# 490d — C502–C511: sequential research record 27 September 2026. Working continuation of C481. File52c’s latest supplied draft controls the rounded investigation. No canonical sources are changed. Second-order inversion remains deferred. Each numbered entry records its question, declared inputs, exact results, finding, and reassessment. These are bounded research actions, not claims of independent discoveries. Parallel agents supplied source extraction and critical review; the main sequence was evaluated in order. ## C502 — Cross-Key coincidence at2926 **Question.** Which part of the Jared-E / Noah-P agreement follows from the calendar ratios? **Source basis.** File52c §3.15.3; C500–C501 **Frozen inputs** ```json { "A": 12026, "Jared_radius": 8372, "Noah_radius": 8970 } ``` **Results** ```json { "E_over_P": "15/14", "source_radius_ratio": "15/14", "Jared_E_endpoint": 2926, "Noah_P_endpoint": 2926, "necessary_relation": "D_E,A(x)=D_P,A(y) iff (A-y)/(A-x)=E/P, for nonzero radii" } ``` **Finding.** The equality is forced by the supplied14:15 radius ratio and E/P=15/14. It is one dependent consequence of the calendar calibration and source placement, not an additional freely selected match. The initial coefficient14 remains a source constraint. **Reassessment.** Test which portions of the equal-step field transfer to LXX and SP at the unchanged anchor. ## C503 — Cross-tradition range of the598 mechanism **Question.** Does the whole J/N/F progression transfer, or only its local Noah–Flood interval? **Source basis.** File18 §§2.1.3,3.1–3.2,4.1; local Gear tables; File52c §3.15 **Frozen inputs** ```json { "Jared_NoahG1_FloodG2": { "MT": [ 3654, 3056, 2458 ], "LXX": [ 4534, 3836, 3238 ], "SP_equalized": [ 3954, 3706, 3108 ] }, "fixed_anchor": 12026 } ``` **Results** ```json { "MT": { "Jared_to_Noah": 598, "Noah_to_Flood": 598, "radii": [ 8372, 8970, 9568 ], "radius_residues_mod23": [ 0, 0, 0 ] }, "LXX": { "Jared_to_Noah": 698, "Noah_to_Flood": 598, "radii": [ 7492, 8190, 8788 ], "radius_residues_mod23": [ 17, 2, 2 ] }, "SP_equalized": { "Jared_to_Noah": 248, "Noah_to_Flood": 598, "radii": [ 8072, 8320, 8918 ], "radius_residues_mod23": [ 22, 17, 17 ] } } ``` **Finding.** The selected NoahG1–FloodG2 interval598 transfers to all three. The preceding interval is598,698,248 respectively, so the complete14/15/16 pattern and its three-Key integrality do not transfer. Local compatibility survives without whole-field identity. **Reassessment.** Check the tenfold529 placement on the same fixed cross-tradition states. ## C504 — Cross-tradition range of the tenfold529 placement **Question.** Does12026=1446+10(Creation−NoahG1) transfer to the declared comparison states? **Source basis.** File18 comparison rows; File52c §§3.14–3.15; C481 **Frozen inputs** ```json { "MT": [ 4114, 3056 ], "LXX": [ 5494, 3836 ], "SP_equalized": [ 4414, 3706 ] } ``` **Results** ```json { "MT": { "Creation_Noah_width": 1058, "tenfold_width": 10580, "residual_against_10580": 0, "width_mod529": 0 }, "LXX": { "Creation_Noah_width": 1658, "tenfold_width": 16580, "residual_against_10580": 6000, "width_mod529": 71 }, "SP_equalized": { "Creation_Noah_width": 708, "tenfold_width": 7080, "residual_against_10580": -3500, "width_mod529": 179 } } ``` **Finding.** Only the selected MT width1058=2×529 satisfies the fixed tenfold placement. LXX1658 and SP708 miss by6000 and−3500 after scaling. The shared529 duration rule can still organize other sourced families without making this MT input universal. **Reassessment.** Derive that duration rule independently of endpoints and compare C480’s smaller ladder with the rounded macro ladder. ## C505 — The529 family is first a duration grammar **Question.