Two Epochs, Two Astronomical Logics

A Second-Stage Computational Study of the Maya Long Count Era Base and the Kali-yuga Epoch

Abstract

This report tests several astronomical propositions that arose from an earlier comparison of the Maya Long Count era base and the conventional Indian Kali-yuga epoch. It reconstructs both the 584,283 and 584,285 Maya correlation variants at eight Mesoamerican sites; compares solar zenith distance, lunar phase, Venus phase, and the visibility of the proposed Three Hearthstones in Orion; reproduces the mean planetary longitudes implied by the Āryabhaṭīya, the Sūryasiddhānta, and the Paitāmaha–Brahmasphuṭasiddhānta system; places those mathematical positions beside DE431 true apparent planetary positions; and propagates five ΔT scenarios through the nearest eclipses.

The expanded calculations sharpen the earlier conclusion. Correlation 584,285 produces an exceptionally close solar-zenith match at Izapa and Copán, with modeled noon zenith distances of 0.016° and 0.099° respectively. Correlation 584,283 produces less exact but still close matches of 0.664° and 0.747°. The same result does not extend uniformly to the regions possessing the earliest complete or proposed Long Count inscriptions: the zenith distance at Tres Zapotes is 2.889° under 584,283 and 3.537° under 584,285. The Orion Hearthstones were already 43–56° above the horizon at nautical dawn and rose approximately 6.5–7.5 hours before the Sun. Neither correlation marks their heliacal return. Venus was near superior conjunction and below the horizon at nautical dawn, while the Moon was waxing gibbous, not at a lunation boundary.

For India, the exact zero-Aries conjunction belongs primarily to mathematical mean-planet systems. The Āryabhaṭīya sunrise and midnight schemes and the extant Sūryasiddhānta set the mean planets to zero by construction at their respective epoch instants. The long-period Paitāmaha–Brahmasphuṭasiddhānta constants place the five planets between 357.408° and 359.460°, with the mean Sun and Moon at zero. DE431 true apparent positions tell a different but still interesting story: in a reconstructed star-relative frame, five bodies fell within roughly ±10° of the sidereal origin, while Mercury lay about 20° west and Saturn about 33° west. The complete seven-body gathering occupied approximately 41°. Thus the physical sky made the mathematical fiction plausible, but did not literally instantiate it.

Eclipse geometry is secure in dynamical time, but local visibility is not. The ΔT scenarios shift the central longitude of the solar eclipses by as much as 81° when an extreme historical extrapolation is included. The first partial lunar eclipse after the Kali epoch has a stable umbral magnitude of approximately 0.636 and is above the Ujjain horizon under four clustered ΔT models, but below it under the most divergent extrapolation. No tested astronomical parameter supplies a technical bridge between the two chronological systems. The strongest result remains a contrast between a geographically selective Maya tropical-solar relation and an Indian calculated sidereal-planetary relation.

Research design and evidentiary discipline

1.1 Questions tested

The study tests four bounded questions rather than searching the ancient sky without limits. First, does changing the Maya correlation constant from 584,283 to 584,285 materially alter the relationship between the era-base date and solar zenith passage at relevant Mesoamerican sites? Second, do the lunar phase, Venus phase, or visibility of Orion's proposed Three Hearthstones distinguish the two correlations or provide an equally strong epoch marker? Third, what exactly do Indian astronomical canons mean when they place the planets at zero longitude at the Kali-yuga epoch, and how do their mean positions compare with reconstructed true apparent positions? Fourth, do plausible alternative treatments of ancient Earth rotation preserve or destroy claims about local eclipse visibility?

This protocol is deliberately narrow. It does not search thousands of years for whichever conjunction, eclipse, star rising, or numerical interval happens to resemble a feature in another culture. Such an unconstrained search would almost guarantee coincidences. Nor is the protocol described as a formal prospective preregistration: the two epoch dates and several candidate phenomena had already been discussed before this calculation began. It is better described as a locked second-stage protocol. The comparison sites, correlation constants, stellar identification, observables, and interpretive thresholds were fixed before introducing any additional dates or astronomical cycles.

