Run B, and what the birth window owes the baptism axis
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In-depth:
Run B couples the birth side to the baptism. Two predicates are added to the model of chapter 8. B-1a is the Tiberius floor: the baptism does not fall before Tiberius' 15th year, so baptism ≥ AD 27, with 27 as the lower bound under inclusive reckoning. G-3 is Luke 3:23, "about thirty" as an age band of 28-32. The baptism is not fixed to a single year but run across 27-29; the age is baptism year + birth year − 1, with no year 0. The sum reproduces the count-back table exactly: at baptism AD 27 a child of 7 BC is 33, of 6 BC 32, of 5 BC 31, of 4 BC 30.
The coupling leaves 149 cells alive and kills 607. For the three 4/3 chains — C, CA, HV — what remains as birth year is 6, 5, 4 BC. SY, on year of death 1 BC, keeps 5 to 1 BC. In the kill matrix, Matthew 2 "after the death" kills 264 cells, G-3 outside the age band 364, Matthew 2 "five years or more before the death" 120.
The difference from run A is two cells. Freestanding, C/CA/HV stood at 8-4 BC, with 8 as a postulate (two years' margin on the Matthew limit) and 7 soft; coupled, they stand at 6-4. The baptism axis removes 8 and 7 — the two softest cells of the early edge. 8 was already a postulate; 7 falls on the age band, for at baptism AD 27 a child of 7 BC is 33, just outside Plummer's band 28-32. The firm core 6-5-4 falls on Matthew and on robustness, not on the baptism. The window thus does not owe its firm core to the baptism/crucifixion axis; but that axis does set the early edge — at 6 instead of 7 — and that hangs on how tightly "about thirty" is read (§9.2).
9.1 The correction
That is not what the first output said. It gave 5-4 BC for run B — 125 living cells against 631 dead, with G-3 as the biggest killer (392) — and on that figure rested two headlines: that the narrowest window was "Sukkot 5 or 4", and that the narrowing from 8-4 to 5-4 was the work's largest measurement result, the window as a derivative of the baptism axis. Both are untenable, and the error can be pinpointed exactly.
In that version G-3 carried two conditions: the age band 27-33, and alongside it a separate floor "birth year ≤ 5", taken from Hoehner's "no earlier than 5 B.C.". That floor is Hoehner's own sum: baptism AD 29 plus age 33 gives 5 BC. It is baptism-AD-29-specific. HV reckons from baptism AD 27, after 20 October of that year, and there the same band gives an earliest birth of 7 BC (baptism 27, age 33, no year 0; on Plummer's band 28-32 this becomes 6 BC). The logigram thus imported a consensus figure from a baptism year that HV itself rejects: HV dismisses Hoehner's baptism year (AD 29) and his crucifixion year (AD 33), yet took Hoehner's outcome along as a cut. The predicate was internally inconsistent — an outcome of baptism 29, applied to a run that inputs baptism 27.
That inconsistency came to light in the testing of the baptism axis; the logigram was recomputed and corrected with a link that runs both runs side by side. The consistent run is the correct one. It widens the window and shrinks the A↔B difference, against the direction that would serve HV's narrow expectation; the correction stands not because it is welcome but because the predicate was wrong. Two consequences: "Sukkot 5 or 4" was too narrow — on HV's own baptism the window runs to 6 BC, with 5 as the most robust cell, not as the only one — and the self-restriction that Drie dagen en drie nachten imposed on itself was more cautious than necessary. Freestanding, the window narrows to 8-4 as well; the baptism axis takes 8 and 7 off that.
9.2 What the baptism axis carries on its own
The baptism axis narrows the early edge by two cells and does not touch the firm core; the question is then what it carries on its own. Three links, each on the same measure.
