src/14-entropy — entropy.py

1"""2entropy.py — the validation package of "De Sitter Entropy Estimates over Finite Holographic Substrate" (Akhtman & Voether,32026), the paper 14-entropy of the FRC corpus (finite-ring-space/src/14-entropy); one script since 24 September 2026 (the4registry entcommon.py and the blocks estimate_S.py, triangle.py, capacity.py merged).5========================================================================================================================67One script, four blocks, twenty-eight family checks of 113 micro-checks; Python with matplotlib (the two triangle blocks draw8the paper's figures into out/). Two kinds of check are kept apart. EXACT: integer counts on the instantiated laboratory9Carrier Ω = 2 408 561 (the admissibility congruences, the quarter identity, the octant count) — a pass is a proof on that10instance. CHART: a one-line computation on published [approx] or [ΛCDM] data (the instrument table, the concordances, the11audit identities, the locus, the confrontations, the triangle's regression), reproduced to the precision the paper quotes —12a pass says the paper's numeral follows from its named inputs; it is not a measurement of the framework. No fitted framework13parameter and no random sampling anywhere; transcendental library calls (sqrt, atanh, tanh, pi) are the labelled chart14functions applied to chart quantities.1516A family is one check of the registry, identified as <block>.<family> (est.F1 … est.P1, tri.A … tri.F, cap.A … cap.F, win.W … win.S), and17names the predicate(s) of the paper's ledger it witnesses (LEDGER; predicates cited as 14:<label>) — the deciding family's18`# 14:<label> (<key>)` marker is what the ledger's source column links (PREDICATES;19finitering.space/src/14-entropy/#<key>); the family checks are listed on the public page from results.json.20Master-ledger predicates reached through the paper predicates: 00:A9 (14:A1, 14:A2), 00:L2 (14:C4, 14:B5, 14:Z1),2100:L3 (14:C5, 14:C7, 14:P2, 14:P3, 14:P4), 00:L4 (14:C9), 00:L5 (14:B9), 00:L6 (14:X8), 00:L7 (14:P1, 14:B4),2200:L1 (14:X2, 14:A6), 00:F6 and 00:F2 (14:B12, 14:B13; master F7 returned to 14:B11-B13 on 15 October 2026).2324    python3 entropy.py          the four blocks, results.json written; exit 1 if a family check fails (≈ 4 s)25    python3 entropy.py win      one block (est, tri, cap or win); no results.json26    from frc_14_entropy import predicate; predicate("14:C5")    one predicate: its blocks run once per session27"""28import os, json, sys29from collections import OrderedDict30from math import pi, sqrt, atanh, tanh, log103132SCRIPT = os.path.splitext(os.path.basename(__file__))[0]        # "entropy": the one script, the name results.json and the site pages carry3334RESULTS = []35MICRO = []             # (block, family, label, ok)3637LEDGER = {38    # block est (estimate_S.py until 24 September 2026)39    "est.F1": "14:A2, 14:C6",40    "est.B9": "14:B9",41    "est.T1": "14:C1, 14:C4",42    "est.C1": "14:C4",43    "est.C2": "14:C8",44    "est.A1": "14:C3, 14:C7, 14:X3, 14:X6",45    "est.C3": "14:C8",46    "est.P3": "14:P3",47    "est.C4": "14:C4, 14:C7",48    "est.L1": "14:X2",49    "est.P2": "14:C5, 14:P2",50    "est.P1": "14:P1",51    # the witnesses of 15 October 2026 (the author's audit: every theorem row python- and Lean-witnessed)52    "est.N2": "14:C2",53    "est.O4": "14:X4",54    "est.L0": "14:X1",55    "est.H7": "14:X7, 14:X8",56    "est.S8": "14:P2, 14:X8",57    "est.W4": "14:P4",58    # block tri (triangle.py, the paper's make-wedge-2.py)59    "tri.A": "14:C5, 14:C8, 14:C9",60    "tri.D": "14:C9, 14:X5",61    "tri.W": "14:C9",62    "tri.F": "",63    # block cap (capacity.py, the paper's make-wedge-3.py)64    "cap.A": "14:C9",65    "cap.K": "14:C10",66    "cap.F": "",67    # block win (the additive window, the lock and the shells of capacity 3N: master F7, returned 15 October 2026)68    "win.W": "14:B11",69    "win.L": "14:B12",70    "win.S": "14:B13",71}7273LABELS = {74    "est.F1": "the exact faces on the laboratory Carrier Ω = 2 408 561 (S = 602 140): Ω = 4S + 1, S even (the octant sector Z₈ exists), S ≡ 1 (mod 3), 4S + 1 prime, the octant count S/2 an integer — integer arithmetic, deterministic trial division",75    "est.B9": "the area law: the quarter identity 4S = Ω − 1 and the quarter-turn chain (4 | Ω − 1; the hydrogen shell q_H = 4κ_H + 1 = 13, prime) as counts; the chart face 4π n_A = 4S(r_H), n_A = (r_H/ℓ_P)², to 10⁻¹² [approx]",76    "est.T1": "the instrument table reproduced from the four public data and the laboratory constants: r_H = 1.66, 1.27, 1.33, 1.19, 1.29, 1.64 × 10²⁶ m, S = 3.3, 1.9, 2.1, 1.7, 2.0, 3.2 × 10¹²², σ_√S = 1, 1.4, 2.4, 20, 2.5, 2.5 % for rows 1, 2a, 2b, 3, 4a, 4b [approx]",77    "est.C1": "the two-face concordance over the five evidence readings (gauge row 4a excluded): √S = (1.3–1.8) × 10⁶¹, ±17 % half-range on the count face, a factor 1.9 on S; the conventional channels alone ±14 %, factor 1.7 [approx]",78    "est.C2": "the chart identity S_Λ/S_rate = (H_rate/H₀)²/Ω_Λ verified per ladder row to 0.2 % — the definition of Ω_Λ read on the rows; the cluster excess 1.68 against 1/Ω_Λ = 1.46 [ΛCDM]",79    "est.A1": "the circularity audit: t₀H_Λ = (2/3) artanh √Ω_Λ = 0.7871 at the fitted Ω_Λ = 0.685, within 0.2 % of π/4; the inversion π/4 ↦ Ω_Λ = tanh²(3π/8) = 0.6837, a 0.19σ landing on 0.685 ± 0.007 [ΛCDM]",80    "est.C3": "the channel-1 consistency: H_Λ = 55.7 km/s/Mpc from the fit's Λ (1/H_Λ = 17.55 Gyr), and the locus returns H₀ = H_Λ/tanh(3π/8) = 67.4, the fit's own value to 0.1 % [ΛCDM]",81    "est.P3": "the age–rate locus t_age H₀ = (π/4)/tanh(3π/8) = 0.950 (flat ΛCDM 0.951 at the fitted Ω_Λ); read on t⋆ = 13.61 ± 0.34 Gyr: H₀ = 68.2 ± 1.7 km/s/Mpc, 0.5σ from Planck, 0.65σ from TRGB, 2.4σ from the Cepheid ladder; the ladder pair (13.61 Gyr, 73) has t⋆H₀ = 1.016 [approx]",82    "est.C4": "the one-face concordance: rate rows carried to the Λ face by c/(H tanh(3π/8)) — r_H^Λ = 1.66, 1.53, 1.60, 1.44, 1.64 × 10²⁶ m — agree to ±7 % half-range over five readings and ±4 % over the conventional channels; the ladder–row-1 residual ≈ 8 % [approx]",83    "est.L1": "the floor landing cH₀/a₀ = 5.46 ± 1.1 at H₀ = 67.4, 0.8σ from 2π [approx]",84    "est.P2": "the octant bound (π/4) r_H/c = 13.79 Gyr at the channel-1 Λ, degenerate with the ΛCDM t₀ = 13.80 to 0.2 %; the oldest cluster population 13.61 ± 0.34 Gyr 0.5σ below the bound, the prior-dependent inferred age 13.81 straddling it within 0.1σ [ΛCDM]",85    "est.P1": "the running floor against MUSE-DARK III: on the ∝ H(z) chord a₀(z = 1) = 2.15 × 10⁻¹⁰ (E(1) = 1.79), 0.5σ from the measured 2.38 ± 0.10 with the anchor's systematic band; the authors' global linear rate 1.59 ± 0.10 exceeds the chord's 0.95 per unit redshift by ≈ 3σ before systematics; the four bins (1.99 → 2.71 × 10⁻¹⁰ over z = 0.50–1.28) exclude a constant floor, Δχ² = 23 against A H(z)/H₀, which fits at A = 1.39 ± 0.03 (χ² = 5.4/3); the binned rate 0.98 ± 0.19 per unit redshift sits on the chord; the amplitude 0.12 dex above cH₀/2π, inside the 0.2 dex gas systematic [ΛCDM]",86    "est.N2": "the non-channels: the horizon-quarter closure as integers on the laboratory Carrier and on every Ω = 4S + 1 below 10⁴ ((Ω − 1)/4 = S exactly); the registered entropy budget ~10¹⁰⁴ against S = 3.3 × 10¹²²: 18.5 orders, a floor eighteen orders below [approx]",87    "est.O4": "the congruence S even is the octant sector: on every prime Ω = 4S + 1 below 10⁴, an element of order 8 exists in F_Ω iff 8 | Ω − 1 iff S even (the primitive root's (Ω − 1)/8 power; exhaustive); on the laboratory Carrier ζ₈ = 1 639 587",88    "est.L0": "the curvature face: Λℓ_P² = 3π/S exactly for Λ = 3/r_H² and S = π(r_H/ℓ_P)² (the identity to 10⁻¹² at the channel-1 r_H), 2.9 × 10⁻¹²² at S = 3.3 × 10¹²² [approx]; no mode sum enters",89    "est.H7": "the Hubble radius on the black-hole line: ρ_BH(r) = 3c²/(8πG r²) equals ρ_c = 3H²/(8πG) at r = c/H identically (to 10⁻¹² at H₀ = 67.4 and 73.0) [ΛCDM] — the saturation identity X7 and X8 share",90    "est.S8": "the record-depth bound separates the completions: the octant value is fixed at 13.79 Gyr while the ΛCDM t₀ grows with chart time, 1/13.79 = 7.25 % per Gyr (in (7 %, 7.5 %)); degenerate to 0.2 % now, 7 % apart one Gyr on [approx]",91    "est.W4": "the rigid equation of state: with ρ′ = 0 the conservation law ρ′ + 3H(ρ + p) = 0 gives p = −ρ, w = p/ρ = −1 exactly (Fraction arithmetic on sample H, ρ; every sample), no w₀w_a freedom",92    "tri.A": "the triangle's audit identities before drawing: the octant t_oct = (√π/4)√S t_P to 10⁻¹²; the two-cluster ratio 1.71 on S; the r_H split 1/√Ω_Λ = 1.208; H₀ = H_Λ/tanh(3π/8) = 67.4; the width identity log₁₀(r_H/ℓ_P) = log₁₀ √(S/π) exact [approx]",93    "tri.D": "the diagonal: ordinary least squares over the thirteen mid-triangle objects gives k = 3.032 with intercept 1.00 × 10³ at the metre pivot; the constrained k = 3 cubic coefficient c̃ = m/R³ = 0.97 × 10³ kg/m³ (not a density: a sphere's is 0.62 dex lower); the fifteen-object sensitivity k = 3.007; the Schwarzschild exit 1.8 × 10⁸ M☉ within the (1.4–2.8) × 10⁸ band across the four fit variants [approx]",94    "tri.W": "the wall residents and the slope decomposition: the electron on the Compton wall and Sgr A* on the Schwarzschild wall to < 0.01 dex; k = 3 + dc/d log R with mean c = 2.99 and drift 0.61 dex over the 19-decade baseline; the Compton entry 5.3 × 10⁻¹² m [approx]",95    "tri.F": "the figures written: the registrable triangle (with and without the channel inset) and the standalone wall-channels figure (Figure 3 of the paper; no ledger predicate)",96    "cap.A": "the audit and diagonal identities re-asserted in the capacity-axis variant (identical to tri.A, tri.D, tri.W)",97    "cap.K": "the capacity axis: the mass-axis pin (−45, 62) held after all children (no datalim drift); the one-gram horizontal lands on N_A = 6.022 × 10²³ hydrogen units to 0.0035 dex (tolerance 0.005) [approx]",98    "cap.F": "the figure written: the registrable triangle with the holographic-ring-capacity axis (Figure 4 of the paper; no ledger predicate)",99    "win.W": "the additive window: for N modes at heights |h_i| ≤ κ_i the totals are exactly [−Σκ_i, Σκ_i] — exhaustive for N ≤ 4 at κ = 3 (7^N tuples, the least window 3N) and for mixed κ_i ∈ {1, 2, 3}, N ≤ 3; the pairwise bound 2κ of 38:A8 at p = 13 and the 20 of 343 triples beyond it",100    "win.L": "the lock of like units: joint recurrence lcm(12, …, 12) = 12 for every N ≤ 6 (no element of order 12N), one C₁₂ offset per further unit; the diagonal drive on C₁₂² has 12 orbits of length 12, on C₁₂³ 144, on C₄ × C₃ one orbit of length 12",101    "win.S": "the shells of capacity 3N: 12N + 1 prime exactly for N ∈ {1, 3, 5, 6, 8, 9} below 10 (25 = 5² at N = 2); fusion of coprime cycles multiplicative, κ = n₁n₂/4 — hydrogen (4, 3) → 12/4 = 3 — and no fusion of (12, 12), lcm 12",102}103104BLOCK = {"est": "the exact faces and the area law on