The STF Process Clock V1.1 — Coefficient-Complete Source-Capacity Timing Law

VERIFICATION RECORD — audit companion to The Selective Transient Field from First Principles (Theory Edition V9.7.1). This section publishes the programme's technical record as it exists: the machine-verified release stack and the post-freeze calculation records, rendered from the frozen source files without editorial rewriting. It is written for auditors, not readers; the reader-facing statement of the theory is the Theory Edition. Every document below is hash-pinned; the packages that carry the checkers and data are listed on the record landing page.

Source file STF_Process_Clock_Law_V1_1_2026-09-03.md  ·  SHA-256 cad109610bf34c27016dff950152e7a7293b8c8d6169694d8c787a866c835276  ·  rendered as-is (GitHub-style ```math fences converted to display math).

A Coefficient-Complete Source-Capacity Timing Law with Declared Inputs

Version: 1.1 (supersedes STF UHECR Process-Anchor Derivation V1.0 of the same date; revision record in Section 15)
Date: 3 September 2026
Author: Z. Paz
Controlling baseline: The Selective Transient Field from First Principles, v9.6.1 Backbone Edition (author final), and v9.6.1 Study 18
Status: standalone derivation record
Claim discipline: theorem / derived on selected comparator / declared input / measured input / conditional bridge / retrospective comparison / not established

Abstract

The frozen v9.6.1 process contains a source-capacity clock for a merging equal-mass phase-star binary: for a declared surface field \(B_*\), dipolar interaction geometry, orbital relative speed, Hillas rigidity capacity, and a Peters-plus-electromagnetic inspiral clock, the first-attainment time of rigidity \(\mathcal R\) obeys

\[ \boxed{\tau_{\rm loc}(\mathcal R)=C_{\rm loc}\,\mathcal R^{-8/5}}, \qquad C_{\rm loc}=\frac{5c^5}{512G^3M^3(1+\chi)} \left[\kappa_H B_* R^3\sqrt{\frac{2GM}{c^2}}\right]^{8/5}, \]

with \(M=2^{1/5}\mathcal M_c\) and \(R=(\bar R/\bar M)\,GM/c^2\). No observed duration enters the construction. The coefficient scales as \(C_{\rm loc}\propto B_*^{8/5}\mathcal M_c^{13/5}/(1+\chi)\), so the law is a two-parameter family: the binary chirp mass is a measured per-event input, the surface field is a declared comparator input, and the rigidity tier at which the clock is read is a declared operational choice.

This paper writes the law in that form and evaluates it on measured inputs. At the GWOSC confident-catalog median chirp mass \(\mathcal M_c=26.5\,M_\odot\), with \(B_*=10^{15}\,\mathrm G\) and \(\mathcal R_\dagger=1\,\mathrm{EV}\), the local first-attainment time is \(1568.99\) d (\(4.30\) yr); over the catalog’s 16th–84th percentile interval \(8.87\le\mathcal M_c/M_\odot\le36.33\) it is \(92.0\)–\(3531.4\) d (\(0.25\)–\(9.67\) yr); this local band is a capacity comparator on a branch whose UHE-ion production audit fails (v9.6.1 §VI.B). The surviving remote production branch multiplies the coefficient by \(\mathcal F^{8/5}\), with \(\mathcal F\) unmeasured (\(\sqrt5\) at the Study 16 reference tuple).

Compared afterwards, the observed \(3.32\pm0.89\) yr UHECR timing lies inside the local population band and, on the remote branch, inside the band for \(0.51\le\mathcal F\le5.01\); at \(B_*=10^{15}\) G and 1 EV it corresponds to a \(24.0\,M_\odot\) source on the local branch and, at the reference tuple, to a \(14.6\,M_\odot\) source on the remote branch, both inside the measured population. On the record’s narrower interval the remote comparator did not contain the value; the containment here is a consequence of the measured interval’s width; at that width it is uninformative, and it is retrospective. The testable content of the law is its scaling: source time \(\propto\mathcal M_c^{13/5}\) at fixed rigidity, \(\propto B_*^{8/5}\), tier ratio \(5^{8/5}=13.13\) between the 1 EV and 5 EV surfaces, and the \(-8/5\) exponent as a diagnostic of dipolar geometry with accelerator size \(b\). The decisive test is the mass scaling, and a preliminary, window-truncated, unreviewed run on the reconstructed association sample does not show it (Section 13, item 6). The conditional scalar-mass bridge is stated as the function \(m_s=h/(c^2C)\) of the same inputs; it is a bridge, not a consequence, and in the memory-gated model class of Study 22 the framework’s own scalar-mass signature is a knee, not a centroid, in the timing distribution.

