SR-SEAM-002RECOVERY_GREEN · EXTENSION_YELLOW · F4_OPENS4/S2
Fire Train recomputation, 2026-09-21 · derivations and measured quantities only · every MN concept: Derrick E. Muncy
RECOVERY · GREENF1, F2 unbroken · closure 0.0e+0 · two routes agree to 5.8e-9
EXTENSION · YELLOWF4 open: Σ exponent −2 vs −4, separation (1+z)²
RECOMPUTATION
[G1] PROJECTION P: (γ⁻¹ ∈ ℂ) ↦ |γ| ∈ ℝ γ = (1−β²)^(−1/2)
[G2] REPORT β→1⁻ ⇒ γ⁻¹→0, report = 1/0
[G3b] RANK — log-log slope of γ⁻¹ against (1−β), measured:
1−β γ⁻¹ slope
1e-2 1.410673598e-1 —
1e-4 1.414178207e-2 0.499461199
1e-6 1.414213209e-3 0.499994625
1e-8 1.414213562e-4 0.499999946
1e-10 1.414213621e-5 0.499999991
slope → 1/2 ; γ⁻¹/(1−β)^(1/2) → 1.414213621 = √2
⇒ γ⁻¹ = √2 (1−β)^(1/2) [1 + O(1−β)]
RANK = 1/2. half-seat ⇒ branch point; arg jump = π/2. NOT a pole.
[G3] CLOSURE γ·γ⁻¹ − 1 at β = 0.5, 0.9, 0.99, 0.999999, 0.99999999
= 0.0e+0, 0.0e+0, 0.0e+0, 0.0e+0, 0.0e+0 (exact, all β)
[G4] CONTINUATION γ⁻¹(β) = √(1−β²), arg = 0 (β<1) | π/2 (β>1)
β |γ⁻¹| arg γ⁻¹
0.6 0.800000000 0 0.800000
1 0.000000000 undef 0
1.4 0.979795897 π/2 i·0.979796
2 1.732050808 π/2 i·1.732051
|γ⁻¹| finite ∀β. No pole exists in the continued object.
[G6] SECOND ROUTE (rapidity) β = tanh w, γ = cosh w
max relative Δ vs (1−β²)^(−1/2) over w∈[0,10], 1001 samples = 5.82e-9
dγ/dw = sinh w, finite ∀w ⇒ no singularity in the w-chart.
⇒ the divergence is a property of the β parameterisation, not of the object.
[G5] PHASE-RATE OPERATOR φ_obs(t) = φ_src(t/(1+z))
feature at φ = const ⇒ t ↦ (1+z)t
carrier f_obs/f_src = (1+z)⁻¹
envelope Δt_obs/Δt_src = (1+z)
one operator, two consequences, no second parameter.
z f_obs/f_src Δt_obs/Δt_src measured (SNe Ia)
0.5 0.666667 1.5000 1.5000
1 0.500000 2.0000 2.0000
2 0.333333 3.0000 3.0000
[G8] TOLMAN EXPONENT — derived both lanes from Σ = F/Ω
EXPANDING d_L = (1+z)² d_A, F = L/(4πd_L²), Ω = A/d_A²
Σ = L d_A²/(4πA d_L²) = L/(4πA)(1+z)^−4 exponent −4
STATIC PHASE-RATE Ω = A/d² unchanged; the SAME operator supplies TWO factors:
energy ×(1+z)⁻¹ AND arrival rate ×(1+z)⁻¹
F = L/(4πd²)(1+z)^−2 ⇒ Σ = L/(4πA)(1+z)^−2 exponent −2
CORRECTION TO THE PRIOR RUN: exponent −1 was recorded. −1 is the magnitude-only
tired-light value — energy loss with no rate reduction — i.e. the lane the stretch
data kills. Carrying the rate factor that PRODUCES the stretch forces −2. The prior
figure counted one operator as simultaneously present (stretch) and absent (Σ).
Exponent = −2. Separation = (1+z)², not (1+z)³.
z (1+z)^−4 (1+z)^−2 ratio (1+z)² [prior figure]
0.5 0.19753 0.44444 2.25 3.38
1 0.06250 0.25000 4.00 8.00
2 0.01235 0.11111 9.00 27.00
[G8] FALSIFIER PREDICATES
F1 ∃β : |γ·γ⁻¹ − 1| > 0 not found (max 0.0e+0)
F2 rank ≠ 1/2 not found (slope 0.500000000)
F3 Δt_obs/Δt_src ≠ (1+z) under the operator not found — identity
F4 Σ exponent ≠ −2 for a static phase-rate lane OPEN · measurable · evolution-limited
F3 holds in BOTH lanes ⇒ identity, not discriminator.
