first order [web, same search]. The wiki's premise (that a moving earth requires a different tilt in water) is exactly what Fresnel's theory was built to avoid, so the null result is the expected one on both the aether and relativistic accounts [read: Airy deep read, section 7 raises the same point]. As an argument for a fixed earth it fails: the premise is wrong, not the test. The converse is also worth stating: to explain aberration with a fixed earth every star must trace a 20.5-arc-second ellipse with period one year whose shape depends on ecliptic latitude, for all stars together, with no mechanism given. That is a numerical coincidence the stationary model has to impose.

2.6 Michelson-Morley, Sagnac, orbital speeds

Dubay item 18 treats Michelson-Morley and Sagnac together as nulls proving a stationary earth [read]. The Sagnac effect is a non-null effect proportional to rotation rate; the page's own sidebar says "light's velocity is indeed affected by detector motion" [read: MGP deep read, sidebar]. The "unthinkable speeds" arguments (19 miles per second orbital, 1,000 mph rotation) are speed in a chosen frame and not a contradiction. Hampden's pamphlet gives rotation as "over 1100 miles per hour" (p.6) and also "700 miles per second" (p.17) [read: Hampden deep read], an internal inconsistency of a factor of 2,300 in the same text. Blount-Smith 1904 shows the same kind of slips: it quotes Ball's 250,000,000,000 miles per year for the sun's motion, says that is "nearly twice" the earth's orbital distance (it is about 2,700 times 93 million miles) [read: Blount-Smith deep read, section 8]; it quotes "180 miles per minute" for the moon (10,800 mph) beside its own correct 2,281 mph [read]; and adds the earth's orbital speed to the moon's as if the moon would be stationary against it, which is a frame error [read: deep read].

2.7 Questions a-d for claim 2

a. Consistent where it makes arithmetic claims (Dubay's 670,568,000 mph sum is arithmetic [calc]); inconsistent in several: sums of non-collinear speeds, the Airy premise, the RLG "twice the rate" reading, the Eotvos argument, Hampden's 1100 mph vs 700 miles per second. b. Inconsistent with Michelson-Gale (0.230 +/- 0.005 observed vs 0.236 +/- 0.002; null is 40 standard errors away), the Foucault sin(latitude) law (Foucault 1851; modern instruments), RLG and fibre gyros (Earth rate 15.04 deg/h and its orientation dependence), the Fresnel-drag reading of Airy (above), computed Coriolis magnitudes that match Reich's drop (27 mm vs "over an inch"). c. Tested: Michelson-Gale 1925 (one site, 269 readings, Michelson and Gale; later ring lasers are beyond this file); Airy 1871 (as reconstructed); Foucault 1851 and later; Benzenberg 1802, Reich 1831 (the page says "Reich's average experimental deflection was almost exactly [the theoretical]" [read]). Not replicated by the TFES: no experiment of their own appears on the pages read; their case is a reading of other people's. d. A fixed earth would require every inertial-frame experiment to be reinterpreted as a rotating cosmos, or the Sagnac phase to arise from a rotating background field with correct latitude dependence; the wiki gives no such model [read: RLG deep read section 8].

Verdict, claim 2 (split): - Earth not rotating relative to inertial frames: Tested, not supported (Michelson-Gale, Foucault, gyros). - Earth does not orbit (Airy/MM as the proof): Refuted by calculation as an inference (the Fresnel coefficient cancels the water effect); the underlying stationary-earth claim remains Never properly tested in the sense of any competing measurement of stellar parallax or aberration done on its terms; physicist to weigh. - "Earth at rest, heavens rotating daily" (the geocentric reading of Sagnac): Not testable as stated by Sagnac-type measurements alone; kinematically equivalent.

