{"id": "c271", "title": "The SP three-Gear radius register", "filename": "490d_C271_SP_Three_Gear_Radius_Register_20260924.md", "role": "Research action", "step": 271, "sha256": "1cff67d282534eb4887a9561a59fc09e073aefda7c6e612c4d49966333616780", "bytes": 11734, "origins": ["C01–C479/packets/C271/490d_C271_SP_Three_Gear_Radius_Register_20260924.md"], "download": "/sources/1cff67d282534eb4887a9561a59fc09e073aefda7c6e612c4d49966333616780.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Inverse", "Counts & NT", "Mirror / 529"], "excerpt": "C271 — The SP three-Gear radius register\n\n24 September 2026. Bounded continuation of C270 in the existing 490d Unified Chronology research. Capture now, canonize later.\n\nThe fixed C270 maps produce six source-bound reflection pairs, with consecutive radii 1 through 6. Primary Noah/Flood-start pairs supply the even radi", "content": "# C271 — The SP three-Gear radius register\n\n24 September 2026. Bounded continuation of C270 in the existing 490d Unified Chronology research. Capture now, canonize later.\n\n**The fixed C270 maps produce six source-bound reflection pairs, with consecutive radii 1 through 6.** Primary Noah/Flood-start pairs supply the even radii 2, 4 and 6; companion Noah/Flood-close pairs supply the odd radii 1, 3 and 5. No offset or comparison scale changes between rows.\n\nEvery mapped endpoint has an existing cumulative LXX OFF counterpart. The correspondence covers annual indices 2–7 of the two cumulative fields; it does not cover their opening indices 0–1. Of all 36 Noah/Flood cross-pairs, exactly six preserve the held local reflection. This both extends C270 and identifies the limits of that extension.\n\n## 1. Fixed domain and inherited maps\n\nThe domain contains twelve regular SP records: six Noah birth records, primary and companion in G1–G3, and six independently typed Flood boundaries, start and close in G1–G3. These are the retained C01 records in the full-430 comparison frame, checked against File_18 §§3.1, 3.1.4 and 3.2.1. Native SP values receive the already declared +215 conversion.\n\n| Gear | Noah primary / companion BC bases | Flood start / close BC bases |\n|---|---|---|\n| G1 | 3706 / 3705 | 3106 / 3105 |\n| G2 | 3708 / 3707 | 3108 / 3107 |\n| G3 | 3710 / 3709 | 3110 / 3109 |\n\nThe person companion and the Flood close are different source mechanisms. The former is the SP-local adjacent Noah station; the latter belongs to the independently defined one-year Flood event. Calling their relation “companion/close” never creates a Flood companion rail.\n\nAs in C270, let y be a reflected regular pair center. A regular BC pair with Nisan base B has reflected pair AD Bt/(B+1)n, center y=B and actual components y±1/4. These are the inherited comparison conventions, not a new claim that an annual source coordinate establishes a historical birthday or an otherwise unstated season.\n\nHold the maps:\n\n\\[\nT(q)=q+8993,\n\\qquad R_N(y)=y-3703,\n\\qquad R_F(y)=3105-y,\n\\]\n\n\\[\nJ_1(x)=2-x,\n\\qquad \\eta(x)=1000(x-1).\n\\]\n\nThe Noah branch corresponds to cumulative Flood; the regular Flood branch corresponds to cumulative Shem translated and reflected. The Noah shift remains 3703=7K, from 17K−10K with K=529. Global Mirror remains M(x)=−x about January AD 1; the local junction x=1 remains January AD 2.\n\nNoah deaths, a new regular Shem map, other regular traditions, and new cumulative or upstream Gears are outside this test.\n\n## 2. The complete six-pair correspondence\n\nLet j be the annual source index from the opening of each C266 LXX OFF field. This OFF field remains the inherited −460 derivative of native ON; membership means a registered annual boundary in that field, not a separately attested date for every member. The signed cumulative source centers are\n\n\\[\nq_{\\mathrm{cum\\ Flood}}=-5290+j,\n\\qquad q_{\\mathrm{cum\\ Shem}}=-5890+j.\n\\]\n\nUnder the held translation and branch maps, their images are\n\n\\[\nN_j=3703+j,\\qquad F_j=3103+j,\n\\]\n\n\\[\nR_N(N_j)=j,\\qquad R_F(F_j)=2-j.