The Tiberius floor, the 15th year from the autumn of AD 27 with the baptism after 20 October, rests on three knobs: epoch August/September AD 14 without co-regency from AD 12, Tishri as year-beginning, inclusive reckoning. That reckoning is on record, by name and with the same dates, on the opposing side. Steinmann gives for the inclusive method "autumn AD 27 to summer AD 28"; Smallwood reckons on the same three knobs that Tiberius' 15th year "ran from October 27". The coin material that rules out inclusive reckoning — RPC I 4270 and 4330, the Alexandrian end-year test, Philo's 23 — measures the pagan counting systems, not the Jewish Tishri framework, and the Judean prefect coins cannot separate the two reckonings. In the only framework in which Luke's 15th year can fall on Tishri AD 27, no one has measured: UNDECIDED, not refuted. The price is there all the same. The pagan default is non-inclusive — RPC I 4270 counts year Α as Actian 45, not as year 2 — so that HV must postulate that Luke uses precisely the Jewish autumn-inclusive framework; Smallwood's own word is "presumably". Within that framework, moreover, Nisan lies as an alternative, which shifts the whole axis by a year, and Smallwood herself reckons on Nisan one note earlier. The cheaper surviving cell is baptism AD 29, with the pagan default and positive coin material behind it. HV rejects it because it demolishes the early edge of the window — a valid internal ground, but circular with respect to the third link.
"About thirty" narrows — but how much hangs on the width of the band. ὡσεί marks a round number with modest latitude: of 18 passages with ὡσεί or ὡς plus a number, eight are measurable; seven give zero difference, one gives two units. The word marks the class of the number, not a wide tolerance. That is why Hoehner's ±3 (age band 27-33) is too wide: age 33 — a birth in 7 BC at baptism 27 — then stays within the band, and read that way "about thirty" narrows nothing. On Plummer's ±2 (28-32) age 33 does fall outside, and 7 BC disappears: the early edge runs to 6, not 7. This work adopts ±2 — defensible on the source, and Hoehner's floor is in any case AD-29-specific. Reading tighter still (±1) would yield 5 BC, precisely HV's own expectation; that would be the mirror asymmetry, and it does not happen. Irenaeus cuts the other way: he reads ἀρχόμενος, "not yet thirty", and on that reading thirty is a ceiling that removes everything before 3 BC, every Herod-4 scenario included. HV thereby drops a primary reading that cuts against its own year of death.
The 69 year-weeks of Daniel 9 — 483 years from Ezra 7, Tishri 457 BC, to Tishri AD 27, with the baptism following after 20 October — add up as arithmetic and carry nothing as an independent constraint: they are by construction the target of the logigram. The terminus a quo hangs on the shemitah chain, and that rests on a synchronism from Seder Olam 28, itself an exposition of Daniel 9:24; the chain that fixes the starting point comes from an ancient exposition of the same 69 weeks. Smallwood, in transmission from North, attacks the whole shemitah class ("cannot be fitted into a coherent chronological scheme"), and on the same structure Kokkinos carries a rival 69 that does not yield HV's AD 27/31.
Net, the baptism axis is the weakest of the axes that carry the window: a defensible minority counting knob, an "about thirty" that on ±2 sets the early edge at 6 but does not touch the firm core, a 69-weeks target with a rival beside it. It narrows the early edge by two cells (8 and 7) and sets no further price. The same measure the other way round: on the year-of-death side SY pays three rejections and eight postulates against C's zero. The consensus does not fall on cost; the baptism axis is a minority reading that survives, not a refuted constraint.
9.3 What the window does owe the baptism
Not its width, but its ranking. The baptism band 27-29 determines how far the early edge runs: 6 BC survives only baptism 27 (at 28 the child is 33, outside the band); 5 BC survives 27 and 28 and falls at 29; 4 BC survives the whole band. The core of the window thus does not hang on the choice of 27 alone. 5 BC has the most surviving combinations — both seasons of death and baptism 27 and 28 — and is therefore the most probable, on robustness; 6 BC leans on one baptism year and 4 BC on one season of death, but neither carries a demonstrable price, they are only less robust. The ages are year cells, baptism year + birth year − 1. With the boundary of Tiberius' 15th year on 20 October 27 the baptism falls late in the autumn of 27 or in the winter of 27/28; only the former is baptism 27. If a baptism in January or February 28 counts as baptism 28, 6 BC drops out in year cells; in completed years it stands, at 32 years.
At the other end of the axis the crucifixion year (AD 31/33) remains unseen. The obstacle there is the weekday: Smallwood and Hoehner exclude AD 31 solely via a Friday filter. The baptism axis ends on an UNDECIDED, not on an exclusion. What it gives the window is two removed cells — a postulate (8) and the soft edge (7) — and a ranking among the cells that remain.