1. Purpose and exact statement

1.1 The question

Does the frozen v9.6.1 process contain a year-scale timing law whose construction uses no observed duration, and if so, what are its inputs and what does it predict?

1.2 The answer

Yes. The law is \(\tau_{\rm loc}=C_{\rm loc}\mathcal R^{-8/5}\) with the coefficient-complete \(C_{\rm loc}\) above. Its inputs are of four kinds and are kept apart throughout:

The controlling distinctions:

1.3 Where this law belongs

In the separation the framework is moving toward — theory on one side, observational anchors on the other — this law is theory-side material: it is written in symbols, its numerical layer is a function of declared and measured inputs, and the observed durations enter only in Sections 11–12, after the law is evaluated. The observational side supplies per-event chirp masses, the composition-dependent rigidity, and the timing record.

2. Timing and selection firewall

No minimization, inverse solve, interval tuning, mass selection, or threshold adjustment against an observed time occurs in Sections 3–10. The observations enter in Section 11.

Two facts about the declared inputs are stated rather than hidden. The comparator (dipolar geometry, \(B_*\), the rigidity tiers) was declared in Study 15 by a session that knew the observed timing; the algebra is non-fitted, the placement of the comparator is not blind. And the V1.0 chirp-mass interval \(18.5\)–\(26\,M_\odot\), labeled “independently inherited” there and in Study 18’s firewall, is replaced here: its endpoints trace to the V7.9 value \(18.54\,M_\odot\) (withdrawn from the corpus on 22 August 2026, V52 edit log) and to the chirp mass \(26.12\,M_\odot\) of the \(30+30\,M_\odot\) example binary. The chirp mass is a measured quantity and is treated as one. The same label stands in Study 18’s TIMING_FIREWALL.csv, in its COMPARATOR_MODEL_CARD.json declared input \([18.5, 26.0]\), and in backbone §VI.A (“the programme’s inherited chirp-mass band”); a versioned correction of those three entries is required and is not performed by this paper.

3. Definitions and clock taxonomy

Three different times are never identified silently:

\[ \text{source capacity time}\neq\text{detector lead}\neq\text{internal phase recurrence}. \]

On the charged small-angle branch \(L(\mathcal R)=\tau_{\rm src}(\mathcal R)-\Delta_{\rm mag}(\mathcal R)\). What is derived here is \(\tau_{\rm src}\) on the local capacity branch, and its remote lift.

4. Frozen inputs

4.1 Constants (cgs)

\[ G=6.67430\times10^{-8}\ \mathrm{cm^3\,g^{-1}\,s^{-2}},\quad c=2.99792458\times10^{10}\ \mathrm{cm\,s^{-1}},\quad M_\odot=1.98847\times10^{33}\ \mathrm g, \]

\[ 1\ \mathrm d=86400\ \mathrm s,\qquad 1\ \mathrm{yr}=365.25\ \mathrm d,\qquad \kappa_H=3.0\times10^{-16}\ \frac{\mathrm{EV}}{\mathrm{G\,cm}}. \]

\(\kappa_H\) is the rounded value of \(e\cdot(1\,\mathrm G)(1\,\mathrm{cm})=299.79\,\mathrm{eV}\) (rounding \(0.07\%\)).

4.2 Phase-star background

The v9.6.1 compact phase-star solution supplies

\[ \bar M=0.10827192328882561,\qquad \bar R=0.7107045862633132,\qquad \lambda_R\equiv\bar R/\bar M=6.564070949099532. \]

4.3 Declared comparator inputs

Surface field \(B_*=10^{15}\,\mathrm G\) (Study 15). The field is not derived; its physical ceiling is the virial field of the phase star, \(B_{\rm vir}=\sqrt{8\pi GM^2/(RV)}\) with \(V=4\pi R^3/3\), \(\approx4.4\times10^{16}\,\mathrm G\) at the reference mass, so the declared value carries \(\sim5\times10^{-4}\) of the binding energy. Rigidity tiers \(1\,\mathrm{EV}\) and \(5\,\mathrm{EV}\) (Study 15). The accelerator size in the Hillas capacity is the separation \(b\) (Study 15).