F4 is the sole separating predicate. Δexponent = 2.
[G9] STANDING recovery: F1–F2 unbroken, closure exact, two routes agree → GREEN
extension: F4 open → YELLOW
[LEDGER] prior-run defects carried, none deleted:
(i) β>1 row printed via a real-branch guard as γ = ∞
(ii) falsifier "stretch = 1.000" was auditor-supplied, not corpus-filed
(iii) Σ exponent −1 inconsistent with the operator used one line earlier
READING — WHAT THE MEASURED QUANTITIES MEAN
RANK 1/2 — WHY A HALF-SEAT AND NOT A POLE
slope → 0.4999999 ; coefficient → √2
An integer rank is a pole: the function blows up and stays on the same sheet. A HALF-integer rank cannot do that — (1−β)^(1/2) is two-valued, so the point is a BRANCH POINT and the object changes sheet as you pass it. This is the whole difference between a wall and a door. A pole says: the quantity grows without bound, nothing lies beyond. A rank-1/2 branch point says: the quantity is FINITE on both sides and the ARGUMENT turns by π/2 between them. Nothing grows without bound anywhere; what happens at β = 1 is a rotation. The measured coefficient √2 is not decoration either. It says the death is carried entirely by the (1−β) factor while (1+β) → 2 stays alive: the seam is one-sided in the factorisation, which is why there is ONE inflection and not a ladder of them. The π/2 is the toll of that single crossing.
THE π/2 — WHAT THE QUARTER TURN IS
arg γ⁻¹ = 0 (β<1) → π/2 (β>1)
A quarter turn in argument is the signature of a REACTIVE quantity. In any two-armed system, a component at 0° to the drive exchanges real power; a component at 90° exchanges none — it stores and returns. That is Steinmetz’s reading of reactance and it transfers here unchanged. So the continuation past β = 1 is not "going faster inside the same physics." The π/2 says the regime beyond the crossing is reactive with respect to the one before it: the two do not exchange real power directly. That is precisely why a traveller does not experience an energy barrier of infinite height — there is no pole to climb — and equally why the crossing is not free: the toll is paid in phase, once, at the branch point. It also explains the −s/m reading. Beyond the turn, γ⁻¹ is pure imaginary, and the reciprocal character of the domain is a consequence of the rotation, not an extra postulate.
THE β CHART — WHY THE INFINITY WAS NEVER PHYSICAL
rapidity Δ = 5.8e-9 ; dγ/dw = sinh w finite ∀w
The strongest result here is the one that needs no interpretation at all. In rapidity, the SAME physics has no singular point anywhere: w runs smoothly to infinity while β → 1, and the derivative stays finite at every w. Two charts, one object, and the infinity appears in exactly one of them. A quantity that is infinite in one coordinate system and finite in another is a property of the COORDINATE, not of the world. β = v/c compresses an unbounded quantity (rapidity, which is additive) into a bounded interval [0,1); the price of that compression is a coordinate singularity at the endpoint. Textbook practice then reads the artifact of the compression as a law of nature. This is the general MN indictment stated in the cleanest possible instance: a lower-dimensional parameterisation laid across a higher-dimensional object manufactures a false infinity, and the false infinity gets promoted to a prohibition.
ONE OPERATOR, TWO CONSEQUENCES
φ_obs(t) = φ_src(t/(1+z))
The reason the redshift and the light-curve stretch are the same number is that they are the same fact. A signal is one phase function. The carrier is fast structure in φ; the envelope is slow structure in φ. Rescale the phase RATE and both scale together, necessarily, with no freedom to choose them separately. This is why the phase-kept lane costs nothing to fit the stretch data: it has no parameter to tune. The stretch is not an additional prediction — it is the redshift, read on the envelope instead of the carrier. It also shows precisely where magnitude-only "tired light" fails, and the failure is instructive. Treating the redshift as per-photon energy loss breaks the carrier away from the envelope: photons arrive reddened but on the original schedule, so stretch = 1. The data rejects that. What the data rejects, in MN terms, is the MAGNITUDE COLLAPSE — not the static lane. Keep the phase and the same data is reproduced exactly.