\n\\]\n\nThus equal annual selectors give a 600 separation on both source sides and a local pair symmetric about 1. The six admitted rows are:\n\n| j | Gear and binding | Reflected Noah / Flood centers | Cumulative Flood / Shem BC bases | Local Noah / Flood centers | Radius ρ |\n|---:|---|---|---|---|---:|\n| 2 | G1 companion / close | 3705 / 3105 | 5288 / 5888 | 2 / 0 | 1 |\n| 3 | G1 primary / start | 3706 / 3106 | 5287 / 5887 | 3 / −1 | 2 |\n| 4 | G2 companion / close | 3707 / 3107 | 5286 / 5886 | 4 / −2 | 3 |\n| 5 | G2 primary / start | 3708 / 3108 | 5285 / 5885 | 5 / −3 | 4 |\n| 6 | G3 companion / close | 3709 / 3109 | 5284 / 5884 | 6 / −4 | 5 |\n| 7 | G3 primary / start | 3710 / 3110 | 5283 / 5883 | 7 / −5 | 6 |\n\nThese are six source-bound comparisons derived through one fixed translation. They are not six independently attested chronologies or six new independent 17K witnesses. The newly systematic comparisons do not by themselves certify stored directed paths in the C127 graph. C269’s inherited path qualifications remain in force.\n\nFor Gear g, write c=0 for primary/start and c=1 for companion/close. The selector and radius reduce to\n\n\\[\nj=2g+1-c,\\qquad \\rho=j-1=2g-c.\n\\]\n\nThis is the useful reduction: Gear supplies the two-unit spacing; the correctly paired companion and closing states fill the intervening radii. It assigns no Gear to the cumulative field itself.\n\n## 3. All 36 bindings are classified\n\nFor any selected reflected Noah center N and Flood center F, define the failure of the held reflection by\n\n\\[\n\\Delta=R_F(F)-J_1(R_N(N))=N-F-600.\n\\]\n\nLet c_N=1 for Noah companion, otherwise 0; let c_F=1 for Flood close, otherwise 0. Then\n\n\\[\n\\boxed{\\Delta=2(g_N-g_F)-c_N+c_F.}\n\\]\n\nBecause c_N and c_F are binary, Δ=0 requires the same Gear and the same selector value. Therefore exactly six of the 36 cross-pairs pass. The other thirty have explicit nonzero residuals; none can be repaired by switching the seasonal labels.\n\nWithin a single Gear:\n\n| Noah / Flood binding | Source difference | Δ | Local pair midpoint |\n|---|---:|---:|---:|\n| Primary / start | 600 | 0 | 1 |\n| Companion / close | 600 | 0 | 1 |\n| Companion / start | 599 | −1 | 1/2 |\n| Primary / close | 601 | +1 | 3/2 |\n\nThe 599 comparison retains its qualified ordinal 600th-year reading. It does not become 600 completed years. The 601 comparison is a cross-boundary span, not a revised Noah age. Their different midpoints are diagnostics; they do not authorize replacement Mirror axes.\n\nAcross all Gears, residuals run from −5 to +5 with counts 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. The complete matrix records every outcome, including failures. These counts summarize a finite arithmetic test, not a statistical measure of historical intentionality.\n\n## 4. Seasonal components and the 25/23 boundary\n\nFor matching reflected component δ=±1/4,\n\n\\[\nR_N(N_j+\\delta)=j+\\delta,\n\\]\n\n\\[\nR_F(F_j+\\delta)=2-j-\\delta=J_1(j+\\delta).\n\\]\n\nThe Flood branch reverses order. Both branches preserve the actual half-year pair width. For every mismatched pairing, the same Δ remains in both component equations. No phase restoration or reassignment is used.\n\n| ρ | Local Noah pair | Local Flood pair, chronological order | Positive η components | Outer / inner ratio |\n|---:|---|---|---|---|\n| 1 | AD 2t/3n | 1t BC / AD 1n | 750 / 1250 | 5/3 |\n| 2 | AD 3t/4n | 2t/1n BC | 1750 / 2250 | 9/7 |\n| 3 | AD 4t/5n | 3t/2n BC | 2750 / 3250 | 13/11 |\n| 4 | AD 5t/6n | 4t/3n BC | 3750 / 4250 | 17/15 |\n| 5 | AD 6t/7n | 5t/4n BC | 4750 / 5250 | 21/19 |\n| 6 | AD 7t/8n | 6t/5n BC | 5750 / 6250 | 25/23 |\n\nEach Flood image supplies the corresponding negative η components, in reversed order. In general, the positive bracket is 1000ρ±250, and its outer/inner ratio is\n\n\\[\n\\frac{4\\rho+1}{4\\rho-1}.\n\\]\n\nAmong the six admitted rows, this equals E=25/23 only at ρ=6. The C270 G3 primary/start closing bracket therefore remains the specific row carrying 5750/6250. The five inner pairs provide a bounded comparison showing that this ratio is not shared by every Gear or companion state. The selection still depends on the inherited maps, radius convention and source register.\n\nThe G1 companion/close row maps the Flood center to x=0. Its components are −1/4 and +1/4, displayed **1t BC / AD 1n**. This is the global Mirror origin in the coordinate system, not a civil year zero. The local reflection center remains x=1.