1.2 Interpretive thresholds

The following thresholds are analytical conventions, not claims about ancient perceptual categories. Their purpose is to prevent flexible language such as “near,” “aligned,” or “visible” from changing after results are known.

These criteria are intentionally conservative. A civilization can attach ritual significance to an event that falls outside them, and a skilled observer may see a bright planet in conditions classified here as marginal. The thresholds govern only the strength of the comparative inference.

Dates, sites, coordinates, and computational method

2.1 Calendar conventions

For the Maya era base, correlation constant 584,283 is represented by Julian Day 584282.5, corresponding to 11 August 3114 BCE in the proleptic Gregorian calendar and 6 September in the proleptic Julian calendar. Correlation 584,285 is two days later: 13 August Gregorian or 8 September Julian. The standard Goodman–Martínez–Thompson family is supported by multiple historical and scientific arguments, including high-precision radiocarbon work, but the two-day Thompson-Lounsbury displacement remains pertinent when phenomena change rapidly with date. Douglas Kennett and colleagues provide modern radiocarbon support for the GMT framework, while Susan Milbrath summarizes the correlation debate and its consequences for early inscriptions and solar interpretation.

The Indian epoch is modeled at local mean midnight at Ujjain at the transition from 17 to 18 February 3102 BCE, proleptic Julian. Converted to UT, the modeled instant is 22 January 3102 BCE, 18:56:51, proleptic Gregorian. The Āryabhaṭīya also has an audayika, or sunrise, scheme; its corresponding modeled sunrise at Ujjain occurred about 6 hours 49 minutes after local midnight. These two instants are calculated separately because the Moon moves several degrees during that interval.

Astronomical year numbering was used internally: astronomical year −3113 is historical 3114 BCE, and astronomical year −3101 is historical 3102 BCE. All dates displayed in the report have been translated back into conventional BCE notation.

2.2 Mesoamerican comparison sites

The site selection separates three different reasons for inclusion. Copán was included because its latitude supplied the strongest result in the first-stage study and because Michael Grofe has examined Orion and solar observations there. Izapa was included because of its importance in discussions of Preclassic iconography and regional solar geography, not because it possesses an uncontested Long Count inscription. Tikal provides the earliest securely documented Initial Series Long Count on a lowland Maya stela. Takalik Abaj, El Baúl, Chiapa de Corzo, Tres Zapotes, and La Mojarra represent Pacific-slope and Isthmian contexts associated with early or proposed Long Count use.

These categories must not be collapsed. Tres Zapotes and La Mojarra belong to an Epi-Olmec or Isthmian scribal setting, and the damaged Chiapa de Corzo inscription has competing reconstructions. Takalik Abaj and El Baúl also present epigraphic uncertainties. Milbrath's survey stresses both the importance and the disputed readings of these monuments. The calculations therefore test geography; they do not assume that every site participated in a single homogeneous “Maya” astronomical system.

Coordinates are approximate monument-area coordinates. At meridian transit the solar-zenith result is governed overwhelmingly by latitude. Moving a site coordinate by 0.1° in latitude changes the idealized zenith distance by approximately 0.1°; small longitude or elevation differences do not alter the ranking reported below.

2.3 Ephemeris and stellar definitions

Planetary, lunar, and solar positions were calculated with Swiss Ephemeris 2.10.03 using its compressed DE431 files for 3601–3002 BCE. The calculations used apparent geocentric ecliptic and equatorial coordinates of date. Fixed-star coordinates included catalogued proper motions and long-term precession. DE431 provides the dynamical geometry; Swiss Ephemeris supplies the compression, coordinate transformations, rise-and-set routines, and eclipse algorithms. Its official documentation explains the relation to JPL ephemerides and the limits of long-range Earth-rotation reconstruction.

The proposed Three Hearthstones were defined as Alnitak, Rigel, and Saiph. This is the triangular identification discussed in modern K'iche' ethnography and in the work of Grofe and Prudence Rice. The Orion Belt—Mintaka, Alnilam, and Alnitak—was calculated separately. The distinction is essential: only Alnitak belongs to both groups. Calling Orion's Belt “the Three Hearthstones” without qualification merges two related but non-identical identifications.