3. Claim 3: sun and moon small and near; sunset as perspective or curved light

3.1 Triangulation: the flat model contradicts itself [calc]

Rowbotham and Blount-Smith: sun and moon about 32 to 30 miles across, 3,000 miles up (Rowbotham, as the deep read recalls [memory, medium]), "six or seven thousand miles" for the highest body (Blount-Smith [read: p.15]). Take the sun above the equator at equinox, height h over a plane. Noon sun elevations on a globe are 90, 60 and 30 deg at latitudes 0, 30 and 60 deg. On a plane, tan(elevation) = h / ground distance, so the distances are 0.577h and 1.732h from the foot, spacing 0.577h and 1.155h, ratio 2.00 [calc]. The surface distance between 0 and 30 deg and between 30 and 60 deg is the same on any map (equal latitude steps are 3,330 km on the ground [memory, high]). A single sun height h cannot give both elevations at one moment at three equal spacings; the measured elevations demand a ratio of 1. Eratosthenes' pair (90 deg and 82.8 deg over 800 km) gives h = 6,300 km on a plane; the same h then predicts elevation 51.6 deg 5,000 km away where the observed is about 45 deg [calc; elevations are textbook geometry, [memory] high]. The test is a pure-geometry consistency check and I do not rely on any single anecdotal observation.

3.2 Angular size [calc]

32 miles at 3,000 miles subtends 0.61 deg (the actual 0.53 needs 3,440 miles, so the sizes are only roughly compatible [calc]). On a plane the sun's distance is h/sin(elevation), so its angular size should shrink as sin(elevation): 0.61 deg overhead, 0.43 at 45 deg, 0.21 at 20 deg, 0.11 at 10 deg, 0.05 at 5 deg [calc]. The observed size is constant to a few percent between noon and sunset. Blount-Smith's reply, "moisture ... tend[s] to magnify the sun's disc, and so prevent the perspective foreshortening" [read: p.20], would need a factor of 5 at 10 deg and 10 at 5 deg elevation, with no calculation. Rowbotham's own perspective law makes the same error. This is the quantitative content of "sunset by perspective": it does not give a disc of constant angular size.

3.3 Electromagnetic Acceleration formula [calc]

The only equation in the wiki: y = (3/4) (beta x^4 / c^2)^(1/3), "a limit of a more complex (and not yet final) expression ... only valid when y is much greater than x" [read: EA deep read]. Units: with beta an acceleration, beta x^4 / c^2 has units of m^3, so y is a length; consistent [calc]. It gives y proportional to x^(4/3), not the parabola y = beta x^2 / (2 c^2) of a constant transverse acceleration at small angle; the derivation is not on the page, so I cannot verify it. For a ray of speed c under a constant acceleration beta the exact path is a circle of radius c^2/beta, not a power law, so the page's power law must come from a position-dependent beta that is not given. Using the formula as written (the deep read gives no value of beta, "attempts will be made to measure this experimentally"): - With beta = g = 9.8 m/s^2 (the scale the UA page uses), a ray rises only y = 7.7 km over x = 10,000 km [calc]. - To place the sun at 4,828 km (3,000 mi) height at x = 10,000 km, beta would have to be 2.4 x 10^9 m/s^2, about 2.4 x 10^8 g [calc]. At x = 5,000 km, 3.8 x 10^10 m/s^2. - A curvature equal to the globe's, 1/R, corresponds to beta = c^2/R = 1.4 x 10^10 m/s^2 (1.4 x 10^9 g) [calc]. So the sunset model requires a "Dark Energy" ray acceleration 10^8 to 10^9 times the gravitational acceleration it is meant to be paired with. The formula is stated not valid for short range, so I do not extrapolate it to a laboratory beam; the physicist should ask why a transverse acceleration of this size is not seen in any horizontal ray. The "Bishop constant" is undetermined; no prediction (sunrise time, horizon dip number) is computed on the page.