4.4 Measured population input

Chirp masses are taken from the GWOSC event list (export of 2 September 2026, Appendix C), restricted to the confident catalogs GWTC-1, 2.1, 3 and 4.0, one row per event (latest catalog preferred), binary black holes selected by \(m_2>3\,M_\odot\): 167 events (of 219 confident events, 43 GWTC-4.0 entries carry no source-frame parameter estimates in the export and are excluded, and 9 are neutron-star-containing). Source-frame chirp-mass quantiles (linear-interpolation, Hyndman–Fan type 7):

The median \(26.5\,M_\odot\) is the reference point of this paper; the 16–84 % and 5–95 % intervals are the population bands. (For the record, 17 % of these events fall inside the V1.0 interval.)

5. From chirp mass to the physical phase-star scale

For \(M_1=M_2=M\), \(\mathcal M_c=(M_1M_2)^{3/5}(M_1+M_2)^{-1/5}=2^{-1/5}M\), so

\[ \boxed{M=2^{1/5}\mathcal M_c.} \]

Matching the dimensionless mass coordinate to \(GM/c^2\),

\[ \boxed{R=\lambda_R\frac{GM}{c^2}.} \]

At the reference point \(M=30.44\,M_\odot\), \(R=295.06\) km; across the 16–84 % band \(M=10.19\)–\(41.73\,M_\odot\), \(R=98.76\)–\(404.51\) km.

6. Rigidity capacity as a function of separation

6.1 Dipolar interaction field

\[ \boxed{B_{\rm int}(b)=B_*\left(\frac Rb\right)^3.} \]

6.2 Relative orbital speed

Each body orbits at radius \(b/2\); \(GM^2/b^2=Mv^2/(b/2)\) gives \(v^2=GM/(2b)\) and \(v_{\rm rel}=2v\):

\[ \boxed{\beta_{\rm rel}(b)=\sqrt{\frac{2GM}{bc^2}}.} \]

6.3 Hillas capacity

With accelerator size \(b\) and field \(B_{\rm int}\),

\[ \mathcal R_{\rm loc}(b)=\kappa_H\,\beta_{\rm rel}\,b\,B_{\rm int} =\kappa_HB_*R^3\sqrt{\frac{2GM}{c^2}}\;b^{-5/2} \equiv K_{\rm loc}\,b^{-5/2}. \]

The exponent is \(b^{-3}\times b^{+1}\times b^{-1/2}\). Both the geometry and the size choice matter: with \(B\propto b^{-n}\) and accelerator size \(b\), \(\mathcal R\propto b^{-(n-1/2)}\); with accelerator size \(R\) instead of \(b\), the dipole gives \(\mathcal R\propto b^{-7/2}\). The \(-8/5\) law below is the diagnostic of the pair \((n=3,\ \text{size }b)\).

6.4 First-attainment separation

\[ \boxed{b_\dagger=\left(\frac{K_{\rm loc}}{\mathcal R_\dagger}\right)^{2/5}.} \]

7. The two-loss inspiral clock

Orbital energy \(E_{\rm orb}=-GM^2/(2b)\). Peters quadrupole power for equal masses, \(P_{\rm GW}=(64/5)G^4M^5/(c^5b^5)\). The selected two-magnetosphere comparator power is \(P_{\rm EM}=B_*^2GMR^6/(cb^5)\), so the ratio is constant along the comparator:

\[ \boxed{\chi=\frac{P_{\rm EM}}{P_{\rm GW}}=\frac{5}{64}\frac{B_*^2c^4R^6}{G^3M^4}}, \qquad \chi=0.0104\ \text{(reference)},\quad 0.0012\text{–}0.0196\ \text{(16–84 \%)}. \]

Energy balance \(-\dot E_{\rm orb}=(1+\chi)P_{\rm GW}\) gives \(\dot b=-(128/5)(1+\chi)G^3M^3/(c^5b^3)\) and, integrating to formal coalescence,

\[ \boxed{\tau(b)=\frac{5c^5b^4}{512G^3M^3(1+\chi)}\equiv K_\tau b^4} \]

(with \(K_\tau\) in days per cm\(^4\) when divided by 86400).

