{"id": "c13", "title": "C13 — Two calibrations control the extended 529 alignment", "filename": "490d_C13_Symbolic_868_460_Calibration_and_529_Closure_20260922.md", "role": "Research action", "step": 13, "sha256": "e67199d029d683c7fc3afb3c347c3ae1c1bf45509318aab707757a25dac38a9c", "bytes": 12570, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/symbolic_868_460_sensitivity_20260922/490d_C13_Symbolic_868_460_Calibration_and_529_Closure_20260922.md"], "download": "/sources/e67199d029d683c7fc3afb3c347c3ae1c1bf45509318aab707757a25dac38a9c.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Inverse", "Mirror / 529"], "excerpt": "C13 — Two calibrations control the extended 529 alignment\n\n22 September 2026 · Bounded symbolic continuation of C12.\n\nThe extended packet has two independent calibration directions. Releasing the measured 868 attachment and the cumulative 460 insertion shows which equations persist by construction and which equalities ", "content": "# C13 — Two calibrations control the extended 529 alignment\n\n22 September 2026 · Bounded symbolic continuation of C12.\n\n**The extended packet has two independent calibration directions.** Releasing the measured 868 attachment and the cumulative 460 insertion shows which equations persist by construction and which equalities require their source values. The 11/2-family alignment fixes the insertion; the 17-family and mixed 20-fold midpoint then fix the attachment. Several prominent checks are therefore different expressions of the same two conditions.\n\nThe **total mixed 20×529 width alone does not fix its midpoint**. Those two requirements together recover both source calibrations in this formal family. A further diagnostic matters: the persistence of 5+5 depends on retaining the equal-interval relation used in C12's compact basis. Holding the directly documented regular +190 profile instead exposes a half-span mismatch away from the 460 calibration.\n\nThis is a sensitivity test of the existing equations. No source lifespan, Gear, profile, event date or canonical file is changed.\n\n**The two released quantities.** In C12's principal N02 packet set\n\n\\[\n\\alpha=b-c=868+x,\n\\qquad\\kappa=c_- -u=460+y.\n\\]\n\nOnly the right-hand sides of R09 and R11 are released. The other C12 coefficients, endpoint assignments, reflection rules and 3/2 and 5 allocation ratios are retained, together with a=−4444. Thus the source packet is **x=y=0**. The symbols x and y do not introduce authorized chronological variants.\n\nThe held Creation/regular core is unchanged: b=−4419, g=2458, n=3056, h=3108 and s=3706. Exact solution of the eighteen-variable system gives the moving coordinates below.\n\n| Endpoint | Signed paired-center expression |\n|---|---|\n| c, cumulative LXX OFF later | −5287−x |\n| c₋, cumulative LXX OFF earlier | −5885−x |\n| u, cumulative LXX ON earlier | −6345−x−y |\n| q, reflected Conquest candidate | 1400−y |\n| v, reflected alternate MT Flood candidate | 2648−y |\n| A, cumulative MT ON Flood candidate | −5293−3y/2 |\n| Y, BC reflection of v | −2648+y |\n| Z, Christ-period landing candidate | −3+7y/2 |\n\nThe actual BC Noah coordinate X=−3706 is held. All entries are formal signed coordinates; off-calibration values do not inherit the named source labels automatically. The paired Mirror axis remains January AD1, with no civil year zero.\n\nThe selected ordinary span remains **L=598**, while\n\n\\[\n\\boxed{E=c-u=1058+y,\\qquad K=E/2=529+y/2.}\n\\]\n\nK is a useful algebraic unit, but it equals the project's fixed **23²=529** only when y=0. The two must be kept distinct during the diagnostic.\n\n**Persistent relations retain the structure that was held.** Throughout the compact symbolic extension,\n\n\\[\nq-b+E=g-b=6877,\n\\]\n\\[\nX-A=3K,\\qquad Y-X=2K,\\qquad Z-Y=5K,\n\\]\n\\[\nY-A=Z-Y=5K,\\qquad Z-X=7K.\n\\]\n\nThe 3+2+5 refinement and the 5+5 equality therefore persist in this system. They preserve the supplied allocations and equal-interval correspondence; their persistence is not a new independent prediction of a particular historical span.\n\nThe original held Creation chain also remains **6877→7475→8125**, with its two 25/23 steps. Those source endpoints were not varied. The interval-image displacement v−q remains **1248**, by the retained correspondence equations.\n\nReleasing the attachment necessarily releases other numerical calibrations that C06's basis derived from it:\n\n\\[\nh-c=8395+x,\\quad q-u=7745+x,\\quad s-c=v-u=8993+x.