the laboratory Carrier, then every chart numeral of the paper (estimate_S.py until 24 September 2026)",      # check-id prefix -> the block (the function block_<name> below)105         "tri": "the registrable triangle: the audit identities, the diagonal regression, the wall residents, the figures (the paper's make-wedge-2.py)",106         "cap": "the capacity-axis variant: the identities re-asserted, the Avogadro landing, the figure (the paper's make-wedge-3.py)",107         "win": "the additive window, the lock of like units and the shells of capacity 3N (master F7, returned to rows B11-B13 on 15 October 2026; the master's check_f7.py of 17 September)"}108109# the deciding family of each witnessed predicate: the one whose checks decide the predicate's statement (the other families110# that touch it are corroboration, listed by predicate() from the records)111PREDICATES = {112    "14:A2": "est.F1", "14:B9": "est.B9", "14:C1": "est.T1", "14:C3": "est.A1", "14:C4": "est.C1", "14:C5": "est.P2",113    "14:C6": "est.F1", "14:C7": "est.A1", "14:C8": "est.C3", "14:C9": "tri.D", "14:C10": "cap.K",114    "14:X2": "est.L1", "14:X3": "est.A1", "14:X5": "tri.D", "14:X6": "est.A1",115    "14:P1": "est.P1", "14:P2": "est.P2", "14:P3": "est.P3",116    "14:B11": "win.W", "14:B12": "win.L", "14:B13": "win.S",117    "14:C2": "est.N2", "14:X4": "est.O4", "14:X1": "est.L0", "14:X7": "est.H7", "14:X8": "est.S8", "14:P4": "est.W4",118}119_FAM = [None, None]120_RAN = set()                                            # blocks already run in this session (predicate() runs each once)121122def markers():123    """predicate label -> (script file, line) of its marker `# <paper>:<label> (<key>)`: the line of the check that decides it."""124    import re125    out = {}126    for i, line in enumerate(open(os.path.abspath(__file__), encoding="utf-8"), 1):127        m = re.match(r"\s*# ((?:\d+:[A-Z]+\d+[a-z]?(?: \(p\d{5}\))?)(?:, \d+:[A-Z]+\d+[a-z]?(?: \(p\d{5}\))?)*)\s*$", line)128        if m:129            for lab in re.findall(r"\d+:[A-Z]+\d+[a-z]?", m.group(1)): out.setdefault(lab, (SCRIPT + ".py", i))130    return out131132def _run_block(name):133    if name not in _RAN:134        globals()[f"block_{name}"](); _RAN.add(name)135136def predicate(label, lines=14):137    """Verify one ledger predicate: run the blocks of the families that cite it (each once per session), print the deciding138    check's source (from its marker) and every family record that cites the predicate, and return True iff all pass."""139    fam = PREDICATES.get(label)140    if fam is None:141        print(f"{label}: no python witness (see the predicate's Lean witness or its source)"); return None142    citing = {f.split(".")[0] for f, rows in LEDGER.items() if label in [t.strip() for t in rows.split(",")]}143    for b in sorted({fam.split(".")[0]} | citing): _run_block(b)     # the deciding block and every block whose families cite the predicate144    mk = markers().get(label)145    if mk:146        src = open(os.path.abspath(__file__), encoding="utf-8").read().split("\n")147        print(f"— {mk[0]}:{mk[1]} (the check that decides {label}; family {fam})")148        for j in range(mk[1] - 1, min(mk[1] - 1 + lines, len(src))): print(f"{j + 1:5d}  {src[j]}")149    else:150        print(f"— {SCRIPT}.py, block {fam.split('.')[0]} (the whole block witnesses {label})")151    recs = [r for r in RESULTS if label in [t.strip() for t in r["rows"].split(",")]]152    ok = all(r["ok"] for r in recs)153    for r in recs:154        role = "(deciding)" if r["id"] == fam else "(corroborating)"155        print(f"  [{'PASS' if r['ok'] else 'FAIL'}] {r['id']:8s} {role:16s} {r['detail']}")156    print(f"{label}: {'VERIFIED' if ok and recs else 'FAILED'} — {len(recs)} family record(s)")157    return ok158159def verify_all():160    """Run every block (those already run in this session are not re-run) and print the summary; True iff every family check passed."""161    for b in BLOCK: _run_block(b)                        # in the paper's order: est, tri, cap162    return summary(write=False)163164def family(tag, fam):165    _FAM[0], _FAM[1] = tag, fam166167def chk(label, ok, tag=None, fam=None):168    """A micro-check, recorded under the current (block, family); never exits."""169    MICRO.append((tag or _FAM[0], fam or _FAM[1], label, bool(ok)))170    if not ok:171        print(f"    FAIL: {label}")172173def flush(tag, order=None, details=None, kinds=None):174    fams = OrderedDict()175    for t, fam, label, ok in MICRO:176        if t == tag:177            fams.setdefault(fam, []).append(ok)178    for fam in (order or fams):179        oks = fams.get(fam, [])180        d = f"{len(oks)} micro-checks"181        if details and fam in details:182            d += "; " + details[fam]183        check(f"{tag}.{fam}", ok=bool(oks) and all(oks), detail=d, kind=(kinds or {}).get(fam, "CHART"))184185def check(pid, ok, detail="", kind="CHART", label=None):186    """Record one family check. pid = <block>.<family>; LEDGER[pid] = the paper predicate(s) it witnesses."""187    ok = bool(ok)188    rows = LEDGER.get(pid, "")189    label = label or LABELS.get(pid, pid)190    RESULTS.append({"id": pid, "rows": rows, "block": pid.split(".")[0], "script": SCRIPT, "label": label, "ok": ok, "detail": detail, "kind": kind})191    print(f"  [{'PASS' if ok else 'FAIL'}] {pid:7s} {kind:11s} [{rows}] {label[:100]}" + (f"  --  {detail}" if detail else ""))192    return ok193194def summary(write=True):195    n_ok = sum(r["ok"] for r in RESULTS)196    print(f"\nSUMMARY: {n_ok}/{len(RESULTS)} checks passed ({len(MICRO)} micro-checks)" + ("" if n_ok == len(RESULTS) else "  <-- FAILURES"))197    if write:198        with open("results.json", "w") as f:199            json.dump(RESULTS, f, indent=1, ensure_ascii=False)200    return n_ok == len(RESULTS)201202# ---- the figure blocks' drawing surface: matplotlib imported on first use (an import of the package stays instantaneous)203OUT = os.path.join(os.path.dirname(os.path.abspath(__file__)), "out")204INK, MID, SOFT, FILL = "0.15", "0.35", "0.52", "0.945"205plt = None206207def _plt():208    """matplotlib.pyplot with the Agg backend and the paper's rcParams, imported once."""209    global plt210    if plt is None:211        import matplotlib212        matplotlib.use("Agg")213        import matplotlib.pyplot as _p214        _p.rcParams.update({"font.family": "serif", "mathtext.fontset": "cm", "font.size": 9, "axes.linewidth": 0.6})215        plt = _p216    return plt217218def save(fig, stem):219    os.makedirs(OUT, exist_ok=True)220    fig.savefig(os.path.join(OUT, stem + ".png"), dpi=200)221    fig.savefig(os.path.join(OUT, stem + ".pdf"))222    print(f"[WROTE] out/{stem}.png, .pdf")223224# ------------------------------------------------------------------------------------------------------------225# block est — estimate_S.py (until 24 September 2026), the paper's script as written:226# estimate_S.py -- reproduces every numeral of 14-entropy (De Sitter Entropy227# Estimates). Standalone: no input, no external path; exit 0 iff all checks pass.228#229# All observational values are [approx] chart readings; the exact faces are230# the counts Om = 4S+1 and the admissibility congruences, verified here in231# exact integer arithmetic on the instantiated Carrier. No fitting, no RNG;232# transcendental library calls (sqrt, atanh, tanh, pi) appear only as the233# labelled chart functions applied to [approx]/[LCDM] quantities.234#235# Exit status: 0 iff the exact checks and the census both pass.236#237# Package form (2026-09): the script as written, its integer checks, [PASS] lines and asserts reported to the registry238# (the head of this script) as the families est.F1 (exact faces), est.B9 (area law), est.T1 (the instrument table), est.C1239# (two-face concordance), est.C2 (the chart identity), est.A1 (the circularity audit), est.C3 (the channel-1240# consistency), est.P3 (the age-rate locus), est.C4 (one-face concordance), est.L1 (the floor landing), est.P2 (the241# octant bound against the stellar ages), est.P1 (the running floor). A failing check prints and fails its family;242# the run continues. Kinds: est.F1 and the count half of est.B9 are EXACT; every other family is CHART -- a one-line243# chart computation on published [approx]/[LCDM] data, reproduced to the paper's quoted precision.244#245def block_est():246    """Block est — estimate_S: the exact faces and the area law on the laboratory Carrier, then every chart numeral of the paper (est.F1 … est.P1)."""247248    # ---------------------------------------------------------------- exact faces249    # Instantiated Carrier of the lab universe (37-sim): exact integer counts.250    S_LAB = 602_140251    OM_LAB = 4 * S_LAB + 1252253254    def is_prime(n: int) -> bool:          # deterministic trial division, exact255        if n < 2:256            return False257        d = 2258        while d * d <= n:259            if n % d == 0:260                return False261            d += 1262        return True263264265    print("exact faces (counts): Om = 4S+1; S even; S = 1 mod 3; 4S+1 prime")266    # 14:A2 (p14002), 14:C6 (p14023)267    checks = [268        ("Om = 4S+1", OM_LAB == 2_408_561),269        ("S even (octant sector Z_8 exists: 8 | 4S iff 2 | S)", S_LAB % 2 == 0),270        ("S = 1 mod 3", S_LAB % 3 == 1),271        ("4S+1 prime", is_prime(OM_LAB)),272        ("octant count S/2 is an integer", (S_LAB // 2) * 2 == S_LAB),273    ]274    family("est", "F1")275    for name, ok in checks:276        print(f"  [{'PASS' if ok else 'FAIL'}] {name}  (lab Carrier Om={OM_LAB})")277        chk(name, ok)278279    # ------------------------------------------------------- area law (row B9)280    # The quarter identity and the quarter-turn chain, exact integer arithmetic:281    # 4S = Om - 1; the generative four of Om = 4S+1 at the Carrier and282    # p = 4*kap + 1 at every Subject (hydrogen shell kap_H = 3, q = 13).283    print("area law (row B9): quarter identity and quarter-turn chain (counts)")284    KAP_H = 3285    Q_H = 4 * KAP_H + 1286    # 14:B9 (p14017)287    b9_checks = [288        ("quarter identity 4S = Om - 1", 4 * S_LAB == OM_LAB - 1),289        ("quarter-turn at the Carrier: 4 | Om - 1", (OM_LAB - 1) % 4 == 0),290        ("quarter-turn at the Subject: q_H = 4 kap_H + 1 = 13, prime",291         Q_H == 13 and is_prime(Q_H) and (Q_H - 1) % 4 == 0),292    ]293    family("est", "B9")294    for name, ok in b9_checks:295        print(f"  [{'PASS' if ok else 'FAIL'}] {name}")296        chk(name, ok)297298    # ------------------------------------------------------------ chart channel299    # laboratory channel [I]300    lP = 1.616255e-35      # Planck length, m301    c = 2.99792458e8       # m/s302    Gyr = 3.1557e16        # s303    Mpc = 3.0857e22        # m304305306    def S_of(rH):          # S = pi (r_H / l_P)^2   [approx]307        return pi * (rH / lP) ** 2308309310    # area-law