3.4 Other arithmetic in Blount-Smith 1904 [read; calc]

Sun speed 1,040 mph at the equator is consistent with 25,000 miles per 24 h [calc]. The moon's 2,281 mph correct; the "nearly 25 hours" lag is correct (24 h 50 min mean). Moon declination limits 18.5 to 19.5 deg in 1904 are plausible for that part of the 18.6-year cycle [read: deep read; the physicist may check]. Equation of a figure-8 for the sun's path: no equation or tables are given, only diagrams, so declination vs date (a sinusoid with 23.4-deg amplitude) is stated qualitatively; the claim that the declination changes by about 24' per day near the equinox is right (23.4' [memory, high], 24' in the text). No sunrise-time or day-length prediction is computed, so I cannot test the model.

3.5 Questions a-d

a. The perspective model gives internally inconsistent triangulation (3.1) and a constant-size sun is not obtained (3.2). The EA formula is dimensionally consistent but underived. b. Contradicts equal-latitude elevation data and constant angular size of the sun. Consistent with Hewitson's 1866 report only as a description of what Rowbotham said [web: via Investigator]. c. The "moon tilt" and the Auckland star-trail photographs are the only observational items in the files (a moon-limb orientation at one time; star trails showing circumpolar motion, which are equally predicted by a rotating earth) [read: Blount-Smith deep read]. No test of the sun-height model by independent triangulation is on record in the files. d. A 30-mile sun at a few thousand miles would require abandoning the measured distances of the planets and light-time delays (physicist); I note only that the model's own angular predictions fail.

Verdict, claim 3: perspective-law form (Rowbotham, Blount-Smith): Refuted by calculation (3.1 and 3.2, from the model's own geometry). Curved-light form (EA): Not testable as stated until beta is fixed; with the formula as written and the sun at 3,000 miles it requires an acceleration of about 10^8 g, which the physicist should weigh.

4. Claim 4: gravity is not a pull from the earth's mass (Universal Acceleration, ether currents)

4.1 Universal Acceleration (UA) [calc]

UA: earth and universe accelerate "upward" at a constant 9.8 m/s^2. Uniform proper acceleration g for a time t gives velocity v = g t / sqrt(1 + (g t/c)^2); relativistic corrections are important after about 0.1 light-year of time: g t/c = 1.03 after one year, so v = 0.72c in the original frame (1 - v/c = 0.24); after 6,000 years, 1 - v/c = 1.3 x 10^-8 and gamma = 6,194; at the cosmic age 13.8 billion years gamma = 1.4 x 10^10 [calc]. The wiki's captions fit this formula and it reaches c only asymptotically [read: UA deep read, section 3]; I infer the formula from the captions as the deep read does. This is not an error. The unanswered questions are which frame the speed is relative to, and what happens to time dilation of signals from "stars" that all accelerate together (the page gives no answer [read: absence]). If earth and every star accelerate uniformly together in a flat spacetime, distant stars stay at fixed separation from the earth, and distances to galaxies would not change with time; the observed redshift-distance relation needs a separate explanation not on the page.

Equivalence-principle check [calc]. The principle is local. A uniformly accelerating (Rindler) frame has acceleration decreasing with height as g(h) = g0 / (1 + g0 h / c^2), a fractional gradient of g/c^2 = 1.1 x 10^-16 per metre. Observed free-air gravity falls 3.086 x 10^-6 s^-2, i.e. 3.1 x 10^-7 per metre [calc; value [memory] high]: a factor of 2.9 x 10^9 larger. Observed g also varies with latitude (about 0.5 percent; the centrifugal term at the equator is 0.0339 m/s^2, 0.35 percent of g) [calc; [memory] high]. UA predicts a uniform g to 10^-16 per metre; gradiometers, torsion balances and satellite geodesy measure 3 x 10^-7 per metre routinely. The deep read confirms the UA page does not mention these [read: UA deep read, section 5]: Cavendish, Eotvos, tides, altitude and latitude variation are either absent or delegated to a separate "Celestial Gravitation" model that restores a small gravitational pull; Gauss's law ("Davis plane") appears as both an explanation and an alternative to UA [read]. For the infinite plane with finite attraction "consistent with Gauss's Law", Gauss's law gives a uniform field g = 2 pi G sigma on both sides of a uniform plane and no altitude gradient [calc/standard]; it fits no 3 x 10^-7 per metre gradient either.