Run B couples the birth side to the baptism. Two predicates are added to the model of chapter 8. B-1a is the Tiberius floor: the baptism does not fall before Tiberius' 15th year, so baptism ≥ AD 27, with 27 as the lower bound under inclusive reckoning. G-3 is Luke 3:23, "about thirty" as an age band of 28-32. The baptism is not fixed to a single year but run across 27-29; the age is baptism year + birth year − 1, with no year 0. The sum reproduces the count-back table exactly: at baptism AD 27 a child of 7 BC is 33, of 6 BC 32, of 5 BC 31, of 4 BC 30.
The coupling leaves 149 cells alive and kills 607. For the three 4/3 chains — C, CA, HV — what remains as birth year is 6, 5, 4 BC. SY, on year of death 1 BC, keeps 5 to 1 BC. In the kill matrix, Matthew 2 "after the death" kills 264 cells, G-3 outside the age band 364, Matthew 2 "five years or more before the death" 120.
The difference from run A is two cells. Freestanding, C/CA/HV stood at 8-4 BC, with 8 as a postulate (two years' margin on the Matthew limit) and 7 soft; coupled, they stand at 6-4. The baptism axis removes 8 and 7 — the two softest cells of the early edge. 8 was already a postulate; 7 falls on the age band, for at baptism AD 27 a child of 7 BC is 33, just outside Plummer's band 28-32. The firm core 6-5-4 falls on Matthew and on robustness, not on the baptism. The window thus does not owe its firm core to the baptism/crucifixion axis; but that axis does set the early edge — at 6 instead of 7 — and that hangs on how tightly "about thirty" is read (§9.2).
9.1 The correction
That is not what the first output said. It gave 5-4 BC for run B — 125 living cells against 631 dead, with G-3 as the biggest killer (392) — and on that figure rested two headlines: that the narrowest window was "Sukkot 5 or 4", and that the narrowing from 8-4 to 5-4 was the work's largest measurement result, the window as a derivative of the baptism axis. Both are untenable, and the error can be pinpointed exactly.
In that version G-3 carried two conditions: the age band 27-33, and alongside it a separate floor "birth year ≤ 5", taken from Hoehner's "no earlier than 5 B.C.". That floor is Hoehner's own sum: baptism AD 29 plus age 33 gives 5 BC. It is baptism-AD-29-specific. HV reckons from baptism AD 27, after 20 October of that year, and there the same band gives an earliest birth of 7 BC (baptism 27, age 33, no year 0; on Plummer's band 28-32 this becomes 6 BC). The logigram thus imported a consensus figure from a baptism year that HV itself rejects: HV dismisses Hoehner's baptism year (AD 29) and his crucifixion year (AD 33), yet took Hoehner's outcome along as a cut. The predicate was internally inconsistent — an outcome of baptism 29, applied to a run that inputs baptism 27.
That inconsistency came to light in the testing of the baptism axis; the logigram was recomputed and corrected with a link that runs both runs side by side. The consistent run is the correct one. It widens the window and shrinks the A↔B difference, against the direction that would serve HV's narrow expectation; the correction stands not because it is welcome but because the predicate was wrong. Two consequences: "Sukkot 5 or 4" was too narrow — on HV's own baptism the window runs to 6 BC, with 5 as the most robust cell, not as the only one — and the self-restriction that Drie dagen en drie nachten imposed on itself was more cautious than necessary. Freestanding, the window narrows to 8-4 as well; the baptism axis takes 8 and 7 off that.
9.2 What the baptism axis carries on its own
The baptism axis narrows the early edge by two cells and does not touch the firm core; the question is then what it carries on its own. Three links, each on the same measure.