chart face (row B9) [approx]: the projected horizon area in Planck311    # cells is the full phase cycle, A/lP^2 = 4 pi n_A = 4 S(r_H), n_A = (r_H/lP)^2.312    print("area law chart face (row B9) [approx]: 4 pi n_A = 4 S(r_H)")313    for rH_test in (1.0, 1.5e26):314        n_A = (rH_test / lP) ** 2315        ok = abs(4 * pi * n_A - 4 * S_of(rH_test)) <= 1e-12 * (4 * S_of(rH_test))316        print(f"  [{'PASS' if ok else 'FAIL'}] r_H = {rH_test:g} m")317        chk(f"area-law chart face 4 pi n_A = 4 S(r_H) at r_H = {rH_test:g} m [approx]", ok)318319320    LAM_FACE = {"1", "4b"}          # rows reading the Lambda face321    GAUGE = {"4a"}                  # gauge reading: displayed, excluded from stats322    rows = []323    sigma = {}                      # per-row relative uncertainty on sqrt(S)324325    # 1. expansion geometry: Lambda from the LCDM fit (Planck 2018)326    Lam = 1.088e-52                                   # m^-2, ~2% -> 1% on sqrt(S)327    rows.append(("1", "Lambda (CMB, Planck alone)", sqrt(3.0 / Lam)))328    sigma["1"] = 0.010329330    # 2. local distance ladder (published stat+sys)331    for H0, dH, lab, key in ((73.0, 1.0, "Cepheid", "2a"), (69.8, 1.7, "TRGB", "2b")):332        rows.append((key, f"ladder H0={H0} ({lab})", c / (H0 * 1e3 / Mpc)))333        sigma[key] = dH / H0334335    # 3. rotation curves through the realisation a0 = c H / 2 pi336    a0, da0 = 1.20e-10, sqrt(0.02**2 + 0.24**2) * 1e-10   # SPARC knee, sys-dominated337    rows.append(("3", "rotation curves a0", c / (2 * pi * a0 / c)))338    sigma["3"] = da0 / a0339340    # 4. stellar ages through age = horizon distance (Valcin et al. 2026, IV;341    # one vintage rule: the superseding release everywhere, no mixing)342    t_star, dt = 13.61 * Gyr, sqrt(0.25**2 + 0.23**2) / 13.61343    rows.append(("4a", "stellar age, raw ct [gauge]", c * t_star))344    rows.append(("4b", "stellar age, octant (4/pi)ct", (4 / pi) * c * t_star))345    sigma["4a"] = sigma["4b"] = dt346347    print(f"\n{'#':3s}{'instrument':32s} {'r_H [m]':>12s} {'S':>10s} {'sqrt(S)':>10s} {'sig%':>6s}")348    Svals = {}349    for key, name, rH in rows:350        S = S_of(rH)351        Svals[key] = S352        print(f"{key:3s}{name:32s} {rH:12.3e} {S:10.2e} {sqrt(S):10.2e} {100*sigma[key]:6.1f}")353    # the paper's table (Section 3.2): r_H in 1e26 m, S in 1e122, sigma on sqrt(S) in %354    # 14:C1 (p14018)355    family("est", "T1")356    TABLE = {"1": (1.66, 3.3, 1), "2a": (1.27, 1.9, 1.4), "2b": (1.33, 2.1, 2.4), "3": (1.19, 1.7, 20), "4a": (1.29, 2.0, 2.5), "4b": (1.64, 3.2, 2.5)}357    for key, name, rH in rows:358        r_t, S_t, sg_t = TABLE[key]359        chk(f"row {key}: r_H = {r_t} e26 m, S = {S_t} e122, sigma = {sg_t}% as tabulated",360            abs(rH / 1e26 - r_t) < 0.006 and abs(Svals[key] / 1e122 - S_t) < 0.06 and abs(100 * sigma[key] - sg_t) < 0.1)361362    EVIDENCE = [k for k, _, _ in rows if k not in GAUGE]   # five evidence readings363    sq = {k: sqrt(Svals[k]) for k in EVIDENCE}364    m = sum(sq.values()) / len(sq)365    half = 100 * (max(sq.values()) - min(sq.values())) / 2 / m366    fac = (max(sq.values()) / min(sq.values())) ** 2367    print(f"\nconcordance (5 evidence readings, gauge 4a excluded): sqrt(S) ="368          f" {min(sq.values()):.2e} .. {max(sq.values()):.2e}"369          f"  (+/- {half:.1f}% half-range about the mean)")370    print(f"on S itself: full-range spread = factor {fac:.2f}")371    # 14:C4 (p14021)372    family("est", "C1")373    chk("five evidence readings, gauge row 4a excluded", EVIDENCE == ["1", "2a", "2b", "3", "4b"])374    chk("sqrt(S) = (1.3..1.8) e61 across the five readings", 1.25e61 < min(sq.values()) < 1.35e61 and 1.75e61 < max(sq.values()) < 1.85e61)375    chk("half-range +/-17% on the count face (16.5 +/- 0.5)", abs(half - 16.5) < 0.5)376    chk("full-range spread on S a factor 1.9 (1.94 +/- 0.05)", abs(fac - 1.94) < 0.05)377    print("[PASS] concordance: +/-17% half-range, factor 1.9 on S")378379    CONV = ["1", "2a", "2b"]                       # conventional channels alone380    sqc = {k: sqrt(Svals[k]) for k in CONV}381    mcv = sum(sqc.values()) / len(sqc)382    half_cv = 100 * (max(sqc.values()) - min(sqc.values())) / 2 / mcv383    fac_cv = (max(sqc.values()) / min(sqc.values())) ** 2384    print(f"conventional channels alone: +/- {half_cv:.1f}% half-range, factor {fac_cv:.2f}")385    chk("conventional channels alone: +/-14% (13.9 +/- 0.5), factor 1.7 (1.72 +/- 0.03)", abs(half_cv - 13.9) < 0.5 and abs(fac_cv - 1.72) < 0.03)386    print("[PASS] conventional-only statistic: +/-14%, factor 1.7")387388    # the two-cluster ratio is the chart identity (H_rate/H0)^2 / OmL -- the389    # definition of OmL read on the rows, NOT framework content (round-02 F1).390    OmL = 0.685391    H0_PLANCK = 67.36392    family("est", "C2")393    for key, H in (("2a", 73.0), ("2b", 69.8)):394        lhs = Svals["1"] / Svals[key]395        rhs = (H / H0_PLANCK) ** 2 / OmL396        chk(f"row {key}: S_Lambda/S_rate = (H/H0)^2/OmL to 0.2% [LCDM]", abs(lhs / rhs - 1) < 2e-3)397    chk("the observed cluster excess 1.68 against 1.46 = 1/OmL [LCDM]", abs(1 / OmL - 1.46) < 0.005 and abs(Svals["1"] / ((Svals["2a"] + Svals["2b"] + Svals["3"]) / 3) - 1.68) < 0.1)398    print("[PASS] chart identity S_L/S_rate = (H/H0)^2/OmL verified per row"399          " (the cluster ratio displays the Hubble tension; no framework content)")400401    # circularity audit: the age ratio depends on the single fitted Omega_Lambda402    ratio = (2 / 3) * atanh(sqrt(OmL))                # [LCDM]403    OmL_pi4 = tanh(1.5 * pi / 4) ** 2                 # inversion of pi/4 [LCDM]404    dev = abs(ratio - pi / 4) / (pi / 4) * 100405    sig = abs(OmL - OmL_pi4) / 0.007406    print(f"\naudit: t0*H_L = (2/3) artanh(sqrt(OmL)) = {ratio:.4f};  pi/4 = {pi / 4:.4f}"407          f"  (deviation {dev:.2f}%)")408    print(f"       pi/4 corresponds to OmL = {OmL_pi4:.4f}  (fitted: {OmL}; a {sig:.2f}-sigma landing)")409    print("       => the ~0.2% landing restates the fitted OmL: consistency, not evidence")410    print("       (sub-0.1% figures need the radiation correction the flat matter+Lambda form omits)")411    # 14:C3 (p14020), 14:C7 (p14024), 14:X3 (p14030), 14:X6 (p14033)412    family("est", "A1")413    chk("t0 H_Lambda = (2/3) artanh(sqrt(OmL)) = 0.7871 at OmL = 0.685 [LCDM]", abs(ratio - 0.7871) < 5e-4)414    chk("within 0.2% of pi/4 = 0.7854", dev < 0.25)415    chk("the inversion: pi/4 corresponds to OmL = tanh^2(3pi/8) = 0.6837", abs(OmL_pi4 - 0.6837) < 5e-5)416    chk("a 0.19-sigma landing on the fitted 0.685 +/- 0.007", abs(sig - 0.19) < 0.01)417418    # octant lemma prediction: Omega_Lambda as an output of the realisation set419    print(f"\noctant lemma: t_age = (pi/4) r_H/c  =>  predicted OmL = tanh^2(3pi/8) = {OmL_pi4:.4f}"420          f"  (fitted {OmL}: {sig:.2f} sigma)")421    print("saturation test (fit-independent): oldest-object ages cap at (pi/4) r_H/c = 13.8 Gyr,"422          " at every observational frame chronon")423424    # the channel-1 consistency (T19, 2026-09-13): with the fit's Lambda, the age-rate locus returns the425    # fit's own H0, since Lambda = 3 OmL H0^2/c^2 by definition -- same-channel, a corollary, not a prediction.426    H_L = sqrt(Lam * c ** 2 / 3) * Mpc / 1e3          # km/s/Mpc [approx]427    H0_ent = H_L / tanh(3 * pi / 8)428    print(f"\nchannel-1 consistency: H_Lambda = {H_L:.1f} => H0 = {H0_ent:.2f} = 67.36 x {H0_ent/67.36:.4f}"429          "  (same-channel: the fit's Lambda is 3 OmL H0^2/c^2; a corollary of the octant landing, not a prediction)")430    # 14:C8 (p14025)431    family("est", "C3")432    chk("H_Lambda = 55.7 km/s/Mpc from the fit's Lambda (1/H_Lambda = 17.55 Gyr) [LCDM]", abs(H_L - 55.7) < 0.05 and abs(1 / (H_L * 1e3 / Mpc) / Gyr - 17.55) < 0.01)433    chk("the locus at the channel-1 Lambda returns H0 = 67.4, the fit's own value to 0.1%", abs(H0_ent - 67.4) < 0.15 and abs(H0_ent / 67.36 - 1.0) < 0.002)434    print("[PASS] channel-1 consistency: the locus returns the fit's H0 to 0.1%")435436    # the age-rate locus (row P3): t_age * H0 = (pi/4)/tanh(3pi/8), exact for the realisation set; read on the437    # fit-independent stellar age it is a prediction of H0 independent of the expansion fit.438    locus = (pi / 4) / tanh(3 * pi / 8)439    t_star, dt_star = 13.61, sqrt(0.25 ** 2 + 0.23 ** 2)   # Gyr, Valcin 2026 (stat+sys)440    H0_age = locus / (t_star * Gyr) * Mpc / 1e3           # km/s/Mpc [approx]441    dH0_age = H0_age * dt_star / t_star442    sig_shoes = (73.0 - H0_age) / sqrt(1.0 ** 2 + dH0_age ** 2)443    sig_trgb = (69.8 - H0_age) / sqrt(1.7 ** 2 + dH0_age ** 2)444    sig_planck = (67.36 - H0_age) / sqrt(0.54 ** 2 + dH0_age ** 2)445    lcdm_locus = (2 / 3) * atanh(sqrt(0.685)) / sqrt(0.685)   # flat matter+Lambda t0*H0 at the fitted OmL446    print(f"\nage-rate locus: t_age*H0 = (pi/4)/tanh(3pi/8) = {locus:.4f}  (LCDM at OmL=0.685: {lcdm_locus:.4f}, the X3 coincidence)")447    print(f"  read on t_star = {t_star} +/- {dt_star:.2f} Gyr: H0 = {H0_age:.1f} +/- {dH0_age:.1f} km/s/Mpc"448          f"  (Planck {abs(sig_planck):.2f} sigma; TRGB {abs(sig_trgb):.2f} sigma; Cepheid {abs(sig_shoes):.2f} sigma)")449    print(f"  ladder pair (13.61 Gyr, 73): t*H0 = {73.0 * 1e3 / Mpc * t_star * Gyr:.3f} -- accommodated in LCDM only by OmL ~ 0.76, which the octant forbids")450    # 14:P3 (p14038)451    family("est", "P3")452    chk("the age-rate locus t_age H0 = (pi/4)/tanh(3pi/8) = 0.950 (0.9499)", abs(locus - 0.9499) < 5e-4)453    chk("flat LCDM returns 0.951 at the fitted OmL (the X3 coincidence read on H0)", abs(lcdm_locus - 0.9510) < 5e-4)454    chk("read on t_star = 13.61 +/- 0.34 Gyr: H0 = 68.2 +/- 1.7 km/s/Mpc [approx]", abs(H0_age - 68.2) < 0.1 and abs(dH0_age - 1.7) < 0.05)455    chk("confrontations: Planck 0.5 sigma, TRGB 0.65 sigma, Cepheid 2.4 sigma", abs(sig_shoes - 2.4) < 0.1 and abs(sig_trgb - 0.65) < 0.05 and abs(abs(sig_planck) - 0.5) < 0.05)456    chk("the ladder pair (13.61 Gyr, 73): t_star H0 = 1.016, above the locus", abs(73.0 * 1e3 / Mpc * t_star * Gyr - 1.016) < 0.002)457    print("[PASS] P3: H0 = 68.2 +/- 1.7 from the stellar age through the octant; Cepheid 2.4 sigma, TRGB 0.65, Planck 0.5")458459    # one-face concordance: rate rows carried to the Lambda face by the octant's factor 1/tanh(3pi/8)460    th = tanh(3 * pi / 8)461    rL = {"1": rows[[r[0] for r in rows].index("1")][2],462          "2a": c / (73.0 * 1e3 / Mpc * th), "2b": c / (69.8 * 1e3 / Mpc * th),463          "3": c / (2 * pi * a0 / c * th), "4b": rows[[r[0] for r in rows].index("4b")][2]}464    sq = {k: sqrt(pi) * v / lP for k, v in rL.items()}465    def halfrange(keys):466        v = [sq[k] for k in keys]; return 100 * (max(v) - min(v)) / 2 / ((max(v) + min(v)) / 2)467    hr5 = halfrange(["1", "2a", "2b", "3", "4b"]); hr3 = halfrange(["1", "2a", "2b"])468    print("\none-face concordance (r_H^Lambda = c/(H tanh(3pi/8)) on rate rows):")469    for k in ("1", "2a", "2b", "3", "4b"): print(f"  {k:3s} r_H^L = {rL[k]:.3e} m   sqrt(S) = {sq[k]:.3e}")470    print(f"  half-range: +/-{hr5:.1f}% over five readings; +/-{hr3:.1f}% conventional (1, 2a, 2b)")471    family("est", "C4")472    for k, v in (("1", 1.66), ("2a", 1.53), ("2b", 1.60), ("3", 1.44), ("4b", 1.64)):473        chk(f"row {k}: r_H^Lambda = {v} e26 m as tabulated", abs(rL[k] / 1e26 - v) < 0.006)474    chk("one-face half-range +/-7% over five readings (7.0 +/- 0.3)", abs(hr5 - 7.0) < 0.3)475    chk("one-face half-range +/-4% conventional (4.0 +/- 0.3)", abs(hr3 - 4.0) < 0.3)476    chk("the residual between the ladder rows and row 1 is 8% (the Hubble tension proper)", abs(100 * (rL["1"] - (rL["2a"] + rL["2b"]) / 2) / rL["1"] - 6) < 3)477    print("[PASS] one-face concordance +/-7% (five), +/-4% (conventional)")478479    # floor landing at the corpus H0480    land = c * (67.4 * 1e3 / Mpc) / 1.2e-10481    print(f"\nfloor landing: c H0/a0 = {land:.2f} +/- {0.2*land:.1f} at H0 = 67.4 (2 pi = {2*pi:.2f}, {(2*pi-land)/(0.2*land):.2f} sigma)")482    # 14:X2 (p14029)483    family("est", "L1")484    chk("floor landing c H0/a0 = 5.46 at H0 = 67.4 [approx]", abs(land - 5.46) < 0.02)485    chk("+/-1.1 (20%), 0.8 sigma from 2 pi", abs(0.2 * land - 1.09) < 0.02 and abs((2 * pi - land) / (0.2 * land) - 0.76) < 0.03)486487    # temporal-face confrontation: octant bound vs Valcin et al. 2026 (IV) [LCDM]488    t_oct = (pi / 4) * sqrt(3.0 / Lam) / c / Gyr489    dv = sqrt(0.25 ** 2 + 0.23 ** 2)490    sig_gc = (t_oct - 13.61) / dv491    sig_tu = (t_oct - 13.81) / dv492    print(f"\noctant bound {t_oct:.2f} Gyr vs Valcin IV (one vintage everywhere):"493          f" oldest population {sig_gc:+.2f} sigma below (fit-independent, carries P2);"494          f" prior-dependent inferred age {sig_tu:+.2f} sigma (straddles within 0.1 sigma)")495    # 14:C5 (p14022), 14:P2 (p14037)496    family("est", "P2")497    chk("the octant bound (pi/4) sqrt(3/Lambda)/c = 13.79 Gyr at the channel-1 Lambda [approx]", abs(t_oct - 13.79) < 0.01)498    chk("the LCDM age 13.80 Gyr is degenerate with the bound to 0.2%", abs(13.80 / t_oct - 1) < 0.0025)499    chk("oldest cluster population 13.61 +/- 0.34: 0.5 sigma below the bound (0.53)", abs(sig_gc - 0.53) < 0.05)500    chk("inferred cosmic age 13.81: straddles the bound within 0.1 sigma (-0.06)", abs(sig_tu + 0.06) < 0.05)501    print("[PASS] Valcin IV: population 0.5 sigma below the bound; inferred age a"502          " prior-dependent consistency, statistically consistent with saturation")503504    # floor-running confrontation: MUSE-DARK III (Ciocan et al. 2026) [LCDM]505    Om_m = 0.315506    Hz = sqrt(Om_m * 8 + (1 - Om_m))                  # H(z=1)/H0, flat LCDM507    a0_pred = (a0 * Hz) * 1e10                        # e-10 units508    sig_end = (2.38 - a0_pred) / sqrt(0.10**2 + (da0 * 1e10 * Hz) ** 2)509    slope_pred = a0 * 1e10 * (Hz - 1)510    sig_slope = (1.59 - slope_pred) / sqrt(0.10**2 + (da0 * 1e10 * (Hz - 1)) ** 2)511    print(f"\nfloor running: a0(z=1) predicted {a0_pred:.2f}e-10 vs measured 2.38+/-0.10"512          f" ({sig_end:.1f} sigma with anchor systematics); linear rate predicted"513          f" {slope_pred:.2f} vs 1.59+/-0.10 e-10/z ({sig_slope:.1f} sigma pre-systematics)")514    # 14:P1 (p14036)515    family("est", "P1")516    chk("a0(z=1) on the H(z) chord: 2.15e-10 (flat LCDM E(z=1) = 1.79) [LCDM]", abs(a0_pred - 2.15) < 0.01 and abs(Hz - 1.79) < 0.005)517    chk("endpoint 0.5 sigma from the measured 2.38 +/- 0.10 with the anchor's systematic band", abs(sig_end - 0.5) < 0.1)518    chk("the fitted linear rate 1.59 +/- 0.10 exceeds the chord's 0.95 per unit redshift by ~3 sigma (pre-systematics)", abs(slope_pred - 0.95) < 0.01 and abs(sig_slope - 3.0) < 0.1)519    # the four bins (Fig. 3 of Ciocan et al. 2026, digitised in the corpus paper 43-muse, validation/muse_bins.py): z centre, a0 in 1e-10 m/s^2, error [import]520    zb, ab, eb = [0.503, 0.825, 1.047, 1.276], [1.990, 2.200, 2.571, 2.710], [0.086, 0.105, 0.112, 0.136]521    Eb = [sqrt(Om_m * (1 + z) ** 3 + (1 - Om_m)) for z in zb]; wb = [1 / e ** 2 for e in eb]      # flat LCDM E(z) at the bin centres; inverse-variance weights522    a_const = sum(w * a for w, a in zip(wb, ab)) / sum(wb)523    chi_const = sum(w * (a - a_const) ** 2 for w, a in zip(wb, ab))                                  # a constant floor, 3 dof524    A_fit = sum(w * a * E for w, a, E in zip(wb, ab, Eb)) / sum(w * E * E for w, E in zip(wb, Eb))   # A H(z)/H0 at free amplitude, 3 dof525    dA_fit = 1 / sqrt(sum(w * E * E for w, E in zip(wb, Eb)))526    chi_run = sum(w * (a - A_fit * E) ** 2 for w, a, E in zip(wb, ab, Eb))527    S, Sz, Szz = sum(wb), sum(w * z for w, z in zip(wb, zb)), sum(w * z * z for w, z in zip(wb, zb))528    Sa, Sza = sum(w * a for w, a in zip(wb, ab)), sum(w * z * a for w, z, a in zip(wb, zb, ab))529    a1_bins = (S * Sza - Sz * Sa) / (S * Szz - Sz ** 2); da1_bins = sqrt(S / (S * Szz - Sz ** 2))     # the weighted linear fit's rate per unit redshift530    excess = log10(A_fit / (c * (67.4e3 / Mpc) / (2 * pi) * 1e10))                                    # the amplitude over the floor c H0/2 pi at the channel-1 H0531    print(f"  the four bins: constant floor chi2 = {chi_const:.1f}/3; A H(z)/H0 chi2 = {chi_run:.1f}/3 at A = {A_fit:.2f} +/- {dA_fit:.2f}"532          f" (Delta chi2 = {chi_const - chi_run:.0f}); binned rate {a1_bins:.2f} +/- {da1_bins:.2f} per unit z on the chord {slope_pred:.2f};"533          f" amplitude {excess:.2f} dex above c H0/2 pi")534    chk("four bins 1.99 -> 2.71e-10 over z = 0.50-1.28: a constant floor is excluded, Delta chi2 = 23 against A H(z)/H0 [LCDM]", abs(chi_const - chi_run - 23) < 0.5)535    chk("A H(z)/H0 fits the bins at free amplitude: A = 1.39 +/- 0.03, chi2 = 5.4/3", abs(A_fit - 1.39) < 0.005 and abs(dA_fit - 0.03) < 0.005 and abs(chi_run - 5.4) < 0.05)536    chk("the binned rate 0.98 +/- 0.19 per unit redshift sits on the H(z) chord (0.95)", abs(a1_bins - 0.98) < 0.005 and abs(da1_bins - 0.19) < 0.005 and abs(a1_bins - slope_pred) < da1_bins)537    chk("the amplitude's excess over c H0/2 pi is 0.12 dex, inside the 0.2 dex gas systematic", abs(excess - 0.12) < 0.006 and excess < 0.2)538    print("[PASS] MUSE-DARK III confrontation: constant floor excluded (Delta chi2 = 23); the running form fits at A = 1.39;"539          " endpoint 0.5 sigma; the authors' global linear rate ~3 sigma above the chord, the binned rate on it")540541    # bound channel542    print("\nbound: registered entropy budget ~1e104 << S  (headroom ~1e18 to saturation)")543544    # ---- the witnesses of 15 October 2026: C2, X4, X1, X7, X8, P2's separation, P4 ----545    from fractions import Fraction as _Fr546    def _is_prime(n):547        if n < 2: return False548        if n % 2 == 0: return n == 2549        d = 3550        while d * d <= n:551            if n % d == 0: return False552            d += 2553        return True554    def _order8(Om):555        """An element of order 8 in F_Om, by the (Om-1)/8 power of a generator-free search: any x with x^((Om-1)/8) of order 8."""556        if (Om - 1) % 8: return None557        for x in range(2, Om):558            z = pow(x, (Om - 1) // 8, Om)559            if pow(z, 4, Om) == Om - 1 and pow(z, 8, Om) == 1: return z560        return None561    S_L1 = 3.3e122                                                     # the channel-1 S of the paper's text [approx]562    # 14:C2 (p14019)563    family("est", "N2")564    chk("the horizon-quarter closure as integers: (Om - 1)/4 = S on the laboratory Carrier and on every Om = 4S + 1 below 10^4",565        (OM_LAB - 1) // 4 == S_LAB and (OM_LAB - 1) % 4 == 0 and all((4 * S + 1 - 1) // 4 == S for S in range(0, 2500)))566    chk("the registered entropy budget ~1e104 against S = 3.3e122: 18.5 orders below, a floor eighteen orders below S [approx]",567        18 <= log10(S_L1 / 1e104) < 19 and abs(log10(S_L1 / 1e104) - 18.52) < 0.01)568    # 14:X4 (p14031)569    family("est", "O4")570    sweep = [(S, 4 * S + 1) for S in range(1, 2500) if _is_prime(4 * S + 1)]571    oct_ok = all((_order8(Om) is not None) == (S % 2 == 0) == ((Om - 1) % 8 == 0) for S, Om in sweep)572    z8 = _order8(OM_LAB)573    chk(f"on every prime Om = 4S + 1 below 10^4 ({len(sweep)} shells): an element of order 8 exists iff 8 | Om - 1 iff S even (exhaustive)", oct_ok and len(sweep) == 609)574    chk("the laboratory Carrier: S = 602 140 even, zeta_8 = 1 639 587 has zeta^4 = -1, zeta^8 = 1", S_LAB % 2 == 0 and pow(1639587, 4, OM_LAB) == OM_LAB - 1 and pow(1639587, 8, OM_LAB) == 1 and z8 is not None)575    # 14:X1 (p14028)576    family("est", "L0")577    rH_1 = sqrt(3.0 / Lam)                                                     # the channel-1 r_H from Lambda = 3/r_H^2578    S_1 = pi * (rH_1 / lP) ** 2579    chk("Lambda l_P^2 = 3 pi / S exactly: Lambda = 3/r_H^2 and S = pi (r_H/l_P)^2 give the identity to 1e-12 at the channel-1 r_H",580        abs(Lam * lP ** 2 - 3 * pi / S_1) <= 1e-12 * (3 * pi / S_1))581    chk("the curvature face's value: 3 pi / S = 2.9e-122 at S = 3.3e122, the observed Lambda l_P^2 [approx]; no mode sum enters",582        abs(3 * pi / S_L1 / 2.86e-122 - 1) < 0.01 and abs(Lam * lP ** 2 / 2.84e-122 - 1) < 0.01)583    # 14:X7 (p14034)584    family("est", "H7")585    G_N = 6.67430e-11586    h7 = []587    for H0_kms in (67.4, 73.0):588        H = H0_kms * 1e3 / Mpc589        rH = c / H; M_bh = rH * c ** 2 / (2 * G_N)590        rho_bh = M_bh / (4 / 3 * pi * rH ** 3); rho_c = 3 * H ** 2 / (8 * pi * G_N)591        h7.append(abs(rho_bh / rho_c - 1) <= 1e-12)592    chk("rho_BH(r_H) = rho_c identically: M = r_H c^2 / 2G in 4/3 pi r_H^3 against 3 H^2 / 8 pi G, to 1e-12 at H0 = 67.4 and 73.0 [LCDM]", all(h7))593    chk("the identity is H-free: rho_BH(r) rho_c^-1 = (c/(H r))^2, so the black-hole line meets the critical density exactly at r = c/H",594        all(abs((3 * c ** 2 / (8 * pi * G_N * r ** 2)) / (3 * (c / r) ** 2 / (8 * pi * G_N)) - 1) <= 1e-12 for r in (1e25, 1.3e26, 4e26)))595    # 14:P2 (p14037), 14:X8 (p14035)596    family("est", "S8")597    sep = 1 / 13.79598    chk("the separation of the completions: the octant bound fixed at 13.79 Gyr against a LCDM t0 growing one Gyr per Gyr of chart time, 1/13.79 = 7.25 % per Gyr, in (7 %, 7.5 %) [approx]",599        0.07 < sep < 0.075 and abs(sep - 0.0725) < 0.0005)600    chk("degenerate to 0.2 % now (13.80 against 13.79), 7 % apart one Gyr on: the bound separates black-hole cosmology's interior completions from the finite reading empirically",601        abs(13.80 / 13.79 - 1) < 0.0025 and abs((13.80 + 1) / 13.79 - 1 - sep) < 0.003)602    # 14:P4 (p14039)603    family("est", "W4")604    w_ok = True605    for Hs, rho in ((_Fr(7, 10), _Fr(3, 2)), (_Fr(1, 3), _Fr(5)), (_Fr(13, 7), _Fr(2, 9))):606        p_fluid = -rho                                                   # rho' = 0 in rho' + 3H(rho + p) = 0 forces rho + p = 0 (3H != 0)607        w_ok = w_ok and (0 + 3 * Hs * (rho + p_fluid) == 0) and (p_fluid / rho == -1)608    chk("the conservation law with rho' = 0: rho' + 3H(rho + p) = 0 forces p = -rho, hence w = p/rho = -1 exactly (three exact samples)", w_ok)609    chk("rigid: w is -1 at every sample, no w0 w_a freedom; a constant Lambda ~ 1/Om has no evolution to fit", w_ok and all(_Fr(-1) == -1 for _ in range(3)))610611    flush("est", order=["F1", "B9", "T1", "C1", "C2", "A1", "C3", "P3", "C4", "L1", "P2", "P1", "N2", "O4", "L0", "H7", "S8", "W4"],612          kinds={"F1": "EXACT", "B9": "EXACT+CHART", "N2": "EXACT+CHART", "O4": "EXACT", "W4": "EXACT"})613614# ------------------------------------------------------------------------------------------------------------615# block tri — triangle.py (until 24 September 2026), the paper's make-wedge-2.py:616# The registrable triangle -- companion figure for the s-estimation note.617#618# Exact layer: the audit identities are asserted before drawing (algebraic619# identities; float tolerance documents CODATA rounding).  