8. Elimination theorem and scaling form

Substituting \(b_\dagger\) into the clock,

\[ \boxed{\tau_{\rm loc}(\mathcal R)=C_{\rm loc}\mathcal R^{-8/5}},\qquad \boxed{C_{\rm loc}=K_\tau K_{\rm loc}^{8/5} =\frac{5c^5}{512G^3M^3(1+\chi)}\left[\kappa_HB_*R^3\sqrt{\frac{2GM}{c^2}}\right]^{8/5}.} \]

Since \(R\propto M\), \(K_{\rm loc}\propto B_*M^{7/2}\) and \(K_\tau\propto M^{-3}/(1+\chi)\), hence

\[ \boxed{C_{\rm loc}\propto\frac{B_*^{8/5}\mathcal M_c^{13/5}}{1+\chi}}, \qquad \frac{\partial\ln C_{\rm loc}}{\partial\ln B_*}=\frac85-\frac{2\chi}{1+\chi}, \qquad \frac{\partial\ln C_{\rm loc}}{\partial\ln\mathcal M_c}=\frac{13}{5}-\frac{2\chi}{1+\chi}. \]

At the reference point the two derivatives are \(1.5793\) and \(2.5793\). The convenient scaling form of the law, valid to the \(\chi\) drift (\(\lesssim2\%\) across the band), is

\[ \boxed{ \tau_{\rm loc}(\mathcal R;\mathcal M_c,B_*) \simeq1569\ \mathrm d\; \left(\frac{\mathcal M_c}{26.5\,M_\odot}\right)^{13/5} \left(\frac{B_*}{10^{15}\,\mathrm G}\right)^{8/5} \left(\frac{\mathcal R}{1\,\mathrm{EV}}\right)^{-8/5}.} \]

The boxed coefficient is in seconds\(\cdot\)EV\(^{8/5}\); the numerical layer quotes \(C_{\rm loc}/86400\) in days. This is the statement the theory side carries: a law with two physical scalings and one declared field. The year scale is not installed; at \(10^{15}\) G and the catalog mass scale the 1 EV surface sits at \(b_\dagger\sim10^{10}\) cm and the fourth-power clock converts it to years.

9. Numerical evaluation before any observed time

9.1 Reference point and population bands (1 EV, \(B_*=10^{15}\) G)

In years: reference \(4.30\); 16–84 % band \(0.252\)–\(9.668\); 5–95 % band \(65.44\)–\(7126.12\) d \(=0.179\)–\(19.51\) yr.

9.2 Threshold dependence

The ratio between the 1 EV and 5 EV surfaces is fixed by the law at \(5^{8/5}=13.13\) for any source.

9.3 Field dependence

At the reference mass, \(B_*=10^{14}\), \(10^{15}\), \(10^{16}\) G give \(C_{\rm loc}=39.8\), \(1569.0\), \(30887.9\) d. The declared field places the band on the time axis; this is the largest single freedom in the numerical layer and is stated as such.

9.4 Phase-star fundamental mode

With \(\omega_0R=0.1889913902\), the reference period \(2\pi R/(\omega_0Rc)=0.0327\) s; a genuine independent dynamical scale, not a year-scale clock.

10. The two production branches

10.1 Local branch

The local capacity surface above is a comparator clock. The v9.6.1 production audit finds that local magnetospheric UHE-ion production is curvature-limited below 1 EV, so the local branch does not supply escaping UHE nuclei: the band of Section 9 is a capacity comparator, not a demonstrated source.

10.2 Remote branch

For a remote Poynting-supported bubble with \(L_B=f_\Omega\beta_wc(B_\phi r)^2\), the time- and coherence-limited capacities give \(\mathcal R_{\rm rem}=\mathcal F\mathcal R_{\rm loc}\) with

\[ \boxed{\mathcal F=\frac{\zeta_{\rm acc}\Phi}{\sqrt{2f_\Omega\beta_w}}},\qquad \Phi=\min\!\left(\frac{\beta_w}{\eta_{\rm acc}},\epsilon_c\right), \qquad \boxed{C_{\rm rem}=C_{\rm loc}\mathcal F^{8/5}.} \]

At the Study 16 reference tuple (\(f_\Omega=0.03\), \(\beta_w=0.3\), \(\epsilon_c=0.3\), \(\eta_{\rm acc}=\zeta_{\rm acc}=1\)), \(\mathcal F=\sqrt5\) and the reference remote time is \(5685.85\) d (\(15.6\) yr); over the 16–84 % band \(333.4\)–\(12797.3\) d. The tuple is declared; the full declared axes give a broad \(\mathcal F\) surface, and an independent determination of \(\mathcal F\) (Study 20’s specification) is required before the remote branch yields an event time.