The Tiberius floor, the 15th year from the autumn of AD 27 with the baptism after 20 October, rests on three knobs: epoch August/September AD 14 without co-regency from AD 12, Tishri as year-beginning, inclusive reckoning. That reckoning is on record, by name and with the same dates, on the opposing side. Steinmann gives for the inclusive method "autumn AD 27 to summer AD 28"; Smallwood reckons on the same three knobs that Tiberius' 15th year "ran from October 27". The coin material that rules out inclusive reckoning — RPC I 4270 and 4330, the Alexandrian end-year test, Philo's 23 — measures the pagan counting systems, not the Jewish Tishri framework, and the Judean prefect coins cannot separate the two reckonings. In the only framework in which Luke's 15th year can fall on Tishri AD 27, no one has measured: UNDECIDED, not refuted. The price is there all the same. The pagan default is non-inclusive — RPC I 4270 counts year Α as Actian 45, not as year 2 — so that HV must postulate that Luke uses precisely the Jewish autumn-inclusive framework; Smallwood's own word is "presumably". Within that framework, moreover, Nisan lies as an alternative, which shifts the whole axis by a year, and Smallwood herself reckons on Nisan one note earlier. The cheaper surviving cell is baptism AD 29, with the pagan default and positive coin material behind it. HV rejects it because it demolishes the early edge of the window — a valid internal ground, but circular with respect to the third link.
"About thirty" narrows — but how much hangs on the width of the band. ὡσεί marks a round number with modest latitude: of 18 passages with ὡσεί or ὡς plus a number, eight are measurable; seven give zero difference, one gives two units. The word marks the class of the number, not a wide tolerance. That is why Hoehner's ±3 (age band 27-33) is too wide: age 33 — a birth in 7 BC at baptism 27 — then stays within the band, and read that way "about thirty" narrows nothing. On Plummer's ±2 (28-32) age 33 does fall outside, and 7 BC disappears: the early edge runs to 6, not 7. This work adopts ±2 — defensible on the source, and Hoehner's floor is in any case AD-29-specific. Reading tighter still (±1) would yield 5 BC, precisely HV's own expectation; that would be the mirror asymmetry, and it does not happen. Irenaeus cuts the other way: he reads ἀρχόμενος, "not yet thirty", and on that reading thirty is a ceiling that removes everything before 3 BC, every Herod-4 scenario included. HV thereby drops a primary reading that cuts against its own year of death.
The 69 year-weeks of Daniel 9 — 483 years from Ezra 7, Tishri 457 BC, to Tishri AD 27, with the baptism following after 20 October — add up as arithmetic and carry nothing as an independent constraint: they are by construction the target of the logigram. The terminus a quo hangs on the shemitah chain, and that rests on a synchronism from Seder Olam 28, itself an exposition of Daniel 9:24; the chain that fixes the starting point comes from an ancient exposition of the same 69 weeks. Smallwood, in transmission from North, attacks the whole shemitah class ("cannot be fitted into a coherent chronological scheme"), and on the same structure Kokkinos carries a rival 69 that does not yield HV's AD 27/31.
Net, the baptism axis is the weakest of the axes that carry the window: a defensible minority counting knob, an "about thirty" that on ±2 sets the early edge at 6 but does not touch the firm core, a 69-weeks target with a rival beside it. It narrows the early edge by two cells (8 and 7) and sets no further price. The same measure the other way round: on the year-of-death side SY pays three rejections and eight postulates against C's zero. The consensus does not fall on cost; the baptism axis is a minority reading that survives, not a refuted constraint.
9.3 What the window does owe the baptism
Not its width, but its ranking. The baptism band 27-29 determines how far the early edge runs: 6 BC survives only baptism 27 (at 28 the child is 33, outside the band); 5 BC survives 27 and 28 and falls at 29; 4 BC survives the whole band. The core of the window thus does not hang on the choice of 27 alone. 5 BC has the most surviving combinations — both seasons of death and baptism 27 and 28 — and is therefore the most probable, on robustness; 6 BC leans on one baptism year and 4 BC on one season of death, but neither carries a demonstrable price, they are only less robust. The ages are year cells, baptism year + birth year − 1. With the boundary of Tiberius' 15th year on 20 October 27 the baptism falls late in the autumn of 27 or in the winter of 27/28; only the former is baptism 27. If a baptism in January or February 28 counts as baptism 28, 6 BC drops out in year cells; in completed years it stands, at 32 years.
At the other end of the axis the crucifixion year (AD 31/33) remains unseen. The obstacle there is the weekday: Smallwood and Hoehner exclude AD 31 solely via a Friday filter. The baptism axis ends on an UNDECIDED, not on an exclusion. What it gives the window is two removed cells — a postulate (8) and the soft edge (7) — and a ranking among the cells that remain.