\n\nThe evidence ledger preserves all 24 components through original regular BC, reflected regular, cumulative and local coordinates. As in C270, original Noah Nisan reaches local Tishri and vice versa; the regular Flood ancestry passes through two reversals and retains its original seasonal label. Shared local labels do not identify the original events.\n\n## 5. Exact coverage and its gaps\n\nEach cumulative field has eight annual boundary records, j=0 through 7. The finite SP register covers six, j=2 through 7, spanning **five elapsed years between the first and last selected centers**. This is not a six-year interval or the complete seven-year cumulative field.\n\nThe omitted source records would require the following reflected regular centers under the unchanged map:\n\n| j | Cumulative Flood / Shem BC bases | Required Noah / Flood centers | Status in held SP register |\n|---:|---|---|---|\n| 0 | 5290 / 5890 | 3703 / 3103 | Both absent |\n| 1 | 5289 / 5889 | 3704 / 3104 | Both absent |\n\nThe cumulative records exist; the corresponding regular source records are not supplied by this register. These calculated images remain generated coordinates. No additional Gear, companion or individual offset is introduced to fill them.\n\nThe selected local center sets are [2,7] on the Noah branch and [−5,0] on the Flood branch, at integer steps. Neither includes the local junction 1. There is a further source distinction: the omitted j=0 images lie at local 0 and 2, which already occur in the selected j=2 row with the cumulative roles reversed. That numerical co-location does not fill the missing source ancestry. Coverage must be counted by source record and branch, not just by distinct plotted positions.\n\n## 6. Assessment and verification\n\nC270’s outer closing pair extends to a finite, orderly register. The admitted source bindings generate radii 1–6 with one rule, ρ=2g−c. Equality of cumulative annual selectors is equivalent to the retained 600 relation and to local J₁ symmetry. The complete cross-pair test establishes where that equivalence fails.\n\nThis is a structural consolidation of existing source data. It does not independently derive C266’s original 5290/5888 translations, establish a universal chronology action, merge cumulative Shem with regular Flood, or turn generated coordinates into historical AD events. The seasonal-ratio comparison singles out the existing outer row within this finite domain; it is not a probability argument.\n\nThe packet freezes ten immediate inputs. **271 exact assertions pass**, and six generated outputs reproduce byte for byte in separate main and independent-review directories. Independent mathematical and source-binding reviews confirm the six matches, all thirty failures, phase transport and coverage limits. Final review found no material mathematical or provenance errors. The report and restart checkpoint are authored documents, not computational replay outputs.\n\nThe certificate continues C270. Integrity records bind the sources and deliverables. File_18, File_09, the C265 checkpoint certificate and the C127 graph remain unchanged; no historical campaign is rerun. Regular Gears remain Noah/Shem/Flood only. Keep K=529, E=25/23, P=70/69 and ρ=radius. Both primers remain quarantined. No canonical amendment, cumulative/upstream Gear, z-law or new global axis is asserted.\n\nC267’s 2500/7500 classifications, the complementary 7000/7500 interpretations, C268’s distinct scaled charts and C269’s source-path qualifications remain retained. Original unprefixed Reports 01–10 and interrupted, unestablished original Reports 11–25 remain distinct from the C-series. The superseded AD 1905–1911 lookup remains unexecuted.\n\n## 7. Best next bounded step\n\n**C272 — compare these six source-bound radius families with C268’s fixed 1196/1300 charts.** Use only the six positive local pairs and their centers, the two inherited C268 anchors, and the frozen regular event register. Classify which chart images land on admitted source records and which remain generated. This tests how the 17K correspondence connects to the 13K/598/650 construction without fitting another anchor, scale or offset. **High is sufficient. C272 has not been executed.**\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-45851004c5864229", "title": "File 09"}]}