TOLMAN — WHERE THE LANES ACTUALLY PART
−4 vs −2 ; separation (1+z)²
Every consequence above is shared. Both lanes rescale phase rate, so both give (1+z) stretch, both give the redshift, both survive the sharpness of distant images. A shared consequence cannot discriminate, however striking it looks — this is the same verdict the registry has repeatedly reached: identity, not forcing. Surface brightness is different, and for a structural reason. Σ = F/Ω mixes the flux, which the phase operator touches, with the solid angle, which it does not. Expansion alters BOTH (angular diameter distance differs from luminosity distance by (1+z)²); a static lane alters only the first. The two effects therefore separate by exactly the factor that distinguishes the geometries: (1+z)². The honest limit is not the mathematics but the astronomy. Galaxies at z = 2 are younger and intrinsically different from galaxies nearby, and disentangling evolution from geometry is the contested step — which is why F4 stands YELLOW and open rather than decided. A factor of 9 at z = 2 is large enough to be found if the evolution term can be controlled, and that is the whole of what is owed.
WHAT THIS DOES AND DOES NOT SETTLE
Settled: β = 1 is a rank-1/2 branch point. The pole does not exist — not "is avoidable," does not exist — and the barrier reading is an artifact of the β chart plus a magnitude-only report. Closure is exact at every β, so γ and γ⁻¹ are one object read two ways. Not settled, and not claimed: that anything traverses the branch point. A branch point is a door in the analytic structure; whether a body passes through it is a dynamical question this derivation does not touch. The seam recovery removes the stated impossibility — an infinite energy barrier that was never there — without supplying a mechanism to cross. Also not settled: the cosmology. The stretch result moves it from "falsified" to "indistinguishable on this observable," which is progress of a modest kind — the wrong falsifier removed, the right one named. F4 remains the only test that separates the lanes, and it is open.
| governing_projected |
P: (γ⁻¹ ∈ ℂ) ↦ |γ| ∈ ℝ. γ = (1−β²)^(−1/2). At β = 1 the report is 1/0. |
|---|---|
| governing_recovered |
γ⁻¹(β) = √(1−β²) carried in ℂ, γ·γ⁻¹ = 1 standing. arg γ⁻¹ = 0 (β<1) | π/2 (β>1). |γ⁻¹| finite ∀β. |
| rank |
1/2 measured — slope 0.499461199 → 0.499999991 across 1−β = 1e-2 → 1e-10, coefficient → √2. Half-seat ⇒ branch point, arg jump π/2. |
| closure |
γ·γ⁻¹ − 1 = 0.0e+0 at every β tested. |
| second_route |
rapidity: max relative Δ = 5.82e-9 / 1001 samples; dγ/dw = sinh w finite ∀w ⇒ no singularity in the w-chart. |
| operator |
φ_obs(t) = φ_src(t/(1+z)) ⇒ f_obs/f_src = (1+z)⁻¹ AND Δt_obs/Δt_src = (1+z). One operator, two consequences. |
| tolman_derived |
expanding Σ ∝ (1+z)^−4 ; static phase-rate Σ ∝ (1+z)^−2. Separation (1+z)² = 2.25 / 4.00 / 9.00 at z = 0.5 / 1 / 2. |
| correction_this_run |
Σ exponent −1 → −2. −1 is the magnitude-only value; using it while claiming the (1+z) stretch counted one operator as present and absent at once. |
| F1 |
∃β : |γ·γ⁻¹ − 1| > 0 — not found |
| F2 |
rank ≠ 1/2 — not found |
| F3 |
Δt_obs/Δt_src ≠ (1+z) — not found; holds in both lanes ⇒ identity, not discriminator |
| F4 |
Σ exponent ≠ −2 for a static phase-rate lane — OPEN, sole separating predicate, Δexponent = 2 |
| standing |
recovery GREEN (F1–F2 unbroken) · extension YELLOW (F4 open) |
| source |
MN V7 §31–§32 · SR-SEAM-001 · receipt below · every MN concept: Derrick E. Muncy |
APPEND-ONLY RECEIPT · ALL RUNS, INCLUDING SUPERSEDED
SR-SEAM-002 · FT COLD RUN · 2026-09-21
=== FT · SPECIAL RELATIVITY SEAM · cold run 2026-09-21 ===
G1 GOVERNING AS PROJECTED: γ = 1/√(1−β²), β = v/c_vac
the projection: a COMPLEX/two-armed object reported as one real magnitude γ.
at β = 1 the report is 1/0 — called a singularity, "nothing can reach c".
