{"id": "c267", "title": "The2500 midpoint and5750/6250 seasonal bracket", "filename": "490d_C267_529_Midpoint_and_Seasonal_Bracket_20260924.md", "role": "Research action", "step": 267, "sha256": "07d7eb37f73ed352637fa31148a4636945bc30ff1914ad679651697ad8579521", "bytes": 12615, "origins": ["C01–C479/packets/C267/490d_C267_529_Midpoint_and_Seasonal_Bracket_20260924.md"], "download": "/sources/07d7eb37f73ed352637fa31148a4636945bc30ff1914ad679651697ad8579521.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Rounded", "Mirror / 529"], "excerpt": "C267 — The 2500 midpoint and the 5750/6250 seasonal bracket\n\n24 September 2026. Continuation of the existing 490d Unified Chronology research through redirected C266. Finite comparison completed; capture now, canonize later.\n\nThe fixed C266 comparison gives 2500 a generated midpoint realization and gives 5750→6250 an e", "content": "# C267 — The 2500 midpoint and the 5750/6250 seasonal bracket\n\n24 September 2026. Continuation of the existing 490d Unified Chronology research through redirected C266. Finite comparison completed; capture now, canonize later.\n\n**The fixed C266 comparison gives 2500 a generated midpoint realization and gives 5750→6250 an exact source-projected seasonal-radius realization of the Key ratio 25/23. The 7500 branch remains a whole seasonal-envelope length in this comparison.** These results strengthen the connection with Dean’s new axis report while preserving the different geometric and source roles.\n\nThe main gain is specific to the admitted LXX OFF fields. Their whole-field midpoints map to ±2500, while their radius-six endpoint pairs contain the seasonal components mapping to ±5750 and ±6250. No additional source dates or historical event identities are needed.\n\n## 1. Fixed question and source domain\n\nC266 established the local comparison chart\n\n\\[\n\\eta(x)=1000(x-1),\\qquad x=1+\\frac{\\eta}{1000}.\n\\]\n\nHere x is the signed center coordinate measured from January AD 1. Center x=1 is January AD 2, displayed by the pair AD 1t/2n. This is the local junction used for comparison; the global Mirror remains M(x)=−x. Its related local reflection is J₁(x)=2−x.\n\nC267 freezes C266’s six core fields. F means translated cumulative Flood; RH means cumulative Shem translated and then reflected by M. Each field has eight annual boundary records and seven years between its first and last pair centers.\n\n| Profile | F centers | RH centers | Selected outer center radius about x=1 |\n|---|---|---|---:|\n| MT ON | [−6,1] | [1,8] | 7 |\n| SP OFF | [−1,6] | [−4,3] | 5 |\n| LXX OFF | [0,7] | [−5,2] | 6 |\n\nThe core therefore contains 48 source-tagged pair records. The 24 previously recorded, unreflected Shem records remain auxiliary. Coincident coordinates retain their separate source tags.\n\nThe primary targets are the signed markers ±2500, ±6250 and ±7500. The companion ±5750 is included because the supplied axis report already explicitly states both 5290→5750→6250 and the 6000±250 bracket in §15.2. Its numerical bracket is inherited, not claimed as a new discovery here.\n\nMidpoint tests are restricted to the six complete fields and the two admitted SP radius-five arms. There is no arbitrary search for pairs that happen to average to a desired number. The classifications are: source-projected pair center, actual seasonal component, generated midpoint, or whole-envelope length.\n\n## 2. Exact classification\n\nFor an integer pair center x, the actual seasonal components are x−1/4 and x+1/4, in chronological Tishri/Nisan order. Applying the fixed chart gives:\n\n| Marker | Negative and positive preimages x | Realization within the fixed comparison |\n|---|---|---|\n| ±2500 | −3/2, 7/2 | Generated midpoints of RH_LXX and F_LXX; also half of the selected SP radius-five arms |\n| ±5750 | −19/4, 27/4 | Actual inner seasonal components of the LXX radius-six endpoint pairs |\n| ±6250 | −21/4, 29/4 | Actual outer seasonal components of those same LXX pairs |\n| ±7500 | −13/2, 17/2 | Outside the core phase hull; 7500 is realized instead as a full seasonal-envelope length |\n\nNone of these four markers is an integer pair-center radius in this chart. The distinctions follow directly from exact coordinates, rather than rounding or approximate visual alignment.\n\nAn arithmetic cross-check is useful. Images of integer centers are congruent to 0 modulo 1000; images of actual quarter-year components are congruent to 250 or 750. Both ±2500 and ±7500 are congruent to 500. They therefore cannot be existing pair centers or actual seasonal components, even when another generated construction supplies their locations.