10.3 Propagation

\(\Delta_{\rm mag}=A\mathcal R^{-2}\) with \(A\) set by the line of sight, so \(L=C\mathcal R^{-8/5}-A\mathcal R^{-2}\), with maximum \(L_{\max}=256C^5/(3125A^4)\) at \(\mathcal R_*=(5A/4C)^{5/2}\). Propagation translates a source time into a detector lead; it supplies no normalization.

11. Posterior comparison

Only now are the observed durations introduced: \(T_{\rm obs}=3.32\pm0.89\) yr \(=1212.63\pm325.07\) d (UHECR channel) and 71 d (GRB channel).

Local branch, 1 EV: \(1212.63\) d lies inside the 16–84 % band \([92.0,\,3531.4]\) d and corresponds to \(\mathcal M_c=23.98\,M_\odot\) at \(B_*=10^{15}\) G — a mid-population source. The 71-day value at 5 EV corresponds to \(\mathcal M_c=21.66\,M_\odot\). The two anchors therefore imply two different sources or a rigidity ratio of \((1212.63/71)^{5/8}=5.9\) rather than 5; the law does not require them to share a source.

Remote branch: the band contains \(1212.63\) d for \(0.513\le\mathcal F\le5.011\) (16–84 % mass band), an interval that includes the reference \(\sqrt5\) and overlaps the robust composition-consistency band \(\mathcal F\in[1.55,\,28.70]\) of Study 20. At \(\mathcal F=\sqrt5\) the observed value corresponds to \(\mathcal M_c=14.59\,M_\odot\) (1 EV) and the 71-day value to \(13.18\,M_\odot\) (5 EV), the 24th and 22nd percentiles of the measured population. Two qualifications. On the record’s interval the same comparator does not contain either anchor (Study 18 RETROSPECTIVE_ANCHOR_AUDIT.csv, contained = false; backbone §VI.B, \(2245.47\)–\(5413.20\) d): the containment here is a consequence of the measured interval’s width, not a change in the branch. And the Study 20 band is a posterior surface that itself scales its rigidity band with the 1,212-day anchor, so the overlap with it is not independent support.

The correct statement is:

At population width, both branches of the forward law contain the observed year-scale timing for a chirp mass inside the measured GW population. The comparison is retrospective, the containment is uninformative at this width, and no observed value was used to construct the law. What the law adds is a per-event prediction — a source time that scales as \(\mathcal M_c^{13/5}\) at fixed rigidity — and that prediction, not the containment, is where the law can fail; a preliminary, window-truncated, unreviewed run does not show it (Section 13, item 6).

12. Conditional scalar-mass bridge

If one full internal recurrence has period \(T_s\), then \(m_s=h/(c^2T_s)\). Identifying \(T_s\) with the process clock is an additional bridge (the Clock-Rate Invisibility Lemma requires an operational observation map between the universal ordering field and a material clock); stated as a function of the same inputs,

\[ \boxed{ m_s(\mathcal M_c,B_*,\mathcal R_\dagger)\stackrel{\rm bridge}{=}\frac{h}{c^2C(\mathcal M_c,B_*)}\,\mathcal R_\dagger^{8/5} \simeq3.05\times10^{-23}\ \mathrm{eV} \left(\frac{\mathcal M_c}{26.5\,M_\odot}\right)^{-13/5} \left(\frac{B_*}{10^{15}\,\mathrm G}\right)^{-8/5} \left(\frac{\mathcal R_\dagger}{1\,\mathrm{EV}}\right)^{8/5}.} \]