\n\\]\n\nThus the 868 release does not simultaneously hold the original 8395 Flood Mirror width fixed. Adding **h−c=8395** as a further fixed constraint would immediately impose x=0. This is a change of which equations are held for the diagnostic, not a new transport rule.\n\n**Exact 529 multiples require specific calibrations.** The following table uses the fixed unit 529, not the moving K.\n\n| Quantity | Symbolic value | Condition for the displayed source multiplier |\n|---|---|---|\n| Held Creation→MT Flood, 13×529 | 6877 | Always held |\n| Creation→Conquest, 11×529 | 5819−y | y=0 |\n| Selected comparison E, 2×529 | 1058+y | y=0 |\n| Cumulative→SP Noah, 17×529 | 8993+x | x=0 |\n| 3+7 outer interval, 10×529 | 5290+5y | y=0 |\n| Mixed cumulative total, 20×529 | 10580+x+3y/2 | 2x+3y=0 |\n| Mixed interval's second half, 10×529 | 5290+x−7y/2 | 2x−7y=0 |\n\nThe literal 3-, 5- and 7-fold lengths likewise require y=0. All departures in this table lie in a two-dimensional linear space. For example, **−y** from the 11-fold width and **x** from the 17-fold width form an independent pair; E supplies the negative of the first departure rather than a third independent condition.\n\n**The mixed 20-fold midpoint supplies a separate condition from total width.** Its two actual halves are\n\n\\[\nH_L=Z-A=5290+5y,\n\\qquad H_R=-c-Z=5290+x-\\frac72y.\n\\]\n\nTheir difference is\n\n\\[\n\\boxed{H_R-H_L=x-\\frac{17}{2}y.}\n\\]\n\nHence the landing Z is the midpoint precisely when **2x−17y=0**. Requiring both the original total width and that midpoint gives\n\n\\[\n2x+3y=0,\\qquad2x-17y=0\n\\quad\\Longrightarrow\\quad\\boxed{x=y=0.}\n\\]\n\nWithin this selected affine family, the total and center jointly recover **α=868, κ=460**. This is an inverse characterization of the known packet, not new independent historical evidence for its inputs.\n\nThe distinction is visible in exact formal controls:\n\n| Diagnostic | α | κ | Creation→Conquest | Noah Mirror width | Mixed total | Mixed halves |\n|---|---:|---:|---:|---:|---:|---|\n| Source packet | 868 | 460 | 5819 | 8993 | 10580 | 5290 / 5290 |\n| Attachment alone +1 | 869 | 460 | 5819 | 8994 | 10581 | 5290 / 5291 |\n| Insertion alone +2 | 868 | 462 | 5817 | 8993 | 10583 | 5300 / 5283 |\n| Preserve total 10580 only | 865 | 462 | 5817 | 8990 | **10580** | **5300 / 5280** |\n| Preserve centering with K=530 | 885 | 462 | 5817 | 9010 | 10600 | 5300 / 5300 |\n\nOnly the first row is the published source calibration. In the fourth, the total is still 20×529 but the landing is not the midpoint. In the last, the Noah and mixed widths equal 17K and 20K for K=530; the held Creation→MT Flood width remains 6877, whereas 13K would be 6890.\n\n**Using a moving unit exposes the shared residual.** Define\n\n\\[\n\\rho=x-\\frac{17}{2}y.\n\\]\n\nThen\n\n\\[\n(s-c)-17K=\\rho,\n\\]\n\\[\n(-A-c)-20K=\\rho,\n\\qquad(-c-Z)-10K=\\rho,\n\\qquad H_R-H_L=\\rho.\n\\]\n\nThese four conditions are equivalent in the compact system. The 17-fold alignment, mixed 20-fold alignment, second 10-fold half and Christ-period midpoint do not supply four independent tests.\n\nThe other independent residual is\n\n\\[\n(q-b)-11K=(g-b)-13K=-\\frac{13}{2}y.\n\\]\n\nTogether these two residuals again force y=0 and x=0. A change of unit can retain some proportional relations, while the independently held Creation chain supplies the remaining calibration condition.\n\n**C12's two equivalent source bases give different off-source tests.** At x=y=0, C12 showed that either of the following may be used as a basis relation:\n\n- The equal-interval correspondence **s−v=c−u**.\n- The directly documented regular displacement **v−g=190=130+60**.\n\nTheir equivalence used the actual cumulative insertion 460 and the retained MT/SP difference 650. If κ is released, they cannot both remain fixed unless y=0.\n\n| Diagnostic choice | Alternate MT coordinate v | Regular profile displacement | Difference between the two 5-fold halves |\n|---|---|---|---|\n| Retain equal-interval R12 | 2648−y | 190−y | 0 |\n| Hold documented MT +190 profile | 2648 | 190 | 2y |\n\nIn the second calculation, the equal-interval equation has residual y and the interval-image displacement becomes 1248+y. The first and second 5-fold halves are **2645+3y/2** and **2645+7y/2**. They coincide at the source calibration.\n\nThis qualifies the robustness claim: **5+5 persists because the compact extension keeps its supporting correspondence**. It is not invariant under every way of varying the insertion while holding source information. The mixed 20-fold total and its Christ-period midpoint condition are unchanged by this choice, since they do not use the intermediate regular Flood point v/Y. The source's MT ON/+60 profile remains 190 throughout the actual research packet.