** What does two-stage E expansion guarantee without any pivot choice? **Source basis.** C480 checkpoint; File52c §§3.8–3.9,6; C481 **Frozen inputs** ```json { "E": "25/23", "small_seed": 1058, "macro_seed": 10580 } ``` **Results** ```json { "small": [ 1058, 1150, 1250 ], "macro": [ 10580, 11500, 12500 ], "general_integer_width_rule": "k*529 -> k*575 -> k*625", "proof": "D_k,a(y)-D_k,a(x)=k*(y-x); second pivot cancels from the width too" } ``` **Finding.** Every supplied529-multiple width follows this ladder under two E expansions. Tenfold scaling commutes with it. Width agreement therefore connects C480 and the rounded macro family while leaving their source roles and pivots distinct. **Reassessment.** Derive the additional congruence condition needed for integral endpoints. ## C506 — Integral endpoints require pivot compatibility **Question.** What extra condition distinguishes an integral width ladder from integral dates? **Source basis.** Strategy §5C; C505; exact algebra **Frozen inputs** ```json { "map": "D_E,a(x)=(25x−2a)/23", "integer_source_and_pivot": true } ``` **Results** ```json { "one_stage_iff": "x congruent a mod23", "two_stage_condition": "first outputs integral; second pivot congruent to every first output mod23", "pair_condition": "if original width belongs to529Z, integral first outputs share one residue mod23", "same_pivot_two_stage_iff": "x−a divisible by529", "proof": "25x−2a ≡2(x−a) mod23 and2 is invertible mod23; for a repeated pivot D_E,a²(x)=a+(625/529)(x−a), gcd(625,529)=1" } ``` **Finding.** Widths lose the pivot term; endpoints retain it. A529-multiple width guarantees a shared second-stage residue after an integral first stage, but it does not supply a licensed pivot of that residue. **Reassessment.** Apply this criterion to the already supplied C480 pivots and the fixed12026 trial, without choosing new pivots. ## C507 — Explain successful and fractional endpoint completions **Question.** Does the pivot-residue theorem explain the two recorded529 realizations? **Source basis.** C480 checkpoint; C481 report; C506 **Frozen inputs** ```json { "C480": { "pair": [ 2148, 3206 ], "first_pivot": 3620, "second_pivot": 2756 }, "Actual_fixed_A": { "pair": [ 4114, 3056 ], "first_pivot": 12026, "second_pivot": 12026 } } ``` **Results** ```json { "C480": { "first": [ 2020, 3170 ], "first_mod23": [ 19, 19 ], "second_pivot_mod23": 19, "second": [ 1956, 3206 ], "fractional_parts": [ 0, 0 ], "second_width": 1250 }, "Actual_fixed_A": { "first": [ 3426, 2276 ], "first_mod23": [ 22, 22 ], "second_pivot_mod23": 20, "second": [ "61598/23", "32848/23" ], "fractional_parts": [ "4/23", "4/23" ], "second_width": 1250 } } ``` **Finding.** C480’s first endpoints and second pivot all have residue19, so the supplied completion is integral. The fixed12026 trial has first residue22 against pivot20, leaving a common4/23 fraction. Both widths still reach1250. This explains the recorded difference without adding a pivot. **Reassessment.** Compare another family where two different Keys produce the same gain: the SP rectangle. ## C508 — SP equal-gain rectangle **Question.** Can the calendar grammar reconstruct the Strategy’s separately declared SP rectangle? **Source basis.** Strategy §4.3 **Frozen inputs** ```json { "Strategy_ReportIV_comparison_heads": [ 4206, 13406 ], "Exodus": 1446, "Conquest": 1406, "state_note": "historical Strategy SP−215 rectangle; not the File18 completion pair used in C483" } ``` **Results** ```json { "input_radii": [ 2760, 11960 ], "P_and_J_outputs": [ 2800, 12000 ], "gains": [ 40, 40 ], "preserved_difference": [ 9200, 9200 ], "E_difference": 10000, "general_rule": "69m ->70m and299m ->300m; common gain m and preserved separation230m" } ``` **Finding.