At the reference point \(m_s=3.0508\times10^{-23}\,\mathrm{eV}\); over the 16–84 % band \(1.36\times10^{-23}\)–\(5.20\times10^{-22}\,\mathrm{eV}\). The v9.6.1 adopted value \(3.94\times10^{-23}\,\mathrm{eV}\) lies inside. This band is a translation of the declared field and the population mass scale; it does not derive \(m_s\), and it makes the scalar mass source-dependent, which the theory’s universal scalar is not. Two facts fix how far the bridge can be pushed. The record’s own scalar-mass audit (Study 21, 2 September 2026) found no admissible binary-independent mechanism fixing \(m_s\), so the mass remains adopted. And in the memory-gated model class examined in Study 22 (Tasks A–B, in-session run of 3 September 2026, unreviewed; Appendix C) the framework’s own signature of \(m_s\) in the timing sector is not a centroid: the recorded memory kernel turns a monotone inspiral drive into a broken power law with a knee at \(\tau_c=\hbar/(m_sc^2)\) (\(0.53\) yr at the adopted mass) and no peak, so a lead-time centroid cannot be inverted for \(m_s\) while a full-shape fit for the knee could. The process clock and the memory clock are different clocks, and the bridge above identifies them by declaration only.

13. Falsifiers and open tests

The law is falsified or materially revised if:

1. the source is not on the equal-mass phase-star branch, or independent mass–radius measurements exclude the scale matching \(R=\lambda_RGM/c^2\);
2. the interaction field is not dipolar over the threshold separation, or the accelerator size is not the separation (either changes the exponent, Section 6.3);
3. the surface field differs substantially from \(10^{15}\) G (coefficient \(\propto B_*^{1.58}\));
4. electromagnetic loss does not share the \(b^{-5}\) dependence (non-constant \(\chi\));
5. the acceleration efficiency carries a separation dependence;
6. per-event source times do not scale as \(\mathcal M_c^{13/5}\) at fixed rigidity. This is the decisive test and it is available in principle from the association sample itself (lead time against catalog chirp mass, with propagation and window truncation modelled). A preliminary in-session run on the reconstructed association sample (2 September 2026; 64 events with GWOSC masses) returned lead-time slopes of \(+0.04\pm0.03\) (detector frame) and \(-0.12\pm0.04\) (source frame) against the predicted \(2.58\); events below \(\mathcal M_c\approx28\,M_\odot\) (\(C_{\rm loc}<5\) yr) lie inside the \(\pm5\)-yr window and should show the full slope. The run is window-truncated and unreviewed and is recorded in the session review, not here; on the remote branch the scaling holds only if \(\mathcal F\) carries no mass dependence. A held-out, per-event test is required before the law is used at the event level;
7. a measured \(\mathcal F\) falls outside \([0.51,\,5.01]\), which would exclude a 1 EV remote-branch source inside the 16–84 % population for the 1,212-day structure (not the law itself);
8. a claimed detector-time association requires a propagation coefficient incompatible with the measured line of sight.

The leading numerical uncertainties are the declared field and, on the remote branch, \(\mathcal F\); the population mass interval is measured, and its width is the width of the population-level statement; a per-event prediction carries the per-event mass uncertainty instead.

14. Claim matrix

No grade or gate status in the v9.6.1 record changes. Three record consequences are stated rather than performed. The provenance label of the chirp-mass input requires a versioned correction (Section 2). The frozen comparator band \(619.63\)–\(1493.75\) d is retained as recorded (Study 18 FREEZE_AMENDMENT_RECORD.json); the population evaluation here is a separate evaluation, not an amendment to the freeze. And the population interval widens the rigidity windows that backbone §VII.D assigns to the 1,212-day structure from \(0.66\)–\(1.14\) EV (local) and \(1.47\)–\(2.55\) EV (remote) to \(0.20\)–\(1.95\) EV and \(0.45\)–\(4.36\) EV; the composition discrimination of backbone §VIII and Studies 19–20 was evaluated on the narrower windows and is not re-evaluated here.

15. Revision record (V1.0 → V1.1)

V1.0 (STF UHECR Process-Anchor Derivation, 3 September 2026, SHA-256 70e932de…) presented the same derivation with the chirp-mass interval \(18.5\)–\(26\,M_\odot\) labeled “independently inherited”, headlined the containment of the 1,212-day value by the resulting \(619.63\)–\(1493.75\) d band, and framed the result as removing the observation from the construction of the timing scale. The session review of the same date (STF_UHECR_Process_Anchor_Derivation_V1_0_REVIEW_2026-09-03.md, SHA-256 53d4d8b9…) reproduced every number and found: the derivation and band are Study 15/18 material already firewalled in v9.6.1; the interval’s endpoints are the withdrawn V7.9 value \(18.54\,M_\odot\) and the \(30+30\,M_\odot\) example binary; the containment is a property of that interval’s width and placement; the containment holds on the branch the record rejects, while the remote branch at the reference tuple did not contain the value for that interval; the cited backbone hash was the pre-final draft; reference 8’s title was wrong.