“Nautical dawn” is defined computationally as the instant when the Sun's geometric center reached 12° below the horizon. Rise lead is the interval between a star's modeled geometric rising and sunrise. Atmospheric extinction, local mountains, humidity, and observer acuity were not reconstructed. An object tens of degrees above the horizon does not require delicate extinction modeling to classify it as established in the sky; an object near the horizon does.

The Maya correlation experiment

3.1 Solar zenith distance

At meridian transit, a Sun and observer sharing nearly the same declination and latitude produce a near-zenith culmination. On 11 August under correlation 584,283, the modeled solar declination at the selected western Mesoamerican sites was approximately +15.58°. Two days later it was approximately +14.94°. Because the Sun was moving southward by roughly one-third of a degree per day, the two-day displacement changed the closest-matching latitude by about 0.65°.

The result is strong but geographically selective. Correlation 584,285 is almost exact for Izapa and exceptionally close for Copán. At Izapa the Sun missed the zenith by only 0.016°, about one arcminute, under the idealized geometric model. Correlation 584,283 remains within one degree at Izapa, Copán, and Takalik Abaj, but it is not nearly as precise.

The same test weakens an overly broad historical inference. Tres Zapotes, the location of the earliest complete five-place Long Count inscription, experienced its descending zenith passage about nine days before the 584,283 date. La Mojarra displays nearly the same result. Tikal's zenith passage came approximately 4.6 days earlier. The astronomical match therefore cannot by itself identify the place where the Long Count was invented. It is compatible with a southern Maya or Pacific-piedmont solar interest, but the best matches occur at Copán and Izapa for reasons largely determined by latitude.

The correlation comparison also illustrates a methodological danger. If Copán or Izapa is selected only after testing many sites, the near-perfect 584,285 match becomes partly post hoc. Its evidentiary value improves only if independent epigraphic or architectural evidence identifies the 14.8–14.9° latitude band as historically relevant to the formulation or later interpretation of the era base. Without that independent evidence, the calculation establishes a geographical fact rather than a causal explanation.

3.2 Lunar phase

The two-day change moves the later correlation substantially closer to full moon, but neither correlation falls on a principal lunar phase. The phase test therefore distinguishes degrees of proximity rather than identifying an exact lunar boundary.

Astronomical full moon occurred on 14 August 3114 BCE Gregorian at approximately 16:45 UT. Correlation 584,285 is therefore a secondary lunar match under the locked two-day threshold, while 584,283 is not. This distinction is real but weaker than the solar result. A date 1.7 days before full moon is not naturally a full-moon epoch unless historical evidence independently permits a broader ritual window.

No solar or lunar eclipse occurred on either era-base date. The nearest subsequent lunar eclipse was penumbral and occurred 33.5 days after 584,283, or 31.5 days after 584,285. Its penumbral magnitude was only about 0.330. The eclipse analysis below shows that even its visibility from the Maya region depends on the assumed history of Earth rotation.

3.3 Venus phase and visibility

At both Maya dates Venus was close to superior conjunction on the far side of the Sun. The two-day displacement changes its elongation only slightly and does not move it into a different observational phase.

Venus was below the horizon when nautical dawn began and rose into rapidly brightening twilight less than an hour before sunrise. Its great brightness may permit observations at surprisingly small elongations under exceptional conditions, but this geometry does not satisfy the study's naked-eye visibility criterion. More importantly, the two-day correlation change does not create a distinct Venus event. Both dates occupy essentially the same portion of the superior-conjunction cycle.

The result provides no support for making Venus the direct determinant of either Maya correlation. This does not diminish the well-established importance of Venus in later Maya astronomy, especially the Dresden Codex. It means only that the Long Count era-base dates tested here do not coincide with a conspicuous Venus station such as inferior conjunction, greatest elongation, first morning visibility, or first evening visibility.

3.4 Three Hearthstones and Orion's Belt

The proposed Hearthstones were not near heliacal emergence. Across all eight sites and both correlations, Alnitak, Rigel, and Saiph had risen approximately 6.5–7.5 hours before sunrise. At nautical dawn they were already high in the southeastern sky.