4.2 Ether currents (Blount-Smith 1904)

No equation: a current in a wire gives circular field lines (Biot-Savart), used as an analogy for the "cogwheel" vortices; no number is derived. Newton's Bentley letter is quoted for dislike of action at a distance [read: p.25]. The claim that a Uranus perturbation "might be accounted for by electric currents" has no calculation [read: p.10]. The two-vortices cogwheel figure has no stated flow law. Not testable as stated.

4.3 Questions a-d

a. UA: the relativistic part is internally consistent; the claim that UA reproduces local gravity everywhere fails by the Rindler gradient (above) and by omitted effects. The page's own "Equivalence Principle ... not possible to locally discern" cuts against its use as a global explanation [read: deep read, point (e)]. b. Inconsistent with the measured vertical gradient (3.1e-7 vs 1.1e-16 per metre), latitude variation, Cavendish attraction between lead spheres (not explained on the page), tides. c. No test by the proponents. d. Newtonian gravitation, tides, the Cavendish and Eotvos results, and the mass-dependence of orbits would all need replacement by something giving 1/r^2 forces; the physicist should say what.

Verdict, claim 4: UA: Tested, not supported (the uniform-g prediction against measured gradient and latitude variation) with an Internally inconsistent component in the page's choice of the equivalence principle as global proof. Ether currents (1904): Not testable as stated. Newton's own "action at a distance" remark is Historically superseded but instructive.

5. Claim 5: agencies conceal the earth's shape, and their documents say "flat, nonrotating"

5.1 What the cited reports contain [read: deep reads; calc]

• NASA TM X-2514 (Preisser 1972): "flat, nonrotating earth" appears four times and means a local tangent-plane frame for a 9-second parachute-test analysis; "this assumption limits the analysis to short-range, short-duration flights" [read: p.2, 7]. The uncertainty from ignoring rotation is stated as 0.25 deg/min [read: p.19], matching the sidereal rate 15.04 deg/h = 0.2507 deg/min [calc], which is about 0.04 deg over 9 s against a 1 to 2 deg gyro error. The same report says earth rotation will "cause readings on the gyro" [read: p.10], an explicit statement that the earth rotates. Its radar transformation uses z = R sin(E) with no curvature term, appropriate over a few kilometres: curvature over 60 km range is d^2/(2R) = 0.28 km [calc].
• NASA CR-4568 (Feeley and Speyer 1993): the real model is a spherical, nonrotating earth with inverse-square gravity, g = g_s r_e^2/(r_e + h)^2, r_e = 2.09256725 x 10^7 ft; the "flat" model is the epsilon = 0 zeroth-order sub-problem for an analytic solution, and the authors report the zeroth-order trajectory "deviated significantly from the optimal trajectory" [read: p.70]. Even in the flat equations r_e appears in the range-angle rate [read: p.44].
• Piester et al. 2007 (PTTI): plane-slab troposphere = path 11 km / sin(elevation): 315 km at 2 deg, 75 km at 8.4 deg, 34 km at 19 deg [calc, agrees with the deep read]; "too large" for low elevation; the paper uses Niell's function, and it also treats the Sagnac correction for Earth rotation (100 ns scale) [read: p.8]. A named approximation rejected because it overestimates.
• WGS 84 (TFES page): the page quotes no WGS 84 or IS-GPS-200 text [read: deep read section 2]. Its own quotations call the earth "approximately spherical", "ellipsoid, not a sphere", "curved surface" [read]. The anecdote "an 11.5-mile arc is only five hundredths of a foot longer than its chord" is a recalled figure by a commenter; the deep read recomputes arc minus chord L^3/(24 R^2) = 0.021 ft [calc/deep read], the spherical-excess figure (75 sq mi gives 0.99 arc-second) checks [calc]. Both numbers are tiny, which is why local plane coordinates work and says nothing against curvature. WGS 84 constants (a = 6,378,137 m, 1/f = 298.257223563, omega = 7.292115e-5 rad/s) [memory, high; deep read]: an ellipsoid whose flattening 1/298 corresponds to equatorial minus polar radius of 21 km, derived from satellite orbits, gravity and rotation together.