V1.1 keeps the derivation (unchanged algebra, regression-checked against the V1.0 endpoints in Appendix A), replaces the mass interval by the measured GWOSC population (Section 4.4), writes the law in scaling form (Section 8), evaluates both branches (Sections 9–11) — on the measured population the remote branch at \(\sqrt5\) also contains the observed value, for a \(14.6\,M_\odot\) source — states the scalar-mass bridge as a function with its limits (Section 12), adds the mass-scaling test as the decisive falsifier (Section 13), corrects the baseline hash and the title of V1.0’s reference 8 (reference 9 here), and removes the “removes the observation” framing: the construction never used it, and this paper documents the law and its declared inputs.

Appendix A. Deterministic reproducer

Standard-library Python. The observed durations are defined only in the posterior block at the end.

import math

# Frozen physical constants, cgs.
G = 6.67430e-8
C_LIGHT = 2.99792458e10
M_SUN = 1.98847e33
DAY_S = 86400.0
YEAR_D = 365.25
H_EV_S = 4.135667696e-15

# Inherited phase-star background (v9.6.1) and declared comparator inputs.
MBAR = 0.10827192328882561
RBAR = 0.7107045862633132
B_SURFACE_G = 1.0e15      # declared
KAPPA_H = 3.0e-16         # EV per (G cm); rounded e(1 G)(1 cm) = 299.79 eV
P = 8.0 / 5.0

# Measured population input (Appendix C): GWOSC confident-catalog
# BBH source-frame chirp masses, 167 events with m_2 > 3 M_sun.
MC_REF = 26.5             # median
MC_16, MC_84 = 8.87, 36.33
MC_05, MC_95 = 7.78, 47.85


def forward(chirp_mass_solar, rigidity_ev=1.0, b_surface=B_SURFACE_G,
            remote_factor=1.0):
    mass = 2.0 ** 0.2 * chirp_mass_solar * M_SUN
    radius = (RBAR / MBAR) * G * mass / C_LIGHT ** 2
    chi = ((5.0 / 64.0) * b_surface ** 2 * C_LIGHT ** 4 * radius ** 6
           / (G ** 3 * mass ** 4))
    k_loc = (KAPPA_H * b_surface * radius ** 3
             * math.sqrt(2.0 * G * mass / C_LIGHT ** 2))
    k_tau = (5.0 * C_LIGHT ** 5
             / (512.0 * G ** 3 * mass ** 3 * (1.0 + chi) * DAY_S))
    c_days = k_tau * k_loc ** P * remote_factor ** P
    separation = (k_loc * remote_factor / rigidity_ev) ** 0.4
    return {
        "component_mass_solar": mass / M_SUN,
        "radius_km": radius / 1.0e5,
        "chi": chi,
        "c_days": c_days,
        "source_days": c_days * rigidity_ev ** (-P),
        "separation_cm": separation,
        "separation_over_radius": separation / radius,
        "beta_rel": math.sqrt(2.0 * G * mass / (separation * C_LIGHT ** 2)),
        "b_interaction_G": b_surface * (radius / separation) ** 3,
        "m_s_eV": H_EV_S / (c_days * DAY_S),
        "dlnC_dlnB": P - 2.0 * chi / (1.0 + chi),
        "dlnC_dlnM": 2.6 - 2.0 * chi / (1.0 + chi),
    }


def chirp_mass_for_time(target_days, rigidity_ev=1.0, remote_factor=1.0):
    lo, hi = 1.0, 200.0
    for _ in range(200):
        mid = 0.5 * (lo + hi)
        if forward(mid, rigidity_ev, B_SURFACE_G,
                   remote_factor)["source_days"] < target_days:
            lo = mid
        else:
            hi = mid
    return 0.5 * (lo + hi)


ref = forward(MC_REF)
lo16, hi84 = forward(MC_16), forward(MC_84)
lo05, hi95 = forward(MC_05), forward(MC_95)