Rigel supplied the lowest altitude and Alnitak the highest at nautical dawn. One hour before sunrise, the stars were still approximately 44–57° high. Orion's Belt showed the same broad conclusion. A two-day displacement altered the altitudes by less than one degree and did not change the visibility classification.

This negative result is unusually secure. Exact heliacal visibility depends on extinction, horizon obstruction, and observer thresholds, but an asterism more than 40° high at nautical dawn is not undergoing a marginal first appearance. The dates belong to the season of Orion's mature predawn visibility. That seasonal presence may still possess symbolic relevance to a creation epoch, especially in conjunction with the Milky Way and solar-zenith season, but such an interpretation is broader than a specific heliacal-rising claim.

3.5 Maya result in historical perspective

The best-supported positive result is the 584,285 zenith relationship at the latitude of Copán and Izapa. Its precision is not reproduced at Tikal, Tres Zapotes, or La Mojarra. The 584,283 correlation offers a somewhat broader but less exact southern-latitude match. Lunar phase modestly favors 584,285, but Venus and Orion do not distinguish the dates.

This pattern favors a limited proposition: Maya scholars working at certain latitudes could experience the mythic era-base anniversary near descending solar zenith passage, and a two-day correlation shift materially affects the closeness of that experience. It does not establish that a remote observation in 3114 BCE founded the Long Count, that Copán originated the calendar, or that the correlation should be selected solely by maximizing zenith precision. Calendar correlation rests on converging epigraphic, ethnographic, historical, radiocarbon, and astronomical evidence, not one optimized celestial match.

Indian mean planets and the physical sky

4.1 What a mean conjunction means

The phrase “all planets at zero Aries” is ambiguous unless the astronomical category is specified. A true apparent longitude describes where a body appears in a defined coordinate frame after accounting for its actual orbital inequality and the observational geometry. A mean longitude belongs to a mathematical model in which irregular motion has been smoothed or distributed between deferents, epicycles, anomalies, apsides, and correction procedures. For Mercury and Venus, the Indian mean motion tabulated in the long-period systems is especially unlike the instantaneous geocentric apparent longitude of the visible planet; it corresponds to a model component comparable to heliocentric mean motion.

The distinction is not a criticism of Indian astronomy. Mean motions are indispensable mathematical tools. Ancient and medieval astronomy everywhere used idealized motions to generate tables, epochs, and corrections. The historical error arises only when a canonical mean conjunction is presented as though seven visible bodies occupied one point in the observed sky.

Dennis Duke's analysis of Indian mean motions shows how the revolution counts combine period relations with longitudes near a later working epoch. In the Āryabhaṭīya schemes, the revolution numbers are constrained so that the mean longitudes are zero at the beginning of the final equal quarter of the mahāyuga. Burgess's translation of the Sūryasiddhānta likewise explains that its revolution numbers, excepting lunar apsis and node, are divisible by four and possess a common 1,080,000-year period. The extant text places the last mean conjunction at the beginning of the current Kali age.

4.2 Reconstructed canonical positions

The table gives internal mean longitudes at the model's Kali epoch. Zero is each canon's own sidereal origin, not the modern tropical equinox.

The first three columns are exact because the systems impose the common epoch. The last column was independently recomputed from the long-period revolution constants discussed by Duke. The Paitāmaha and Brahmasphuṭasiddhānta share a 4,320,000,000-year framework in which 1,972,944,000 years, exactly 0.4567 of the period, had elapsed at the current Kali epoch. For each body, the calculation was

[
\lambda = 360^{\circ},\operatorname{frac}(0.4567R),
]

where (R) is the canon's integral revolution count. The recomputation gives Sun and Moon exactly at zero, while the five planets fall from 0.540° to 2.592° west of zero. This is a remarkably tight mean grouping but not an exact one.

The contrast between these systems is historically instructive. The “Indian tradition” did not preserve one immutable numerical model. Different canons adopted different total periods, day counts, revolution numbers, epoch conventions, and correction procedures. Agreement at or near zero reflects a shared epoch-making strategy; the exactness and the computational path vary by text.