5.2 Dubay's flight-time argument (items 25-26, 179-184)

Under the post's own model (airliner 500 mph relative to the ground, ground and air co-moving with the earth's surface) the rotation does not alter ground speed, as item 23 itself says; the "half the time" prediction is unsupported by either model. The actual asymmetry (LA-NYC 5.5 h vs 6 h, about 9 percent) matches a jet-stream headwind of about 40 to 50 mph on a 500-mph airliner [calc, order of magnitude]. The deep read lists a second internal contradiction: Tokyo-LA is eastbound [read].

5.3 Questions a-d

a. The claim's quoted documents are consistent with what they say; the claim misreads them. b. The documents themselves carry the opposite: spherical gravity, rotation terms, Sagnac correction. c. "Tested" here means the quoted passages were read in context by the deep readers (five reports) and checked by fact-checks (Lead Stories, AAP) [read]; what remains unread are the full texts of the other Reddit-list reports (digest and phrase search only). That means claim 5's list of NASA "flat earth" statements is partly unchecked: Step 1 flags this [read: claims.md coverage note]; I record that the five read are all misreadings, and a mathematician cannot exclude a sixth. d. The claim of concealment itself is outside mathematics: no document, dataset or witness for the faked-space claims is cited in Dubay's class (d) items [read: Dubay deep read]. The Investigator's finds do not bear on it.

Verdict, claim 5 (split): "These documents state a flat, nonrotating earth as a description": Refuted by calculation / Tested, not supported (the five read documents say the opposite and use the flat model as a stated local approximation with quantified error). "Agencies conceal the shape and space travel is faked": Not testable as stated from the files (no records offered; the physicist and contrarian may weigh the evidence).

6. Claim 6: scripture describes a flat, stationary, enclosed earth

A theological claim; mathematics offers no verdict on interpretation. I checked the numerical statements around it. Hampden counts "world" over 200 times and "round" never applied to the earth [read: Hampden deep read]; a verse count is arithmetic that I did not check. Hampden's Flood account has water covering a plane. Blount-Smith's prime meridian at Eden near the Euphrates "about 45 deg east of Greenwich" [read: p.17] is a stated choice. No physical prediction follows. Not testable as stated (as a physical claim); Historically superseded but instructive for the argument from textual absence.

7. Weighing the Investigator's findings (kept whatever my verdict)

Corrections to Step 1 (earlier documents)