\n\n## 3. The 2500 midpoint realization\n\nThe LXX translated Flood field runs from center 0 to 7. Its midpoint is 7/2, whose distance from the junction at 1 is 5/2. The reflected Shem field runs from −5 to 2; its midpoint is −3/2, at the opposite distance of 5/2.\n\n\\[\n\\eta\\!\\left(\\frac{0+7}{2}\\right)=2500,\n\\qquad\n\\eta\\!\\left(\\frac{-5+2}{2}\\right)=-2500.\n\\]\n\nThe source endpoints are retained:\n\n| Field | Source endpoint pairs, before the inherited operations | Operation | Resulting center endpoints |\n|---|---|---|---|\n| LXX Flood | 5291t/5290n BC; 5284t/5283n BC | Translate by 5290 | 0, 7 |\n| LXX Shem | 5891t/5890n BC; 5884t/5883n BC | Translate by 5888, then reflect | 2, −5 |\n\nAmong the six whole-field midpoints, MT supplies ±3500, SP supplies ±1500, and LXX supplies ±2500. LXX is thus the unique whole-field match in this fixed six-field comparison. This is not a corpus-wide uniqueness claim.\n\nThe same generated locations are halfway along the selected SP arms [−4,1] and [1,6]. That is a second description of the same coordinate result, not an independent chronological witness.\n\nSeasonal labels need care at these half-integer centers. Averaging the corresponding lower and upper endpoint components produces:\n\n| Generated field midpoint | Phasewise midpoint coordinates | Physical seasonal labels |\n|---|---|---|\n| F_LXX, center 7/2 | 13/4, 15/4 | AD 4n, AD 4t |\n| RH_LXX, center −3/2 | −7/4, −5/4 | 2n BC, 2t BC |\n\nThese are July-centered Nisan-to-Tishri pairs. They must not be relabeled as January-centered t/n pairs merely because they were constructed as center±1/4. Their components individually coincide with admitted seasonal points, but the generated midpoint center is not thereby a newly sourced event.\n\nThis supplies a geometric realization of the converted 2500 marker from the inherited ladder 2116→2300→2500. It does not identify the midpoint’s source history with that macro conversion history.\n\n## 4. The 5750/6250 bracket realizes 25/23 in seasonal radii\n\nThe positive LXX outer pair has center 7 and physical components AD 7t and AD 8n. Relative to the junction at 1, their radii are 23/4 and 25/4. The reflected negative pair has center −5 and the same absolute radii in reverse chronological order.\n\n| Signed chart marker | Exact x | Current physical component | LXX source component and inherited operation |\n|---:|---:|---|---|\n| −6250 | −21/4 | 6t BC | Shem 5883n BC; translate 5888, then reflect |\n| −5750 | −19/4 | 5n BC | Shem 5884t BC; translate 5888, then reflect |\n| +5750 | 27/4 | AD 7t | Flood 5284t BC; translate 5290 |\n| +6250 | 29/4 | AD 8n | Flood 5283n BC; translate 5290 |\n\nBoth pairs derive from closing source members, index 7. Reflection reverses the inherited component order: the negative Tishri component comes from source Nisan, and the negative Nisan component comes from source Tishri.\n\nConsequently,\n\n\\[\n5750=23\\times250,\\qquad6000=24\\times250,\\qquad6250=25\\times250,\n\\]\n\n\\[\n\\boxed{\\frac{6250}{5750}=\\frac{25/4}{23/4}=\\frac{25}{23}=E.}\n\\]\n\nDean’s macro ladder remains\n\n\\[\n5290=10\\times529\\xrightarrow{E}5750\\xrightarrow{E}6250.\n\\]\n\nThe new connection is now explicit: **its second conversion ratio is the outer-to-inner seasonal-radius ratio at the source-projected LXX radius-six pair.** The source pair has chart center 6000 and component separation 500. The negative pair supplies the bilateral counterpart.\n\nEach of these four components also coincides with one MT interior component. The evidence ledger preserves both source histories per match. Those co-locations neither merge the underlying events nor create additional independent witnesses.\n\nThere is a precise limit to this realization. The radial dilation corresponding to E in this chart would be\n\n\\[\ng(x)=1+E(x-1).\n\\]\n\nIt sends the positive inner component 27/4 to the positive outer component 29/4. But applying it to the entire half-year pair changes that pair’s width from 1/2 to 25/46, and its next image leaves the actual quarter-year phase lattice. The matching radius ratio therefore does not establish an E action preserving all six source fields or their phase spacing.\n\nThe values 5750 and 6250 remain comparison labels here. Their presence does not assert historical events in AD 5750 or AD 6250, or identify the successive macro conversion states as one historical seasonal pair.