4.3 DE431 true apparent positions at Ujjain midnight

The physical sky was reconstructed at Ujjain local mean midnight. Two star-relative diagnostics are supplied. “True-Citrā” is the modern Swiss Ephemeris convention fixing Spica/Citrā to 180°; it is not claimed as the ancient canonical ayanāṃśa. The “Burgess-origin” column uses Burgess's reconstruction placing the Hindu origin 50°22′29″ west of the vernal equinox in 3102 BCE. Changing the sidereal zero shifts every longitude together and does not change the 41° width of the gathering.

Five bodies—Mars, Sun, Moon, Venus, and Jupiter—lay within about 8.6° of the reconstructed sidereal origin. Mercury lay about 20° west and Saturn about 33° west. The minimum circular arc containing all seven bodies was approximately 41.01°. This is a real broad gathering and much tighter than an ordinary distribution around the ecliptic, but it is not the exact canonical conjunction.

The physical sky therefore occupies a middle position between dismissal and literalism. The epoch was not unrelated to planetary geometry: the gathering was genuine, it followed astronomical new moon by about 17.6 hours, and Venus and Jupiter reached a true angular separation of roughly 0.13° about 8.7 hours later. Yet the exact zero belongs to the mean-planet model. The mathematical scheme compresses a 41° true-apparent configuration to zero or, in the long-period Brahmagupta system, to less than 2.6°.

4.4 Sunrise sensitivity

At modeled Ujjain sunrise the Sun's true-Citrā longitude had advanced to 351.388°, and the Moon to 4.242°. In the Burgess-origin frame, the Moon, Venus, and Jupiter occupied 8.021°, 7.951°, and 8.035° respectively—an exceptionally close longitude grouping—while Mars was −8.393°, Mercury −19.497°, and Saturn −33.004°. The complete arc remained about 41°.

The sunrise calculation confirms two points. First, changing the daily convention cannot transform the physical sky into an exact conjunction. Saturn and Mercury remain far from zero, while the Moon's rapid motion carries it farther east. Second, the distinction between the audayika and ardharātrika systems is not a trivial clock adjustment. Each scheme chooses an epoch appropriate to its own day count and mean-motion construction. A historically responsible comparison should not mix an audayika canonical zero with a midnight DE431 sky without stating the difference.

4.5 What the Indian comparison establishes

The Indian “zero Aries” configuration is not merely an invention of modern popular astrology. It is genuinely embedded in siddhāntic mathematical practice. The strongest formulation, however, is that one or more systems define the mean planets to be at their sidereal origin at the epoch. The extant physical sky contains an unusual broad grouping that makes this definition astronomically intelligible, while the exact mean conjunction provides an elegant common radix for chronological calculation.

This finding also clarifies why religious or cosmological use does not invalidate the mathematical method. The cosmological magnitude of a mahāyuga is not an observed time span, but integral revolution counts within it encode ratios of mean motions. Those ratios can be used operationally to calculate calendars and planetary positions. The metaphysical frame, idealized epoch, and effective mathematics coexist; they should neither be collapsed into modern observational science nor dismissed as mathematically empty.

Eclipses under alternative ΔT models

5.1 Why ΔT dominates ancient local visibility

DE431 determines the relative dynamical positions of Earth, Moon, and Sun in a uniform time argument. An observer's terrestrial longitude, however, depends on how far Earth had rotated at that dynamical instant. ΔT is the difference between Terrestrial Time and Universal Time derived from Earth rotation. Before telescopic observations and reliable timed eclipse records, ΔT must be extrapolated.

The earliest reasonably useful historical eclipse constraints are thousands of years later than the two epochs studied here. The following scenarios therefore are not confidence intervals. They are published long-term parabolas extended to approximately 3100 BCE in order to measure sensitivity. Stephenson, Morrison, and Hohenkerk's modern reconstruction covers 720 BCE onward and cannot directly determine ΔT in 3100 BCE; NASA likewise cautions that pre-700-BCE values rest on tidal extrapolation.