Display layer620# [approx]: log10 chart of the bounded-window mass--radius plane.  FRC621# reading: the two walls are the two lattice generators (Compton = the622# crossing hbar; Schwarzschild = the normalisation G); the apex is the Planck623# anchor; the right wall is the totality on its own horizon; the record-depth624# ruler saturates at the octant (pi/4) r_H/c, the same at every observational625# frame chronon, while the rival chart's cosmic time runs on -- the paper's626# saturation falsifier drawn.627#628# Package form (2026-09): make-wedge-2.py of the paper, renamed triangle.py, wrapped in run(); its asserts report to the629# registry (the head of this script) as the families tri.A (audit identities), tri.D (the diagonal), tri.W (wall residents and the slope630# decomposition), tri.F (the figures, written to out/). The wall-channels annotation reads "channel-1 consistency rate"631# (the wording of the 13 September 2026 revision) where the July figure read "entailed rate H0 = 67.4 +/- 0.7".632def block_tri():633    """Block tri — triangle: the audit identities before drawing, the diagonal, the wall residents, the figures (tri.A, tri.D, tri.W, tri.F)."""634    plt = _plt()635    # ---- imports [approx]: laboratory constants and the note's table ----636    c, G, hbar = 2.99792458e8, 6.67430e-11, 1.054571817e-34637    lP, mP = 1.616255e-35, 2.176434e-8638    tP = lP / c                            # identity anchor; CODATA agrees to 1e-7639    rH_L = 1.659e26                        # Lambda face (channel 1)640    rows = [  # (label, r_H [m], cluster) from the note, Sec. 3; 4a gauge (excluded)641        ("1", 1.659e26, "L"), ("2a", 1.27e26, "R"), ("2b", 1.33e26, "R"),642        ("3", 1.19e26, "R"), ("4a", 1.29e26, "G"), ("4b", 1.64e26, "L"),643    ]644    S_L = pi * (rH_L / lP) ** 2645646    # ---- audit asserts ----647    family("tri", "A")648    t_oct = pi / 4 * rH_L / c649    chk("octant identity t_oct = (sqrt(pi)/4) sqrt(S) t_P to 1e-12", abs(t_oct - sqrt(pi) / 4 * sqrt(S_L) * tP) / t_oct < 1e-12)650    mean_R = sum(r for _, r, k in rows if k == "R") / 3    # evidence rows only651    mean_L = sum(r for _, r, k in rows if k == "L") / 2652    chk("two-cluster ratio on S (Valcin IV rows)", abs((mean_L / mean_R) ** 2 - 1.71) < 0.02)653    chk("chart-identity r_H split", abs(1 / sqrt(0.685) - 1.208) < 0.002)654    # entailed rate from the octant + measured Lambda (round-02 F1)655    from math import tanh656    _HL = sqrt(1.088e-52 * c ** 2 / 3) * 3.0857e22 / 1e3657    chk("H0 = 67.4 km/s/Mpc", abs(_HL / tanh(3 * pi / 8) - 67.4) < 0.15)658    # wedge width: the count face sqrt(S) in its chart dress sqrt(S/pi) -- the659    # solid angle of the area law is the declared register->chart transport (0.25 dex)660    chk("width identity log10(r_H/l_P) = log10 sqrt(S/pi) to 1e-12", abs(log10(rH_L / lP) - log10(sqrt(S_L / pi))) < 1e-12)661662    # ---- chart ----663    xg, yg = log10, log10664    comp = lambda x: log10(hbar / c) - x        # Compton:  m R = hbar/c665    schw = lambda x: x - log10(2 * G / c**2)    # Schwarzschild: m = R c^2/2G666    x_apex = (log10(hbar / c) + log10(2 * G / c**2)) / 2667    xW = xg(rH_L)668    x_oct = xg(pi / 4 * rH_L)669670    fig, ax = plt.subplots(figsize=(6.4, 8.9))671    ax.set_aspect("equal", adjustable="box")672673    ax.fill([x_apex, xW, xW], [comp(x_apex), schw(xW), comp(xW)],674            color=FILL, zorder=0, lw=0)675    ax.plot([x_apex, xW], [schw(x_apex), schw(xW)], color=INK, lw=1.1, zorder=3)676    ax.plot([x_apex, xW], [comp(x_apex), comp(xW)], color=INK, lw=1.1, zorder=3)677    ax.plot([xW, xW], [comp(xW) - 2.5, schw(xW) + 3.2], color=MID, lw=1.0,678            ls=(0, (5, 3)), zorder=3)679680    ax.text(-20, schw(-20) + 2.0, r"Schwarzschild $\;R=2Gm/c^{2}$ — the normalisation $G$",681            rotation=45, ha="center", va="center", fontsize=9.5, color=INK)682    ax.text(-14, comp(-14) - 2.0, r"Compton $\;mRc=\hbar$ — the crossing $\hbar$",683            rotation=-45, ha="center", va="center", fontsize=9.5, color=INK)684    ax.text(xW - 1.1, -26, r"$r_H$ — the totality wall", rotation=90,685            ha="center", va="center", fontsize=9.5, color=MID)686    ax.text(3, 39, "over-closed (horizon)", fontsize=8.5, color=SOFT,687            ha="center", style="italic", rotation=0)688    ax.text(-19, -42, "delocalised (no registration)", fontsize=8.5, color=SOFT,689            ha="center", style="italic")690    ax.text(13.5, -9, "the registrable triangle", fontsize=11.5, color="0.45",691            ha="center")692693    # Planck apex and the totality point694    ax.scatter([xg(lP)], [yg(mP)], s=22, color=INK, zorder=5)695    ax.annotate(r"Planck apex $(\ell_P,\,m_P)$", (xg(lP), yg(mP)),696                textcoords="offset points", xytext=(0, -14), ha="left",697                fontsize=8.5, color=INK)698    ax.scatter([xW], [schw(xW)], s=26, color=INK, zorder=5)699    ax.annotate(r"$\Omega$ — the totality on its own horizon",700                (xW, schw(xW)), textcoords="offset points", xytext=(-10, -11),701                ha="right", fontsize=9, color=INK)702703    # reference marks [approx]704    objs = [  # reference marks along the typical-object diagonal [approx]705        ("electron", 3.86e-13, 9.109e-31, (7, -3), "left"),706        ("proton", 8.4e-16, 1.673e-27, (5, 4), "left"),707        ("H atom", 5.29e-11, 1.674e-27, (6, -8), "left"),708        ("protein", 5.0e-9, 1e-22, (5, -3), "left"),709        ("virus", 1.0e-7, 1e-18, (5, -3), "left"),710        ("bacterium", 1.0e-6, 1e-15, (5, -3), "left"),711        ("human cell", 1.0e-5, 1e-12, (5, -3), "left"),712        ("ant", 4.0e-3, 3e-6, (-5, -2), "right"),713        ("human", 1.0, 7e1, (5, -9), "left"),714        ("blue whale", 2.5e1, 1.5e5, (5, -2), "left"),715        ("comet", 5.0e3, 1e13, (5, -3), "left"),716        ("asteroid", 2.6e5, 2.6e20, (-6, -2), "right"),717        ("Moon", 1.74e6, 7.35e22, (4, -11), "left"),718        ("Earth", 6.37e6, 5.97e24, (5, -9), "left"),719        ("Sun", 6.96e8, 1.99e30, (5, -9), "left"),720        ("white dwarf", 7.0e6, 1.2e30, (0, -13), "center"),721        ("solar system", 6.0e12, 2e30, (5, -3), "left"),722        ("neutron star", 1.2e4, 2.8e30, (-6, -6), "right"),723        (r"$10M_\odot$ hole", 2.95e4, 1.99e31, (-6, 4), "right"),724        ("Sgr A*", 1.2e10, 8.2e36, (7, -3), "left"),725        ("globular cluster", 2.5e17, 1e36, (5, -8), "left"),726        ("galaxy", 4.7e20, 2e42, (2, -10), "left"),727        ("galaxy cluster", 5.0e22, 2e45, (0, -12), "right"),728        ("supercluster", 4.0e24, 5e47, (-9, -12), "center"),729    ]730    # the object diagonal: ordinary least-squares regression over the ON-DIAGONAL731    # scale-typical single observable objects, MID-WEDGE ONLY.  Wall residents732    # are excluded by the same rule as every other exclusion: the electron sits733    # on the Compton wall by construction (its R is its reduced Compton734    # wavelength) and Sgr A* on the Schwarzschild wall (its R is its horizon735    # radius), so both are wall-displaced, drawn hollow, and become736    # out-of-sample checks of the diagonal's wall tangencies.  Also hollow:737    # the proton (the H atom stripped to its Compton-scale core), the compact738    # remnants (white dwarf, neutron star, stellar hole: collapsed ends hugging739    # the Schwarzschild wall), the solar system (the Sun's mass gravitationally740    # diluted), and the bound aggregates (globular cluster through supercluster:741    # collections, not single typical objects, bending toward the totality point).742    FIT = ["H atom", "protein", "virus", "bacterium",743           "human cell", "ant", "human", "blue whale", "comet", "asteroid",744           "Moon", "Earth", "Sun"]745    FIT_FULL = FIT + ["electron", "Sgr A*"]      # sensitivity population746    pos = {n: (xg(R), yg(m)) for n, R, m, _, _ in objs}747748749    def ols(names):750        n = len(names)751        sx = sum(pos[m][0] for m in names); sy = sum(pos[m][1] for m in names)752        sxx = sum(pos[m][0] ** 2 for m in names)753        sxy = sum(pos[m][0] * pos[m][1] for m in names)754        k = (n * sxy - sx * sy) / (n * sxx - sx * sx)755        return k, (sy - k * sx) / n756757758    def ols_k3(names):                       # constrained k = 3 fit759        n = len(names)760        b = sum(pos[m][1] - 3 * pos[m][0] for m in names) / n761        return 3.0, b762763764    k_fit, b_fit = ols(FIT)765    k_full, b_full = ols(FIT_FULL)766    k_c13, b_c13 = ols_k3(FIT)767    k_c15, b_c15 = ols_k3(FIT_FULL)768    x_lo = (log10(hbar / c) - b_fit) / (1 + k_fit)              # Compton-wall entry769    x_hi = min((-log10(2 * G / c**2) - b_fit) / (k_fit - 1), xW)  # exit: wall or r_H770    M_exit = 10 ** (x_hi - log10(2 * G / c**2)) / 1.989e30        # exit mass [Msun]771772    # ---- diagonal asserts + pass lines [approx] ----773    # 14:C9 (p14026), 14:X5 (p14032)774    family("tri", "D")775    chk("slope 3 within 1.1%", abs(k_fit - 3.032) < 0.005)776    chk("coefficient 1.00e3 kg/m^3", abs(10 ** b_fit / 1.00e3 - 1) < 0.05)777    chk("sensitivity: walls included", abs(k_full - 3.007) < 0.005)778    chk("over-closure bound [Msun]", abs(M_exit / 1.8e8 - 1) < 0.05)779    chk("constrained k=3 coefficient", abs(10 ** b_c13 / 0.97e3 - 1) < 0.05)780781782    def exit_of(k, b):                                # exit mass of a fit variant783        xh = (-log10(2 * G / c**2) - b) / (k - 1)784        return 10 ** (xh - log10(2 * G / c**2)) / 1.989e30785786787    band = sorted(exit_of(k, b) for k, b in788                  ((k_fit, b_fit), (k_full, b_full), (k_c13, b_c13), (k_c15, b_c15)))789    chk("over-closure band (1.4-2.8)e8", band[0] > 1.3e8 and band[-1] < 2.9e8)790    print(f"[PASS] constrained k=3: c~ = m/R^3 = {10 ** b_c13:.2e} kg/m^3 (a cubic coefficient, not a density); over-closure band"791          f" ({band[0]:.2e} .. {band[-1]:.2e}) Msun across the four fit variants")792    family("tri", "W")793    chk("electron on the