\n\n**Divisibility alone has an infinite formal lattice; the held source fields select its origin.** Exact named multipliers are stronger than merely being divisible by 529. For whole-year Nisan endpoints, x is integral and y must be even because A and Z have half-integer coefficients of y. Divisibility of E and the Noah Mirror width requires y and x to be multiples of 529. Consequently the complete locus for the listed base-stage 529 widths is\n\n\\[\n\\boxed{x=529m,\\qquad y=1058n,\\qquad m,n\\in\\mathbb Z.}\n\\]\n\nThe multipliers become:\n\n| Width | Multiplier of 529 on this formal lattice |\n|---|---|\n| Held Creation→MT Flood | 13 |\n| Creation→Conquest | 11−2n |\n| E | 2+2n |\n| Noah Mirror | 17+m |\n| 3+7 outer interval | 10+10n |\n| Mixed total | 20+m+3n |\n| Mixed second half | 10+m−7n |\n\nMixed centering additionally requires **m=17n**. This concerns the base 529 carrier; the held 7475 and 8125 widths retain their separate 575 and 625 carrier roles.\n\nThe source fields sharply limit these formal alternatives. The cumulative OFF later member is **5287+x**, and the held field 5283–5290 requires **−4≤x≤3**. The reflected Conquest candidate is **1400−y**, and its held field 1399–1406 requires **−6≤y≤1**. These intervals contain no nonzero multiples of 529 or 1058, respectively. Thus the lattice intersects these two source bounds only at **m=n=0**, recovering the published calibration.\n\nMembership without the divisibility or specified-member conditions is weaker. At y=0, all three cumulative endpoints c/c₋/u remain inside their fields for integer **−2≤x≤3**—six boundary selections, α=866 through 871. Their Noah Mirror widths are 8991 through 8996, with 8993 the sole multiple of 529. These are a bounded membership control, not six new admitted joins. Only x=0 retains N02's specified cumulative members. The original source certificate and the fixed Cainan lifespan already identify the actual values; the modular calculation does not replace that provenance.\n\n**What this establishes.** The fixed Creation chain, the selected interval correspondences and the chosen partition ratios account for the identities that persist in the compact algebra. The exact square-of-23 lengths and the mixed midpoint require two calibrated inputs. Several family checks share one residual, so their number should not be treated as a count of independent constraints. The conditional full-week 300→299 compression is not used or established by this calculation.\n\nAll source restrictions remain active: regular Gears only at Noah/Shem/Flood; SP companions only at Noah/Shem; independent Flood start/close; fixed Creation fields; distinct regular and cumulative Cainan operations; no promotion of generated points to historical events. C12's N04 apparent-boundary and companion-expansion exclusions remain intact. The symbolic calculation does not revise the five sourced mixed cumulative intervals at x=y=0.\n\n**Evidence and checkpoint.** `evidence/check_c13.py` passes **114 exact checks**. It solves the frozen C12 matrix with affine right-hand sides, verifies polynomial coefficients rather than inferring identities from samples, compares the two basis choices, derives the residual ranks and divisibility lattice, and checks the finite source bounds. `C13_SYMBOLIC_CERTIFICATE.json` records every endpoint, width and residual coefficient. `C13_RESULTS.json` contains the formal controls and bounded membership diagnostic. Eight frozen inputs include controlling File_18/File_09 and the relevant C11/C12 material and original field register. Both result files reproduce byte for byte in an isolated replay; SHA256 manifests accompany the packet.\n\nNo canonical source or historical graph was changed, and no completed endpoint campaign was rerun. The completed continuation is **C01–C13**, alongside recovered original Reports **01–10**. Interrupted original Reports **11–25** remain unestablished as completed; C13 is a continuation supplement. The previously recorded seasonal interpretation and 5292 typo confirmation remain unresolved.\n\n**Best next bounded step:** test whether the **12×529 family** can attach through C10/C12's **2 BC landing**, supplying a contiguous source path for **2+10=12** and clarifying its relation to the companion branch. Keep its later 6900/7500 realizations separately qualified. **High is sufficient.**\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-45851004c5864229", "title": "File 09"}]}