The first four models cluster within about one hour. The Stephenson–Houlden extrapolation differs by about 4.4–5.4 hours. Because Earth rotates 15° per hour, even the clustered models can displace an eclipse path by roughly 15° in longitude, while the stress-test spread approaches 81°.

5.2 Eclipse sequence after the Maya date

The first relevant lunar event was a penumbral eclipse on 13 September 3114 BCE Gregorian, 33.5 days after correlation 584,283. Its dynamical penumbral magnitude remained 0.330 in every model, but the modeled lunar altitude at maximum varied sharply.

This is a textbook model-dependent result. The eclipse itself and its shallow penumbral magnitude are secure. A claim that Copán or Tikal saw maximum eclipse is not. Even under models placing the Moon barely above the horizon, a penumbral magnitude of 0.330 would be visually subtle and vulnerable to haze and twilight.

The following solar eclipse occurred on 27 September and was dynamically hybrid. Its maximum central latitude was stable near 3.62° south, but its central longitude was not.

Across the four clustered models, every selected Mesoamerican site sees a substantial partial eclipse, with modeled magnitudes ranging approximately from 0.60 to 0.83. Under the divergent 26.38-hour scenario, the region sees no eclipse. The full model band for the central longitude extends from 57.08° W to 24.36° E, about 81.44°. The narrower four-model band extends about 14.62°.

This event cannot plausibly define the era-base day because it occurred almost seven weeks later. It may be relevant to the astronomical environment of the season, but treating it as an epoch determinant would require independent textual evidence for an interval connecting the dates.

5.3 Eclipse sequence after the Kali epoch

The first full moon after the Kali epoch produced a partial lunar eclipse on 6 February 3102 BCE Gregorian. Its umbral magnitude was approximately 0.636 in every scenario.

The local result is more stable than the Maya penumbral case within the clustered set of long-term models. Four scenarios put an easily eclipsed portion of the Moon 26–39° above Ujjain's horizon. The divergent stress test reverses the conclusion. It is therefore reasonable to say that the eclipse was probably visible from Ujjain under the family of modern approximately 21–22-hour extrapolations, but not defensible to call that local visibility model-independent.

The following new moon generated an annular solar eclipse on 20 February 3102 BCE Gregorian, 28.74 days after the epoch. Its central latitude was approximately 47.78° south. None of the five ΔT scenarios produced visibility from Ujjain or any selected Mesoamerican site. The central longitude ranged from 8.15° W to 72.74° E across the full model set, but the far-southern latitude keeps the tested sites outside the partial zone. This is a robust local negative result.

The Indian epoch thus occurred at new moon during an eclipse season, and its first full moon was eclipsed. That is a stronger eclipse relationship than anything found at the Maya era base. It remains an Indian result, not a shared eclipse signature.

Cycle comparison and contact-hypothesis tests

6.1 The interval between the epochs

Under the adopted instants, the Maya 584,283 era date and Ujjain-midnight Kali epoch are separated by 4,182.789 days. The interval fails to equal a whole number of the most obvious cycles.

Moving the Maya date two days later reduces each residual by two days but does not create an integer recurrence. No familiar planetary, lunar, or eclipse cycle carries one epoch into the other.

6.2 Venus as a convergence test

The Maya conventional Venus period of 584 days is necessarily close to Indian values because all sustained naked-eye traditions observe the same physical synodic cycle. The Sūryasiddhānta revolution counts imply approximately 583.900 days per Venus synodic cycle; the two Āryabhaṭīya day-count schemes imply approximately 583.897 days. Five such Indian cycles total about 2,919.49–2,919.50 days, roughly half a day short of the simple Maya equation of five times 584, or 2,920 days.

This agreement is astronomically meaningful but historically weak as evidence for contact. A period near 584 days is forced by Venus's orbit and can be measured independently. A stronger diffusion test would require a shared non-obvious correction rule: for example, the same sequence of unequal interval corrections, the same table length for reasons not dictated by visibility, or the same error pattern relative to the physical planet. No such correspondence has been demonstrated here.