Compton wall to < 0.01 dex", abs(pos["electron"][1] - (log10(hbar / c) - pos["electron"][0])) < 0.01)794    chk("Sgr A* on the Schwarzschild wall to < 0.01 dex", abs(pos["Sgr A*"][1] - (pos["Sgr A*"][0] - log10(2 * G / c**2))) < 0.01)795    mean_c = sum(pos[m][1] - 3 * pos[m][0] for m in FIT) / len(FIT)796    drift = k_fit - 3.0                       # slope = 3 + d c / d logR identically797    span = max(pos[m][0] for m in FIT) - min(pos[m][0] for m in FIT)798    chk("slope decomposition: mean c = 2.99, drift 0.61 dex over the baseline", abs(mean_c - 2.99) < 0.02 and abs(drift * span - 0.61) < 0.02)799    print(f"[PASS] slope decomposition: k = 3 + dc/dlogR, drift {drift:+.4f} "800          f"({drift * span:.2f} dex over {span:.1f} dex), mean c = {mean_c:.2f}")801    print(f"[PASS] diagonal: k = {k_fit:.4f} (13 mid-wedge), intercept "802          f"{10 ** b_fit:.2e} at the metre pivot (density units only at k = 3); "803          f"sensitivity k = {k_full:.4f} (15 incl. walls)")804    print(f"[PASS] boundaries: Compton entry R = {10 ** x_lo:.2e} m; "805          f"over-closure exit M = {M_exit:.2e} Msun")806    chk("Compton entry at the atomic core scale, R = 5.3e-12 m (the electron 1.1 dex beyond)", abs(10 ** x_lo / 5.3e-12 - 1) < 0.05 and abs(x_lo - pos["electron"][0] - 1.1) < 0.1)807    chk("Sgr A* 1.6 dex before the over-closure exit", abs(x_hi - pos["Sgr A*"][0] - 1.6) < 0.1)808    print(f"[PASS] wall residents on-wall to <0.01 dex: electron (Compton, "809          f"{x_lo - pos['electron'][0]:+.1f} dex beyond entry), "810          f"Sgr A* (Schwarzschild, {x_hi - pos['Sgr A*'][0]:+.1f} dex before exit)")811812    ax.plot([x_lo, x_hi], [b_fit + k_fit * x_lo, b_fit + k_fit * x_hi],813            color="0.62", lw=0.9, ls=(0, (1, 2.6)), zorder=2)814    ax.text(-1.6, 12.5, r"typical objects: $m=\tilde c\,R^{3}$,"815            r"  $\tilde c\approx10^{3}\,$kg$\,$m$^{-3}$",816            rotation=71.6, ha="center", va="center", fontsize=7.5, color="0.5")817818    for name, R, m, off, ha in objs:819        if name in FIT:820            ax.scatter([xg(R)], [yg(m)], s=11, color="0.4", zorder=4)821        else:822            ax.scatter([xg(R)], [yg(m)], s=11, facecolors="white",823                       edgecolors="0.45", linewidths=0.9, zorder=4)824        ax.annotate(name, (xg(R), yg(m)), textcoords="offset points", xytext=off,825                    fontsize=7.5, color="0.4", ha=ha)826827    # record-depth ruler (top interior): solid to the octant, dashed on, arrow828    yr = 58.0829    ax.plot([xg(c * tP), x_oct], [yr, yr], color=INK, lw=1.2, zorder=4)830    ax.plot([x_oct, xW + 2.8], [yr, yr], color=SOFT, lw=0.9, ls=(0, (3, 3)), zorder=4)831    ax.annotate("", (xW + 3.6, yr), (xW + 2.8, yr),832                arrowprops=dict(arrowstyle="->", color=SOFT, lw=0.9))833    for tval, lab in ((tP, r"$t_P$"), (1.0, r"$1\,$s"), (3.15576e7, r"$1\,$yr"),834                      (3.15576e16, r"$1\,$Gyr")):835        xt = xg(c * tval)836        ax.plot([xt, xt], [yr - 0.8, yr + 0.8], color=INK, lw=0.8)837        ax.annotate(lab, (xt, yr - 2.0), ha="center", va="top", fontsize=8, color=INK)838    ax.plot([x_oct, x_oct], [yr - 1.3, yr + 1.3], color=INK, lw=1.8)839    ax.annotate("record depth saturates:  " + r"$(\pi/4)\,r_H/c = 13.8\;$Gyr",840                (x_oct - 0.8, yr + 2.0), ha="right", va="bottom", fontsize=8.5, color=INK)841    ax.annotate("chart time\n(unbounded)", (xW + 1.3, yr - 1.4), ha="left", va="top",842                fontsize=7.2, color=SOFT)843    ax.annotate(r"registrable depth $\;t = R/c$", (xg(c * tP) + 5.0, yr - 2.0),844                ha="left", va="top", fontsize=8.5, color=INK)845846    # axes cosmetics847    ax.set_xlim(-37, 33.5)848    ax.set_ylim(-45, 62)849    ax.set_xlabel(r"$\log_{10} R\;$ [m]$\;$ [approx]", fontsize=9.5)850    ax.set_ylabel(r"$\log_{10} m\;$ [kg]$\;$ [approx]", fontsize=9.5)851    ax.set_xticks(range(-30, 31, 10))852    ax.set_yticks(range(-60, 61, 20))853    ax.tick_params(labelsize=8, color=MID, labelcolor=MID, length=3, width=0.6)854    for s in ax.spines.values():855        s.set_color(MID)856857    # ---- inset: the wall, measured ----858    axi = ax.inset_axes([-35.5, 31.5, 27.0, 20.0], transform=ax.transData)859    axi.set_title("the wall, measured — six channels [approx]",860                  fontsize=8, color=INK, pad=4)861    for a, b in ((1.17, 1.35), (1.59, 1.68)):862        axi.axvspan(a, b, color=FILL)863    order = {"3": 0, "2a": 1, "4a": 2, "2b": 3, "4b": 4, "1": 5}864    for lab, r, k in rows:865        y = 0.16 + 0.105 * order[lab]866        axi.scatter([r / 1e26], [y], s=17, color=INK if k == "R" else "0.5", zorder=4)867        axi.annotate(lab, (r / 1e26, y), textcoords="offset points",868                     xytext=(4, -2.5), fontsize=7, color="0.3")869    axi.axvline(pi / 4 * rH_L / 1e26, color=INK, lw=1.1, ls=(0, (4, 2)))870    axi.annotate(r"$(\pi/4)\,r_H^{\Lambda}$", (pi / 4 * rH_L / 1e26, 0.80),871                 ha="center", va="top", fontsize=7.5, color=INK)872    axi.annotate("registered rate", (1.26, 0.035), ha="center", va="bottom",873                 fontsize=7.5, color=MID)874    axi.annotate(r"$\Lambda$ face", (1.635, 0.035), ha="center", va="bottom",875                 fontsize=7.5, color=MID)876    axi.annotate(r"split: the ladder--CMB (Hubble-tension) split; entailed $H_0=67.4$",877                 (1.425, 0.875), ha="center", fontsize=7.2, color=INK)878    axi.set_xlim(1.1, 1.75)879    axi.set_ylim(0, 0.95)880    axi.set_yticks([])881    axi.set_xticks([1.2, 1.4, 1.6])882    axi.set_xlabel(r"$r_H\;$ [$10^{26}\,$m]", fontsize=7.5, labelpad=2)883    axi.tick_params(labelsize=7, color=MID, labelcolor=MID, length=2.5, width=0.5)884    for s in axi.spines.values():885        s.set_color(MID); s.set_linewidth(0.5)886887    fig.subplots_adjust(left=0.11, right=0.97, top=0.985, bottom=0.05)888    ax.set_xlim(-37, 33.5)889    ax.set_ylim(-45, 62)   # re-assert: pinned after all children890    save(fig, "registrable-wedge")891892    # ---- plain version: no inlay (the channel strip ships as its own figure) ----893    axi.remove()894    save(fig, "registrable-wedge-plain")895896    # ---- standalone channels figure (the former inlay, proportioned for print) ----897    fig2, ax2 = plt.subplots(figsize=(5.2, 2.3))898    for a, b in ((1.17, 1.35), (1.59, 1.68)):899        ax2.axvspan(a, b, color=FILL)900    for lab, r, k in rows:901        y = 0.14 + 0.135 * order[lab]902        ax2.scatter([r / 1e26], [y], s=26, color=INK if k == "R" else "0.5", zorder=4)903        ax2.annotate(lab, (r / 1e26, y), textcoords="offset points", xytext=(5, -3),904                     fontsize=8.5, color="0.3")905    ax2.axvline(pi / 4 * rH_L / 1e26, color=INK, lw=1.2, ls=(0, (4, 2)))906    ax2.annotate(r"$(\pi/4)\,r_H^{\Lambda}$" + "  (depth face)",907                 (pi / 4 * rH_L / 1e26 + 0.012, 0.97), ha="left", va="top",908                 fontsize=8.5, color=INK)909    ax2.annotate("registered-rate cluster", (1.26, 0.035), ha="center", va="bottom",910                 fontsize=8.5, color=MID)911    ax2.annotate(r"$\Lambda$-face cluster", (1.635, 0.035), ha="center", va="bottom",912                 fontsize=8.5, color=MID)913    ax2.annotate(r"cluster split on $r_H$: the ladder--CMB split (Hubble tension); channel-1 rate $67.4$",914                 (1.425, 1.05), ha="center", fontsize=8.5, color=INK,915                 annotation_clip=False)916    ax2.set_xlim(1.1, 1.75)917    ax2.set_ylim(0, 0.97)918    ax2.set_yticks([])919    ax2.set_xticks([1.2, 1.3, 1.4, 1.5, 1.6, 1.7])920    ax2.set_xlabel(r"$r_H\;$ [$10^{26}\,$m]   [approx]", fontsize=9)921    ax2.tick_params(labelsize=8, color=MID, labelcolor=MID, length=3, width=0.6)922    for sp in ax2.spines.values():923        sp.set_color(MID); sp.set_linewidth(0.6)924    fig2.subplots_adjust(left=0.03, right=0.985, top=0.86, bottom=0.19)925    save(fig2, "wall-channels")926    family("tri", "F")927    for f in ("registrable-wedge", "registrable-wedge-plain", "wall-channels"):928        chk(f"{f}.png and .pdf written to out/", os.path.exists(os.path.join(OUT, f + ".png")) and os.path.exists(os.path.join(OUT, f + ".pdf")))929    plt.close("all")930    print("ok")931    flush("tri", order=["A", "D", "W", "F"])932933# ------------------------------------------------------------------------------------------------------------934# block cap — capacity.py (until 24 September 2026), the paper's make-wedge-3.py:935# The registrable triangle, capacity-ruler variant (20260815).936#937# Derived from make-wedge-2.py; adds the object-capacity ruler on the outer938# right axis: log10 capacity in the format of the mass axis, kappa primary,939# zero at the H atom (kappa_H = 3, the first Object), one dex per chart dex.940#941# Exact layer: the audit identities are asserted before drawing (algebraic942# identities; float tolerance documents CODATA rounding).  Display layer943# [approx]: log10 chart of the bounded-window mass--radius plane.  FRC944# reading: the two walls are the two lattice generators (Compton = the945# crossing hbar; Schwarzschild = the normalisation G); the apex is the Planck946# anchor; the right wall is the totality on its own horizon; the record-depth947# ruler saturates at the octant (pi/4) r_H/c, the same at every observational948# frame chronon, while the rival chart's cosmic time runs on -- the paper's949# saturation falsifier drawn.  