** At m=40, two distinct Keys add the same40, preserving9200 while terminating at the shared1406 target with the two heads held. This is a reusable equal-gain construction, separate from the newer File18 completion comparison. **Reassessment.** Derive how anchor changes affect otherwise identical expansions, needed to relate the14726 family’s two presentations. ## C509 — Anchor-change and composition laws **Question.** Which affine relations explain a change of anchor without choosing a new ratio? **Source basis.** Strategy §5C; File52c §3.11 **Frozen inputs** ```json { "E": "25/23", "anchors": [ 14726, 6 ], "sample_inverse_Shem": 5526 } ``` **Results** ```json { "difference_law": "D_k,a(x)-D_k,b(x)=(1-k)(a-b)", "conjugacy": "T_t D_k,a = D_k,a+t T_t", "same_anchor_composition": "D_l,a D_k,a = D_lk,a", "different_anchor_order": "D_l,b D_k,a − D_k,a D_l,b = (1-l)(1-k)(b-a)", "Shem_Creation_held": 4726, "Shem_Nativity_held": 6006, "Nativity_minus_Creation": 1280 } ``` **Finding.** Changing only the anchor shifts every output by the same amount. The14726 versus6 E-presentations differ1280 throughout. Same-anchor ratios compose multiplicatively; different anchors generally make order matter. **Reassessment.** Keep this affine geometry distinct from the two coordinate conventions used by Mirror comparisons. ## C510 — Mirror requires a declared coordinate chart **Question.** How can Mirror join the affine grammar without confusing civil and rounded crossings? **Source basis.** File51a coordinate clarification; File52c §§1,3.12,6 **Frozen inputs** ```json { "civil": "q(BC B)=1−B; q(AD Y)=Y", "rounded": "q(BC B)=1−B; q(AD Y)=Y−1", "sample_pair": [ 601, 1351 ] } ``` **Results** ```json { "civil_crossing": 1951, "rounded_crossing": 1950, "rounded_reflection": "q -> −q carries BC B to AD B in this display", "affine_compatibility": "M D_k,a = D_k,M(a) M for M(q)=−q", "source_roles": "ordinary same-side reflection, cross-polarity Mirror, and decimal reversal remain distinct operators" } ``` **Finding.** Reflection is affine once its coordinate chart is declared. Civil and rounded opposite-side spans differ by one; they cannot be pooled as one metric. The rounded origin is a comparison convention, not a historical year zero. **Reassessment.** Reconstruct the rounded macro529 endpoint ladder in that declared chart. ## C511 — The rounded macro529 endpoint realization **Question.** Does the reflected12026 anchor complete the macro ladder with the declared rounded coordinates? **Source basis.** File52c §§3.8–3.9; C505,C510 **Frozen inputs** ```json { "anchor_AD": 12026, "starting_AD": 1446, "widths": [ 10580, 11500, 12500 ] } ``` **Results** ```json { "anchor_q": 12025, "outputs_q": [ 1445, 525, -475 ], "display": [ "AD 1446", "AD 526", "476 BC" ], "successive_widths": [ 10580, 11500, 12500 ], "civil_first_width_if_substituted": 10580 } ``` **Finding.** The supplied rounded Mirror ladder is AD1446→AD526→476BC, with AD12026 held. This is a licensed endpoint realization of the tenfold duration ladder. The last crossing uses the rounded convention; a civil substitution would produce a different final display. **Reassessment.** Use the14726 anchor-change law to reconstruct the complete ten-member family rather than selected coincidences. ## Evidence The accompanying evidence checkpoint contains the sequential journal, exact-arithmetic step scripts, source snapshots and SHA256 manifest. Latest step: C511. Assertions verify arithmetic or explicit scope decisions; they are not independent historical or statistical witnesses.