6.3 Technical predictions and outcomes

A contact hypothesis is not disproved merely because these seven predictions fail; contact could occur without transmitting astronomy. The narrower claim—that the two era bases encode a shared astronomical configuration—is strongly disfavored. Future diffusion arguments should state in advance which arbitrary technical feature is expected to match and why independent observation would not produce it.

Established results, reasonable interpretations, and speculation

7.1 Established by the calculations

The 584,285 correlation places the era-base anniversary extremely close to descending solar zenith passage at Izapa and Copán. The 584,283 correlation remains within one degree there but is less exact. Neither correlation creates a comparable match at Tres Zapotes, La Mojarra, or Tikal. This is a geometrical result conditioned on the stated correlation and site coordinates.

The Moon was waxing gibbous on both Maya dates. Correlation 584,285 fell about 1.70 days before full moon, while 584,283 fell about 3.70 days before. Venus was nearly full, about 9° west of the Sun, and below the horizon at nautical dawn. The proposed Three Hearthstones were more than 42° high at nautical dawn across every selected site and rose more than six and a half hours before sunrise.

The Āryabhaṭīya and Sūryasiddhānta mean-planet systems enforce an exact common longitude at their Kali epoch. The Paitāmaha–Brahmasphuṭasiddhānta revolution constants produce a near-zero grouping within 2.592°. DE431 true apparent positions occupy a 41.01° arc. Five bodies fall within about ±10° of the reconstructed Hindu origin, with Mercury and Saturn farther west.

The lunar-eclipse magnitudes and solar-eclipse dynamical types do not change with ΔT scenario. Local circumstances do. The full scenario spread moves the central longitude of the tested solar eclipses by approximately 81°. The annular solar eclipse following the Kali epoch remains invisible from all selected sites under every tested model.

7.2 Reasonable interpretations

The Maya calculations justify continued attention to solar zenith passage as one layer of era-base commemoration at southern Maya and Pacific-piedmont latitudes. Correlation 584,285 strengthens that local relationship. Because the earliest complete Long Count occurs farther north at Tres Zapotes, the result is better interpreted as a possible regional appropriation or astronomical resonance than as proof of the calendar's point of invention.

The Indian epoch combines an idealized mathematical reset with a real broad gathering, new-moon timing, and a close Venus–Jupiter conjunction. It is reasonable to infer that the physical clustering made a zero-longitude epoch attractive or retrospectively credible to calculators. The exact conjunction, however, belongs to the mean model, not to simultaneous naked-eye observation.

The first partial lunar eclipse after the Kali epoch was probably observable from Ujjain if remote-past ΔT lay near the approximately 21–22-hour cluster favored by several modern extrapolations. The word “probably” is essential because no direct fourth-millennium-BCE Earth-rotation measurement exists.

7.3 Speculation not established

The calculations do not show that the Long Count was invented at Copán or Izapa, that correlation 584,285 is correct because it optimizes a zenith passage, or that Maya creation narratives encode an observation made in 3114 BCE. They do not show that the Orion Hearthstones were undergoing heliacal birth on either correlation date.

The calculations do not prove that Indian astronomers directly observed the seven bodies in conjunction. Most occupied twilight or solar glare, and the true positions were not coincident. Nor do they establish that the partial lunar eclipse following the epoch was used to choose it.

Finally, the calculations do not supply evidence of Maya–India contact, a transmitted epoch, or a shared secret chronology. Similar interest in Venus, lunar phases, solar cycles, and bright stellar patterns is expected wherever trained observers monitor the same sky. Historical contact requires archaeological, linguistic, genetic, iconographic, or technical evidence with explanatory specificity beyond celestial resemblance.

Limitations and genuinely productive next steps

The Mesoamerican solar calculation should next be joined to architectural horizon measurements. A site-specific zenith passage is different from an architectural alignment, and neither establishes calendrical causation alone. Digital elevation models could quantify local horizon obstruction for Orion and Venus. Archaeological dating must also be kept in view: Classic monuments cannot straightforwardly identify where a mythic epoch was first formulated.