assert math.isclose(ref["c_days"], 1568.986569537098, rel_tol=2e-13)
assert math.isclose(forward(MC_REF, 5.0)["source_days"],
                    119.47229851566442, rel_tol=2e-13)
assert math.isclose(lo16["c_days"], 91.99869728776754, rel_tol=2e-13)
assert math.isclose(hi84["c_days"], 3531.37521721786, rel_tol=2e-13)
assert math.isclose(lo05["c_days"], 65.4401366005507, rel_tol=2e-13)
assert math.isclose(hi95["c_days"], 7126.118120773689, rel_tol=2e-13)
assert math.isclose(ref["m_s_eV"], 3.050792478955473e-23, rel_tol=2e-13)
# legacy V1.0 endpoints, retained only as a regression check of the algebra
assert math.isclose(forward(18.5)["c_days"], 619.6285245816075, rel_tol=2e-13)
assert math.isclose(forward(26.0)["c_days"], 1493.7513654949166, rel_tol=2e-13)

print("reference (Mc = 26.5, B* = 1e15 G, 1 EV): C = %.6f d = %.5f yr"
      % (ref["c_days"], ref["c_days"] / YEAR_D))
print("16-84%% band: %.4f .. %.4f d ; 5-95%% band: %.4f .. %.4f d"
      % (lo16["c_days"], hi84["c_days"], lo05["c_days"], hi95["c_days"]))
print("conditional m_s at reference = %.6e eV ; 16-84%% band %.6e .. %.6e eV"
      % (ref["m_s_eV"], hi84["m_s_eV"], lo16["m_s_eV"]))
print("remote reference at F = sqrt(5): %.2f d"
      % forward(MC_REF, 1.0, B_SURFACE_G, math.sqrt(5.0))["c_days"])

# Posterior block: observed durations are defined only here.
OBS_DAYS = 3.32 * YEAR_D
print("posterior: 1,212.63 d inside 16-84%% local band: %s"
      % (lo16["c_days"] <= OBS_DAYS <= hi84["c_days"]))
print("posterior: chirp mass implied by 1,212.63 d at 1 EV: local %.2f, "
      "remote(sqrt5) %.2f M_sun"
      % (chirp_mass_for_time(OBS_DAYS),
         chirp_mass_for_time(OBS_DAYS, 1.0, math.sqrt(5.0))))
print("posterior: chirp mass implied by 71 d at 5 EV: local %.2f, "
      "remote(sqrt5) %.2f M_sun"
      % (chirp_mass_for_time(71.0, 5.0),
         chirp_mass_for_time(71.0, 5.0, math.sqrt(5.0))))
print("posterior: remote F containing 1,212.63 d over the 16-84%% band: "
      "[%.3f, %.3f]" % ((OBS_DAYS / hi84["c_days"]) ** (1.0 / P),
                        (OBS_DAYS / lo16["c_days"]) ** (1.0 / P)))

Expected output:

reference (Mc = 26.5, B* = 1e15 G, 1 EV): C = 1568.986570 d = 4.29565 yr
16-84% band: 91.9987 .. 3531.3752 d ; 5-95% band: 65.4401 .. 7126.1181 d
conditional m_s at reference = 3.050792e-23 eV ; 16-84% band 1.355464e-23 .. 5.202957e-22 eV
remote reference at F = sqrt(5): 5685.85 d
posterior: 1,212.63 d inside 16-84% local band: True
posterior: chirp mass implied by 1,212.63 d at 1 EV: local 23.98, remote(sqrt5) 14.59 M_sun
posterior: chirp mass implied by 71 d at 5 EV: local 21.66, remote(sqrt5) 13.18 M_sun
posterior: remote F containing 1,212.63 d over the 16-84% band: [0.513, 5.011]

Appendix B. Dependency ledger

No observed duration enters B-01 through B-10.

Appendix C. Source-control record

Population statistics in Section 4.4 are computed from the GWOSC export by selecting rows whose catalog is one of GWTC-1-confident, GWTC-2.1-confident, GWTC-3-confident, GWTC-4.0; keeping one row per commonName (latest catalog preferred); requiring mass_2_source \(>3\,M_\odot\) and a finite chirp_mass_source; and taking the 5th, 16th, 50th, 84th and 95th percentiles of chirp_mass_source (167 events). The SHA-256 of this paper is reported externally.

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End of paper.