Top axis (20260819): the same chart read as950# time, t = R/c, in decades matching the length axis -- the registrable span951# is the same 61 dex on both faces, t_P to r_H/c.952#953# Package form (2026-09): make-wedge-3.py of the paper, renamed capacity.py, wrapped in run(); its asserts report to the954# registry (the head of this script) as the families cap.A (the audit and diagonal identities, identical to triangle.py's), cap.K (the955# capacity axis: the pinned mass axis, the Avogadro landing), cap.F (the figure, written to out/).956def block_cap():957    """Block cap — capacity: the same identities re-asserted, the pinned mass axis and the Avogadro landing, the figure (cap.A, cap.K, cap.F)."""958    plt = _plt()959    # ---- imports [approx]: laboratory constants and the note's table ----960    c, G, hbar = 2.99792458e8, 6.67430e-11, 1.054571817e-34961    lP, mP = 1.616255e-35, 2.176434e-8962    tP = lP / c                            # identity anchor; CODATA agrees to 1e-7963    rH_L = 1.659e26                        # Lambda face (channel 1)964    rows = [  # (label, r_H [m], cluster) from the note, Sec. 3; 4a gauge (excluded)965        ("1", 1.659e26, "L"), ("2a", 1.27e26, "R"), ("2b", 1.33e26, "R"),966        ("3", 1.19e26, "R"), ("4a", 1.29e26, "G"), ("4b", 1.64e26, "L"),967    ]968    S_L = pi * (rH_L / lP) ** 2969970    # ---- audit asserts ----971    family("cap", "A")972    t_oct = pi / 4 * rH_L / c973    chk("octant identity t_oct = (sqrt(pi)/4) sqrt(S) t_P to 1e-12", abs(t_oct - sqrt(pi) / 4 * sqrt(S_L) * tP) / t_oct < 1e-12)974    mean_R = sum(r for _, r, k in rows if k == "R") / 3    # evidence rows only975    mean_L = sum(r for _, r, k in rows if k == "L") / 2976    chk("two-cluster ratio on S (Valcin IV rows)", abs((mean_L / mean_R) ** 2 - 1.71) < 0.02)977    chk("chart-identity r_H split", abs(1 / sqrt(0.685) - 1.208) < 0.002)978    # entailed rate from the octant + measured Lambda (round-02 F1)979    from math import tanh980    _HL = sqrt(1.088e-52 * c ** 2 / 3) * 3.0857e22 / 1e3981    chk("H0 = 67.4 km/s/Mpc", abs(_HL / tanh(3 * pi / 8) - 67.4) < 0.15)982    # wedge width: the count face sqrt(S) in its chart dress sqrt(S/pi) -- the983    # solid angle of the area law is the declared register->chart transport (0.25 dex)984    chk("width identity log10(r_H/l_P) = log10 sqrt(S/pi) to 1e-12", abs(log10(rH_L / lP) - log10(sqrt(S_L / pi))) < 1e-12)985986    # ---- chart ----987    xg, yg = log10, log10988    comp = lambda x: log10(hbar / c) - x        # Compton:  m R = hbar/c989    schw = lambda x: x - log10(2 * G / c**2)    # Schwarzschild: m = R c^2/2G990    x_apex = (log10(hbar / c) + log10(2 * G / c**2)) / 2991    xW = xg(rH_L)992    x_oct = xg(pi / 4 * rH_L)993994    fig, ax = plt.subplots(figsize=(7.2, 8.9))995    ax.set_aspect("equal", adjustable="box")996997    ax.fill([x_apex, xW, xW], [comp(x_apex), schw(xW), comp(xW)],998            color=FILL, zorder=0, lw=0)999    ax.plot([x_apex, xW], [schw(x_apex), schw(xW)], color=INK, lw=1.1, zorder=3)1000    ax.plot([x_apex, xW], [comp(x_apex), comp(xW)], color=INK, lw=1.1, zorder=3)1001    ax.plot([xW, xW], [comp(xW) - 2.5, schw(xW) + 3.2], color=MID, lw=1.0,1002            ls=(0, (5, 3)), zorder=3)10031004    ax.text(-18, schw(-18) + 2.0, r"Schwarzschild $\;R=2Gm/c^{2}$ — the normalisation $G$",1005            rotation=45, ha="center", va="center", fontsize=9.5, color=INK)1006    ax.text(-14, comp(-14) - 2.0, r"Compton $\;mRc=\hbar$ — the crossing $\hbar$",1007            rotation=-45, ha="center", va="center", fontsize=9.5, color=INK)1008    ax.text(xW - 1.1, -26, r"$r_H$ — the totality wall", rotation=90,1009            ha="center", va="center", fontsize=9.5, color=MID)1010    ax.text(3, 39, "over-closed (horizon)", fontsize=8.5, color=SOFT,1011            ha="center", style="italic", rotation=0)1012    ax.text(-19, -42, "delocalised (no registration)", fontsize=8.5, color=SOFT,1013            ha="center", style="italic")1014    ax.text(12, -9, "the registrable triangle", fontsize=11.5, color="0.45",1015            ha="center")10161017    # Planck apex and the totality point1018    ax.scatter([xg(lP)], [yg(mP)], s=22, color=INK, zorder=5)1019    ax.annotate(r"Planck apex $(\ell_P,\,m_P)$", (xg(lP), yg(mP)),1020                textcoords="offset points", xytext=(6, -6), ha="left",1021                fontsize=8.5, color=INK)1022    ax.scatter([xW], [schw(xW)], s=26, color=INK, zorder=5)1023    #ax.annotate(r"$\Omega$ — the totality on its own horizon",1024    #            (xW, schw(xW)), textcoords="offset points", xytext=(-18, -11),1025    #            ha="right", fontsize=9, color=INK)10261027    # reference marks [approx]1028    objs = [  # reference marks along the typical-object diagonal [approx]1029        ("electron", 3.86e-13, 9.109e-31, (7, -3), "left"),1030        ("proton", 8.4e-16, 1.673e-27, (-4, 5), "left"),1031        ("H atom", 5.29e-11, 1.674e-27, (4, -6), "left"),1032        ("protein", 5.0e-9, 1e-22, (5, -3), "left"),1033        ("virus", 1.0e-7, 1e-18, (5, -3), "left"),1034        ("bacterium", 1.0e-6, 1e-15, (5, -3), "left"),1035        ("human cell", 1.0e-5, 1e-12, (5, -3), "left"),1036        ("ant", 4.0e-3, 3e-6, (-5, -2), "right"),1037        #("1 mole", 1.0e-1, 1e-3, (-5, -2), "right"),1038        ("human", 1.0, 7e1, (4, -6), "left"),1039        ("blue whale", 2.5e1, 1.5e5, (5, -2), "left"),1040        ("comet", 5.0e3, 1e13, (5, -3), "left"),1041        ("asteroid", 2.6e5, 2.6e20, (5, -9), "left"),1042        ("Moon", 1.74e6, 7.35e22, (4, -11), "left"),1043        ("Earth", 6.37e6, 5.97e24, (5, -9), "left"),1044        ("Sun", 6.96e8, 1.99e30, (5, -9), "left"),1045        ("white dwarf", 7.0e6, 1.2e30, (0, -13), "center"),1046        ("solar system", 6.0e12, 2e30, (5, -3), "left"),1047        ("neutron star", 1.2e4, 2.8e30, (-6, -3), "right"),1048        (r"$10M_\odot$ hole", 2.95e4, 1.99e31, (-3, 4), "right"),1049        ("Sgr A*", 1.2e10, 8.2e36, (7, -3), "left"),1050        ("globular cluster", 2.5e17, 1e36, (-17, -8), "left"),1051        ("galaxy", 4.7e20, 2e42, (-2, -10), "left"),1052        ("galaxy cluster", 5.0e22, 2e45, (2, -8), "right"),1053        ("supercluster", 4.0e24, 5e47, (-9, -8), "center"),1054    ]1055    # the object diagonal: ordinary least-squares regression over the ON-DIAGONAL1056    # scale-typical single observable objects, MID-WEDGE ONLY.  Wall residents1057    # are excluded by the same rule as every other exclusion: the electron sits1058    # on the Compton wall by construction (its R is its reduced Compton1059    # wavelength) and Sgr A* on the Schwarzschild wall (its R is its horizon1060    # radius), so both are wall-displaced, drawn hollow, and become1061    # out-of-sample checks of the diagonal's wall tangencies.  Also hollow:1062    # the proton (the H atom stripped to its Compton-scale core), the compact1063    # remnants (white dwarf, neutron star, stellar hole: collapsed ends hugging1064    # the Schwarzschild wall), the solar system (the Sun's mass gravitationally1065    # diluted), and the bound aggregates (globular cluster through supercluster:1066    # collections, not single typical objects, bending toward the totality point).1067    FIT = ["H atom", "protein", "virus", "bacterium",1068           "human cell", "ant", "human", "blue whale", "comet", "asteroid",1069           "Moon", "Earth", "Sun"]1070    FIT_FULL = FIT + ["electron", "Sgr A*"]      # sensitivity population1071    pos = {n: (xg(R), yg(m)) for n, R, m, _, _ in objs}107210731074    def ols(names):1075        n = len(names)1076        sx = sum(pos[m][0] for m in names); sy = sum(pos[m][1] for m in names)1077        sxx = sum(pos[m][0] ** 2 for m in names)1078        sxy = sum(pos[m][0] * pos[m][1] for m in names)1079        k = (n * sxy - sx * sy) / (n * sxx - sx * sx)1080        return k, (sy - k * sx) / n108110821083    def ols_k3(names):                       # constrained k = 3 fit1084        n = len(names)1085        b = sum(pos[m][1] - 3 * pos[m][0] for m in names) / n1086        return 3.0, b108710881089    k_fit, b_fit = ols(FIT)1090    k_full, b_full = ols(FIT_FULL)1091    k_c13, b_c13 = ols_k3(FIT)1092    k_c15, b_c15 = ols_k3(FIT_FULL)1093    x_lo = (log10(hbar / c) - b_fit) / (1 + k_fit)              # Compton-wall entry1094    x_hi = min((-log10(2 * G / c**2) - b_fit) / (k_fit - 1), xW)  # exit: wall or r_H1095    M_exit = 10 ** (x_hi - log10(2 * G / c**2)) / 1.989e30        # exit mass [Msun]10961097    # ---- diagonal asserts + pass lines [approx] ----1098    chk("slope 3 within 1.1%", abs(k_fit - 3.032) < 0.005)1099    chk("coefficient 1.00e3 kg/m^3", abs(10 ** b_fit / 1.00e3 - 1) < 0.05)1100    chk("sensitivity: walls included", abs(k_full - 3.007) < 0.005)1101    chk("over-closure bound [Msun]", abs(M_exit / 1.8e8 - 1) < 0.05)1102    chk("constrained k=3 coefficient", abs(10 ** b_c13 / 0.97e3 - 1) < 0.05)110311041105    def exit_of(k, b):                                # exit mass of a fit variant1106        xh = (-log10(2 * G / c**2) - b) / (k - 1)1107        return 10 ** (xh - log10(2 * G / c**2)) / 1.989e30110811091110    band = sorted(exit_of(k, b) for k, b in1111                  ((k_fit, b_fit), (k_full, b_full), (k_c13, b_c13), (k_c15, b_c15)))1112    chk("over-closure band (1.4-2.8)e8", band[0] > 1.3e8 and band[-1] < 2.9e8)1113    print(f"[PASS] constrained k=3: c~ = m/R^3 = {10 ** b_c13:.2e} kg/m^3 (a cubic coefficient, not a density); over-closure band"1114          f" ({band[0]:.2e} .. {band[-1]:.2e}) Msun across the four fit variants")1115    chk("electron on the Compton wall to < 0.01 dex", abs(pos["electron"][1] - (log10(hbar / c) - pos["electron"][0])) < 0.01)1116    chk("Sgr A* on the Schwarzschild wall to < 0.01 dex", abs(pos["Sgr A*"][1] - (pos["Sgr A*"][0] - log10(2 * G / c**2))) < 0.01)1117    mean_c = sum(pos[m][1] - 3 * pos[m][0] for m in FIT) / len(FIT)1118    drift = k_fit - 3.0                       # slope = 3 + d c / d logR identically1119    span = max(pos[m][0] for m in FIT) - min(pos[m][0] for m in FIT)1120    chk("slope decomposition: mean c = 2.99, drift 0.61 dex over the baseline", abs(mean_c - 2.99) < 0.02 and abs(drift * span - 0.61) < 0.02)1121    print(f"[PASS] slope decomposition: k = 3 + dc/dlogR, drift {drift:+.4f} "1122          f"({drift * span:.2f} dex over {span:.1f} dex), mean c = {mean_c:.2f}")1123    print(f"[PASS] diagonal: k = {k_fit:.4f} (13 mid-wedge), intercept "1124          f"{10 ** b_fit:.2e} at the metre pivot (density units only at k = 3); "1125          f"sensitivity k = {k_full:.4f} (15 incl. walls)")1126    print(f"[PASS] boundaries: Compton entry R = {10 ** x_lo:.2e} m; "1127          f"over-closure exit M = {M_exit:.2e} Msun")1128    print(f"[PASS] wall residents on-wall to <0.01 dex: electron (Compton, "1129          f"{x_lo - pos['electron'][0]:+.1f} dex beyond entry), "1130          f"Sgr A* (Schwarzschild, {x_hi - pos['Sgr A*'][0]:+.1f} dex before exit)")11311132    ax.plot([x_lo, x_hi], [b_fit + k_fit * x_lo, b_fit + k_fit * x_hi],1133            color="0.62", lw=0.9, ls=(0, (1, 2.6)), zorder=2)1134    ax.text(-1.6, 12.5, r"typical objects: $m=\tilde c\,R^{3}$,"1135            r"  $\tilde c\approx10^{3}\,$kg$\,$m$^{-3}$",1136            rotation=71.6, ha="center", va="center", fontsize=7.5, color="0.5")11371138    for name, R, m, off, ha in objs:1139        if name in FIT:1140            ax.scatter([xg(R)], [yg(m)], s=11, color="0.4", zorder=4)1141        else:1142            ax.scatter([xg(R)], [yg(m)], s=11, facecolors="white",1143                       edgecolors="0.45", linewidths=0.9, zorder=4)1144        ax.annotate(name, (xg(R), yg(m)), textcoords="offset points", xytext=off,1145                    fontsize=7.5, color="0.4", ha=ha)11461147    # named times, ticked inward from the top axis with the label beneath each1148    ytop = 62.01149    for tval, lab in ((tP, r"$t_P$"), (1.0, r"$1\,$s"), (3.15576e7, r"$1\,$yr")):1150        xt = xg(c * tval)1151        ax.plot([xt, xt], [ytop - 0.7, ytop], color=INK, lw=0.5, zorder=4)1152        ax.annotate(lab, (xt, ytop - 1.2), ha="center", va="top",1153                    fontsize=8, color=INK)1154    ax.plot([x_oct, x_oct], [ytop - 2.6, ytop], color=INK, lw=1.2, zorder=4)1155    ax.annotate(r"$(\pi/4)\,r_H/c = 13.8\;$Gyr$=10^{61}t_P$", (x_oct+8.0, ytop-3.2),1156                ha="right", va="top", fontsize=8.5, color=INK)1157    # the time face carries the length face's own 61 decades1158    #ax.annotate(r"$61$ dex, $t_P$ to $r_H/c$ — the length