The Indian comparison should be extended from mean longitudes to complete canon-specific true-place algorithms. That task requires implementing manda and śīghra corrections, apsidal and nodal parameters, canon-specific day counts, and each text's ayanāṃśa treatment. It would then be possible to compare not only canonical mean zeroes but the true places each historical algorithm predicts for its own epoch. The present report deliberately stops before pretending that one generalized “Sūryasiddhānta algorithm” represents all recensions and bīja corrections.

For eclipses, the correct output is an uncertainty surface rather than a single line on a modern map. A future numerical study could assign priors to tidal acceleration and non-tidal length-of-day variation, then produce probabilistic path-density maps. The published parabolas used here measure model sensitivity but are not probability distributions.

For cultural contact, the next test should begin with documentary or material evidence and derive an astronomical prediction from it. Searching astronomical databases first and constructing a transmission story afterward reverses the logic of historical proof. A useful candidate would be an arbitrary correction sequence or tabular error pattern whose complexity makes independent invention unlikely. No such candidate emerged from the present comparison.

Conclusion

The second-stage study makes the contrast between the epochs more precise. The Maya correlation question is genuinely sensitive to two days because the descending Sun moves rapidly across the latitude band of southern Mesoamerica in August. Correlation 584,285 gives an almost exact zenith culmination at Izapa and Copán; 584,283 gives a broader, less precise match. Yet the earliest complete Long Count region does not share that exact geometry, and neither correlation is marked by a principal Venus station, a lunar boundary, an eclipse, or the heliacal return of Orion's Hearthstones.

The Indian epoch exhibits the inverse structure. Its tropical solar circumstances are unremarkable, but its mathematical astronomy is organized around a star-relative planetary origin. Several canonical systems place mean planets exactly or nearly at zero, while DE431 reconstructs an uncommon physical gathering spread across approximately 41°. The mean conjunction is therefore neither a simple record of observation nor an empty fiction. It is a mathematical idealization anchored loosely but intelligibly in the actual sky.

Alternative ΔT scenarios further demonstrate why archaeoastronomy must distinguish celestial dynamics from terrestrial visibility. Eclipse type and magnitude remain stable while modeled paths move across tens of degrees of longitude. Claims about what a particular ancient city saw require much greater caution than claims that an eclipse occurred.

No shared astronomical state connects the two chronological systems. Their most compelling relationship is comparative rather than genealogical. The Maya case points toward a tropical, latitude-sensitive solar geometry embedded in regional sacred landscapes. The Indian case points toward a sidereal, computational planetary gathering embedded in mathematical cosmology. Both traditions transformed celestial order into chronology, but they did so through different astronomical logics. That difference is not an obstacle to comparison; it is the most historically meaningful result the comparison has produced.

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Reproducibility note

The Maya epoch calculations used Julian Days 584282.5 and 584284.5. Solar zenith distance was computed at the next meridian transit after 00:00 UT on each proleptic Gregorian civil date. Nautical dawn was solved numerically for solar geometric altitude −12°. Fixed-star altitudes used apparent equatorial coordinates of date and standard sea-level pressure; elevations were included but no local horizon mask was applied.

The Indian midnight calculation used Julian Day 588465.2894763888, obtained by converting Ujjain local mean midnight to UT. The modeled sunrise was Julian Day 588465.5735971448. DE431 true apparent positions used Swiss Ephemeris flags FLG_SWIEPH | FLG_SPEED; the modern diagnostic sidereal positions used SIDM_TRUE_CITRA. The Burgess-origin diagnostic added 50°22′29″ to tropical longitude. The Paitāmaha–Brahmasphuṭasiddhānta longitudes were recomputed directly from the revolution counts and the elapsed fraction 4567/10000.

For each eclipse and ΔT scenario, the same dynamical event was recalculated after assigning the model ΔT. Solar central coordinates were evaluated at global maximum. Local solar circumstances were sampled across an eight-hour interval and maximized while the Sun was above the conventional rising horizon. Lunar magnitude was evaluated at global maximum; local lunar altitude was recomputed separately at each site. Numerical precision in the tables exceeds historical certainty and is retained to permit replication, not to imply that fourth-millennium-BCE civil time is known to minutes.

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