1. Deep read of the TFES "Bedford Level Experiment" page: the line "the wiki's 'eleven feet' is not derived on the page ... 8 in x 25 = about 16.7 ft, not 11 ft" is wrong. 16.7 ft is the drop at the water line; the wiki's 11 ft (Rowbotham's 11 ft 8 in) is the hidden height of the top of a 5-ft flag, 16.7 - 5 = 11.7 ft [calc; read: Rowbotham 1881 Exp.1, which states "11 feet 8 inches below the horizon" for the flag top].
2. Same deep read, Clifton check: "8 in x 18.2 = about 15 ft" should be 12.1 ft (18.2 miles^2 x 8 in = 145.6 in) for a 2-ft camera. The "more than 20 feet" in the wiki matches only the no-horizon drop; the geometry with the camera's own horizon gives 12.1 ft at 2 ft eye and 13.5 ft at 1.5 ft [calc]. The conclusion (that refraction is not addressed on the page) stands.
3. Deep read of the TFES "Michelson-Gale-Pearson" page: "Table I's set sizes (12 x 20 + 23) sum to 263 but the table gives 269" is a misreading of the table. Set 6 is 26 observations (as the other deep read from the page image says), and 11 x 20 + 26 + 23 = 269 [calc]. There is no inconsistency in the paper. claims.md already says this; the deep read file contradicts it and should be corrected in the record.
4. Deep read of Rowbotham 1881, section 3 item 3: "[calc] Hekkengen at 16 mi ... would be about 32 ft hidden, not 380 ft, so the 1865 entry carried an arithmetic error." Wrong: the printed 1865 text (p.171) reads "It ought to be sunk beneath the horizon 30 feet !"; "380" is an OCR misreading, as the 1865 deep read says. I checked the page image [read: Rowbotham 1865 PDF p.176]. The two 1865 slips that are real and printed are Whitby ("102 feet", should be 112) and Lower Farn ("67 feet", should be 21.7): both present in the page image.
5. claims.md claim 1: "...the globe rule of 8 inches times miles squared says they should be hidden" is imprecise: for a low observer the rule must be applied beyond the eye's horizon (quantity 2 in my section 0), and for equal-height marks it gives a sagitta (quantity 3). Suggested wording: "the globe's curvature (about 8 inches times the square of the distance beyond the observer's horizon) says they should be hidden, unless refraction (coefficient k) lifts them".
6. claims.md claim 1 Counter-records: "Hirt et al. 2010 on variable refraction" is under-described. The measured range (k -4 to +16 at 1.8 m over grass; +12 to +16 around sunset) is the key figure for the Bedford sightings, and it is also the one that shows the standard 0.13 is not an upper limit. The paper's measurements were over land, not water, on 800-m lines [read: Hirt deep read]. Keep both parts.
7. Hirt deep read, sign convention caution: the deep read says the Fig. 1 caption is inconsistent. With k = R/r as defined (eq. 1), apparent surface curvature is (1 - k)/R, so k greater than 1 reverses the sign; the paper's text on pp.4-5 is the reliable one and the caption wording is the odd one. I agree with the deep read's resolution.
8. Dubay deep read (section 3b), "Items 81-93 ... not reproduced by either convention": items 81-86 are Rowbotham's 1865/1881 figures, reproducible only by 8 in x (d - 4 mi)^2 less the light height for a 10-ft eye with k = 0 (the 1865 deep read's table, "Recomputed by his rule"), consistent with the Rowbotham deep reads. The Dubay deep read's note that they "are not reproduced" holds for the exact horizon method.
9. Koreshan deep read, section 4e check: the deep read compares a concave tangent slope at 1 mile (0.030 in per 12 ft) with the book's 0.042 in and calls the order of magnitude agreed. The ratio 1.4 is within the instrument's stated reading precision of 0.075 in graduations only if the reading has an uncertainty of at least 0.01 in, which the authors do not give. I flag that these instrument readings (bubble 0.0375 in, 0.077 in, 0.089 in at 1, 2, 2.5 mi) are 40 to 90 percent of one scale division and are not independently informative.
10. Aberration/Airy: the TFES page's premise that a moving earth requires a bigger tilt in water is the premise Fresnel's coefficient removes; the Airy deep read already asks the reviewers to test it. I have, and the premise fails (section 2.5). The deep read's [memory] note on Boscovich's 1760s proposal is unchecked (low importance).

Open points and what would settle them (for the arbiter)

• Claim 1: a measurement of the vertical temperature gradient (hence k) simultaneously with a long-water sighting, repeated at several times of day, and the equal-height ratio test over water at 3 and 6 miles. A control run of the Rectilineator on an independently known baseline. The Florida archive notebooks.
• Claim 2: nothing beyond published ring-laser and gyro data; the geocentric-rotating-heavens reading is kinematically equivalent and not decidable by Sagnac.
• Claim 3: the triangulation consistency of section 3.1 is a calculation any reader can repeat.
• Claim 4: a gravimeter reading at two heights 100 m apart. The measured free-air difference is 3.086 x 10^-6 s^-2 x 100 m = 3.1 x 10^-4 m/s^2 (about 31 milligal); the uniform-acceleration (Rindler) picture predicts about 1 x 10^-13 m/s^2 [calc].

Summary of verdicts