face's own span",1159    #            ((xg(c * tP) + xW) / 2, ytop - 5.0), ha="center", va="top",1160    #            fontsize=8, color=MID)11611162    # axes cosmetics1163    ax.set_xlim(-37, 35.0)1164    ax.set_ylim(-45, 62)1165    ax.set_xlabel(r"registration depth $\log_{10} R\;$ [m]$\;$ [approx]", fontsize=9.5)1166    ax.set_ylabel(r"$\log_{10} m\;$ [kg]$\;$ [approx]", fontsize=9.5)1167    ax.set_xticks(range(-30, 31, 10))1168    ax.set_yticks([-60, -40, -20, -3, 0, 20, 40, 60])1169    ax.tick_params(labelsize=8, color=MID, labelcolor=MID, length=3, width=0.6)1170    for s in ax.spines.values():1171        s.set_color(MID)11721173    # top axis: the same chart read as time, t = R/c -- one decade of time per1174    # decade of length, the 61-dex registrable span identical on the two faces1175    tax = ax.secondary_xaxis("top", functions=(lambda x: x - log10(c),1176                                               lambda t: t + log10(c)))1177    tax.set_xlabel(r"registration depth $\log_{10} t\;$ [s],$\;t = R/c\;$ [approx]", fontsize=9.5,1178                       labelpad=6)1179    tax.set_xticks(list(range(-40, 21, 10)))1180    tax.tick_params(labelsize=8, color=MID, labelcolor=MID, length=3, width=0.6)1181    tax.spines["top"].set_color(MID)11821183    # ---- inset: the wall, measured ----1184    axi = ax.inset_axes([-35.5, 31.5, 27.0, 20.0], transform=ax.transData)1185    axi.set_title("the wall, measured — six channels [approx]",1186                  fontsize=8, color=INK, pad=4)1187    for a, b in ((1.17, 1.35), (1.59, 1.68)):1188        axi.axvspan(a, b, color=FILL)1189    order = {"3": 0, "2a": 1, "4a": 2, "2b": 3, "4b": 4, "1": 5}1190    for lab, r, k in rows:1191        y = 0.16 + 0.105 * order[lab]1192        axi.scatter([r / 1e26], [y], s=17, color=INK if k == "R" else "0.5", zorder=4)1193        axi.annotate(lab, (r / 1e26, y), textcoords="offset points",1194                     xytext=(4, -2.5), fontsize=7, color="0.3")1195    axi.axvline(pi / 4 * rH_L / 1e26, color=INK, lw=1.1, ls=(0, (4, 2)))1196    axi.annotate(r"$(\pi/4)\,r_H^{\Lambda}$", (pi / 4 * rH_L / 1e26, 0.80),1197                 ha="center", va="top", fontsize=7.5, color=INK)1198    axi.annotate("registered rate", (1.26, 0.035), ha="center", va="bottom",1199                 fontsize=7.5, color=MID)1200    axi.annotate(r"$\Lambda$ face", (1.635, 0.035), ha="center", va="bottom",1201                 fontsize=7.5, color=MID)1202    axi.annotate(r"split: the ladder--CMB (Hubble-tension) split; entailed $H_0=67.4$",1203                 (1.425, 0.875), ha="center", fontsize=7.2, color=INK)1204    axi.set_xlim(1.1, 1.75)1205    axi.set_ylim(0, 0.95)1206    axi.set_yticks([])1207    axi.set_xticks([1.2, 1.4, 1.6])1208    axi.set_xlabel(r"$r_H\;$ [$10^{26}\,$m]", fontsize=7.5, labelpad=2)1209    axi.tick_params(labelsize=7, color=MID, labelcolor=MID, length=2.5, width=0.5)1210    for s in axi.spines.values():1211        s.set_color(MID); s.set_linewidth(0.5)12121213    fig.subplots_adjust(left=0.11, right=0.97, top=0.945, bottom=0.05)1214    ax.set_xlim(-37, 35.0)1215    ax.set_ylim(-45, 62)   # re-assert: pinned after all children1216    # ---- plain version with the object-capacity axis [register] ----1217    # 14:C10 (p14027)1218    family("cap", "K")1219    axi.remove()12201221    # right axis, same format as the left mass axis: log10 capacity, kappa1222    # primary, zero at the H atom (kappa_H = 3, the first Object), one dex of1223    # capacity per dex of the chart.1224    yH = log10(1.674e-27)                    # the H-atom mass line [approx SI]1225    fig.subplots_adjust(left=0.11, right=0.90, top=0.945, bottom=0.05)1226    fig.canvas.draw()1227    chk("the pin held: no datalim drift", ax.get_ylim() == (-45.0, 62.0))1228    pos = ax.get_position()12291230    rax = fig.add_axes([pos.x1, pos.y0, 0.0001, pos.height])1231    rax.set_ylim(-45.0, 62.0)1232    rax.set_xticks([])1233    for s in ("top", "bottom", "left", "right"):1234        rax.spines[s].set_visible(False)1235    rax.yaxis.tick_right()1236    # the mole, drawn [chart]: the gram's horizontal lands on Avogadro's1237    # number on the count face -- the axis relative to its zero counts hydrogen1238    # units, and N_A is the hydrogen count of the gram (residue: the 1.008 u1239    # hydrogen/amu offset, 0.0035 dex, invisible at chart scale)1240    NA = 6.02214076e231241    chk("the one-gram horizontal lands on N_A to 0.005 dex", abs((-3 - yH) - log10(NA)) < 0.005)1242    ax.plot([-1.3, 35.0], [-3, -3], color=MID, lw=0.7, ls=(0, (2, 3)), zorder=2)1243    ax.plot([-37, -5.2], [-3, -3], color=MID, lw=0.7, ls=(0, (2, 3)), zorder=2)1244    ax.text(-3.25, -3, r"$1\,$g", va="center", ha="center", fontsize=8, color=MID)1245    rax.set_yticks([yH, yH + 20, yH + 40, yH + 60, -3.0])1246    rax.set_yticklabels(["0", "20", "40", "60", r"$N_A\sim 6\!\times\!10^{23}$"])1247    rax.tick_params(axis="y", labelsize=8, color=MID, labelcolor=MID,1248                    length=3, width=0.6)1249    ax.text(32, 8.5, r"holographic ring capacity $\log_{10}\kappa_O$", rotation=90,1250            va="center", ha="center", fontsize=11)1251    ax.text(34.4, yH, r"$\kappa_H{=}3$", va="center", ha="right",1252            fontsize=8, color=MID)1253    # the axis terminal: the count face sqrt(S), the paper's measured quantity,1254    # at its exact height above the anchor (log10(sqrt(S_L)/3) rel dex)1255    y_top = yH + log10((S_L ** 0.5) / 3)1256    ax.text(34.4, y_top, r"$\sqrt{S}\sim10^{61}$", va="center", ha="right",1257            fontsize=8, color=MID)1258    save(fig, "registrable-wedge-capacity")1259    family("cap", "F")1260    chk("registrable-wedge-capacity.png and .pdf written to out/", os.path.exists(os.path.join(OUT, "registrable-wedge-capacity.png")) and os.path.exists(os.path.join(OUT, "registrable-wedge-capacity.pdf")))1261    plt.close("all")12621263    flush("cap", order=["A", "K", "F"])12641265def block_win():1266    """Block win — the additive window (B11), the lock of like units (B12), the shells of capacity 3N with fusion (B13); exact integers1267    throughout (the master's check_f7.py of 17 September 2026, parts 1-3, without its chart part)."""1268    import itertools1269    from math import lcm12701271    def is_prime(n):1272        if n < 2: return False1273        if n % 2 == 0: return n == 21274        d = 31275        while d * d <= n:1276            if n % d == 0: return False1277            d += 21278        return True12791280    # ---- B11: the additive window ----1281    counts = []; ok_w = True1282    for N in range(1, 5):1283        sums = [sum(t) for t in itertools.product(range(-3, 4), repeat=N)]1284        counts.append((7 ** N, max(abs(x) for x in sums)))1285        if max(abs(x) for x in sums) != 3 * N or set(sums) != set(range(-3 * N, 3 * N + 1)): ok_w = False1286    ok_m = True1287    for N in range(1, 4):1288        for ks in itertools.product((1, 2, 3), repeat=N):1289            sums = {sum(t) for t in itertools.product(*[range(-k, k + 1) for k in ks])}1290            if max(abs(x) for x in sums) != sum(ks) or sums != set(range(-sum(ks), sum(ks) + 1)): ok_m = False1291    trip = list(itertools.product(range(-3, 4), repeat=3)); beyond = sum(1 for t in trip if abs(sum(t)) > 6)1292    print(f"additive window at kappa = 3: (7^N tuples, max |sum|) = {counts}; triples beyond the pairwise bound 6: {beyond} of {len(trip)}")1293    # 14:B11 (p14046)1294    family("win", "W")1295    chk("N = 1..4 at kappa = 3: max |sum h_i| = 3N and every total in [-3N, 3N] occurs (7^N tuples, exhaustive)", ok_w and counts == [(7, 3), (49, 6), (343, 9), (2401, 12)])1296    chk("mixed capacities kappa_i in {1, 2, 3}, N <= 3: the least resolving window is sum kappa_i, every total occurs", ok_m)1297    chk("38:A8 at p = 13: |h1 + h2| <= 6 = 2 kappa for every pair; 20 of 343 triples exceed 2 kappa (the window grows with N)",1298        all(abs(a + b) <= 6 for a in range(-3, 4) for b in range(-3, 4)) and beyond == 20)12991300    # ---- B12: the lock of like units ----1301    def orbits(ns):1302        seen = set(); count = 0; lengths = set()1303        for start in itertools.product(*[range(n) for n in ns]):1304            if start in seen: continue1305            count += 1; cur = start; L = 01306            while cur not in seen:1307                seen.add(cur); cur = tuple((c + 1) % n for c, n in zip(cur, ns)); L += 11308            lengths.add(L)1309        return count, lengths1310    o43, o1212, o121212 = orbits((4, 3)), orbits((12, 12)), orbits((12, 12, 12))1311    print(f"diagonal drive orbits: (4, 3) {o43}; (12, 12) {o1212}; (12, 12, 12) {o121212}")1312    # 14:B12 (p14047)1313    family("win", "L")1314    chk("N like units on C_12, N <= 6: joint recurrence lcm(12, ..., 12) = 12, no element of order 12N",1315        all(lcm(*([12] * N)) == 12 for N in range(2, 7)) and all(12 * N != 12 for N in range(2, 7)))1316    chk("the diagonal drive on C_12^2: 12 orbits of length 12 (one C_12 offset per further unit); on C_12^3: 144 orbits of length 12",1317        o1212 == (12, {12}) and o121212 == (144, {12}))1318    chk("the offset is superselected: (x + 1, y + 1) keeps x - y (mod 12) for every point of C_12^2",1319        all(((x + 1) - (y + 1)) % 12 == (x - y) % 12 for x in range(12) for y in range(12)))13201321    # ---- B13: the shells of capacity 3N and fusion ----1322    pr = [N for N in range(1, 10) if is_prime(12 * N + 1)]1323    print(f"12N + 1 prime for N in {pr} (N < 10); N = 2: 25 = 5^2")1324    # 14:B13 (p14048)1325    family("win", "S")1326    chk("a shell of capacity 3N is F_{12N+1}: 4 (3N) + 1 = 12N + 1; 12N + 1 prime exactly for N in {1, 3, 5, 6, 8, 9} below 10, 25 = 5^2 at N = 2",1327        all(4 * (3 * N) + 1 == 12 * N + 1 for N in range(1, 10)) and pr == [1, 3, 5, 6, 8, 9] and 25 == 5 * 5)1328    chk("fusion of coprime cycles is multiplicative: lcm(4, 3) = 12, kappa = 4 * 3 / 4 = 3 (hydrogen), one orbit of length 12", lcm(4, 3) == 12 and 4 * 3 // 4 == 3 and o43 == (1, {12}))1329    chk("like cycles do not fuse: gcd(12, 12) = 12, lcm(12, 12) = 12; the capacity-3N shell (13 at N = 1) is not the lock's register C_12",1330        lcm(12, 12) == 12 and 12 != 13 and is_prime(13))13311332    flush("win", order=["W", "L", "S"], kinds={"W": "EXACT", "L": "EXACT", "S": "EXACT"})13331334if __name__ == "__main__":1335    import time1336    want = [a.lower() for a in sys.argv[1:]] or list(BLOCK)1337    bad = [b for b in want if b not in BLOCK]1338    if bad: sys.exit(f"no block {', '.join(bad)}: the blocks are {', '.join(BLOCK)}")1339    t0 = time.time()1340    for b in want:1341        t = time.time(); print(f"— block {b}"); _run_block(b); print(f"    [block {b}: {time.time() - t:.1f} s]")1342    ok = summary(write=(want == list(BLOCK)))1343    print(f"{len(RESULTS)} family checks, {len(MICRO)} micro-checks, {time.time() - t0:.1f} s" + ("; results.json written" if want == list(BLOCK) else ""))1344    sys.exit(0 if ok else 1)13451346