\n\n## 5. The 7500 branch retains its length role\n\nAll six fields span seven years between first and last centers. Adding their outer quarter-year components gives each a full seasonal-envelope length of 7.5 years, which scales to 7500.\n\nFor the two MT fields adjoining at center 1, the envelopes are\n\n\\[\n[-6.25,1.25],\\qquad[0.75,8.25].\n\\]\n\nEach has length 7.5. Together their lengths sum to 15, but they overlap by 0.5 and their outer union has length 14.5. At the declared 1000:1 scale these quantities are 15000, 500 and 14500 respectively.\n\nThe core phase hull is [−6.25,8.25], or [−7250,7250] after applying η. The preimages of ±7500 lie a further quarter-year beyond each end. They cannot be core centers, actual components, or midpoints of any core points, because midpoints remain inside that hull.\n\nThe auxiliary unreflected MT Shem field does contain a generated adjacent-center midpoint at the negative preimage −6.5. It supplies no positive counterpart at 8.5. This separately recorded one-sided observation does not extend the symmetric core domain.\n\nThe 7500/Year 15 reading and the axis report’s separately centered macro diagram remain available on their stated constructions. Within this finite source comparison, 7500 has the whole-envelope role; 7000 remains the pair-center radius associated with the 14-year center span. The sum of two envelope lengths must retain its distinction from their union.\n\n## 6. What the combined ladder now supports\n\n| Value | Role in the fixed comparison | Status |\n|---:|---|---|\n| 2500 | LXX whole-field midpoint radius; half SP radius-five arm | Generated location |\n| 5000 | Selected SP outer pair-center radius | Source-projected center |\n| 5750 | LXX inner seasonal radius | Source-projected component |\n| 6000 | Selected LXX outer pair-center radius | Source-projected center |\n| 6250 | LXX outer seasonal radius | Source-projected component |\n| 7000 | Selected MT outer pair-center radius | Source-projected center |\n| 7500 | Whole seven-year field’s seasonal-envelope length | Length |\n\nThis is a connected comparison with several geometric roles. Treating every entry as the same kind of event endpoint would lose the specific result. The LXX midpoint and phase bracket are the added source-traceable connections; the inherited arithmetic and the local/global distinction remain intact.\n\n## 7. Evidence and controls\n\nThe packet freezes eight immediate dependencies, including C266’s certificate, report, checkpoint, endpoint ledger and results; Dean’s new axis report; and the controlling File_09 and File_18. Their original paths, byte counts and SHA-256 values are recorded in `SOURCE_MANIFEST.json`.\n\nThe verifier uses exact rational arithmetic. **138 assertions pass, and all six generated output files reproduce byte for byte in a separate output directory.** The evidence includes the full source-tagged phase ledger, marker classification, midpoint provenance, envelope calculation, executable verifier and independent mathematical review. Assertion counts describe verification coverage, not independent witnesses. The report is authored interpretation; the six replayed files are explicitly identified in `REPLAY.json`.\n\nThe new branch report’s separately mentioned 855-check package was not supplied and is not claimed as replayed. No historical expansion campaign was rerun. The C267 certificate links to C266; the artifact manifest covers the packaged payloads. Current source/control and prior graph hashes remain unchanged.\n\nKeep K=529, E=25/23, P=70/69 and ρ for radius. Regular Gears remain Noah/Shem/Flood only; no Gear is used in this step. No cumulative Gear, new global axis, z-law, canonical source amendment or primer reopening is introduced. The original unprefixed Reports 01–10 remain distinct from C01 onward; interrupted original Reports 11–25 remain unestablished. Existing source qualifications and missing joins are unchanged.\n\n## 8. Best next bounded step\n\n**C268 — compare the LXX seasonal-radius realization with the two retained source-qualified 6877→7475→8125 chains.** Retrieve their existing C01 and C206 anchor/center records selectively. Test source roles, fixed anchors and component phases alongside the shared 25/23 ratios, and distinguish a coordinate rescaling from a shared chronological endpoint map. This reconnects the new local result to the established 13×529 core without reopening the broader search. **High is sufficient. C268 has not been executed.**\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-45851004c5864229", "title": "File 09"}]}