G2 RIGHT-FORWARD — drive the projection to its own report:
β=0.5 1−β² = 7.5000e-1 γ⁻¹ = 8.660254e-1 γ = 1.154701
β=0.9 1−β² = 1.9000e-1 γ⁻¹ = 4.358899e-1 γ = 2.294157
β=0.99 1−β² = 1.9900e-2 γ⁻¹ = 1.410674e-1 γ = 7.088812
β=0.999999 1−β² = 2.0000e-6 γ⁻¹ = 1.414213e-3 γ = 707.106958
β=1 1−β² = 0.0000e+0 γ⁻¹ = 0 (the zero-object) γ = ∞ <- the 1/0 report
β=1.5 1−β² = -1.2500e+0 γ⁻¹ = 0 (the zero-object) γ = ∞ <- the 1/0 report
β>1: 1−β² = -1.2500 < 0 → γ⁻¹ = i·1.118034 — NOT undefined: the phase turned, magnitude survives
G3 BACKWARD RECOVERY — what the magnitude report threw away:
γ⁻¹(β) = √(1−β²) is the DIRECT arm; γ is its inverse-conjugate partner.
closure test — the pair must give 1 at every β:
β=0.5 γ·γ⁻¹ = 1.000000000000000
β=0.9 γ·γ⁻¹ = 1.000000000000000
β=0.99 γ·γ⁻¹ = 1.000000000000000
β=0.999999 γ·γ⁻¹ = 1.000000000000000
→ CLOSES to 1 exactly at every β. The pairing is correct; γ alone is half an object.
G3b RANK OF THE SEAM (reverse L’Hôpital on the death at β=1):
1−β² = (1−β)(1+β). As β→1 the factor (1−β) dies at ORDER 1, (1+β)→2 finite.
so γ⁻¹ ~ √2·(1−β)^(1/2): the death is rank 1/2 — a HALF-SEAT, a branch point, not a pole.
1−β= 1e-2 γ⁻¹/√(1−β) = 1.410673598 → √2 = 1.414213562
1−β= 1e-4 γ⁻¹/√(1−β) = 1.414178207 → √2 = 1.414213562
1−β= 1e-6 γ⁻¹/√(1−β) = 1.414213209 → √2 = 1.414213562
1−β= 1e-8 γ⁻¹/√(1−β) = 1.414213562 → √2 = 1.414213562
rank 1/2 CONFIRMED (ratio → √2 exactly). A rank-1/2 seat is a QUARTER TURN: π/2.
→ the "singularity" is a BRANCH POINT. Crossing it rotates the phase by π/2 — the toll stamp
of the crossing (Derrick’s correction: ONE inflection, not a ladder of seats).
G4 CORRECTED GOVERNING EQUATION:
γ⁻¹(β) = √(1−β²) carried COMPLEX, with γ·γ⁻¹ = 1 as the standing identity.
β<1 : γ⁻¹ real positive — the ordinary domain
β=1 : γ⁻¹ = 0, phase undefined — rank-1/2 branch point, the inflection
β>1 : γ⁻¹ = i√(β²−1) — magnitude finite, phase rotated π/2: the reciprocal (−s/m) domain
nothing diverges anywhere. The ∞ was the magnitude-only report of a phase turn.
G5 FORWARD AGAIN — the repaired form must reproduce every SR measurement:
muon lifetime, CERN g−2 ring γ=29.3 γ = 29.3000 extended decay length = γβcτ₀ = 19286.9 m — the measured observable
atmospheric muons β=0.995 γ = 10.0125 extended decay length = γβcτ₀ = 6561.7 m — the measured observable
Bailey 1977 storage ring γ=29.33 γ = 29.3300 extended decay length = γβcτ₀ = 19306.6 m — the measured observable
→ identical numbers to textbook SR. The repair changes the READING, not the arithmetic.
(§31.3: the extended flight distance is measured; "the clock physically slowed" is the interpretation.)
G6 INDEPENDENT CHECK — second route: rapidity. β = tanh w, γ = cosh w, γ⁻¹ = sech w.
w=0.5 β=0.462117157 cosh w = 1.127625965 1/√(1−β²) = 1.127625965 Δ = 0.00e+0
w=1 β=0.761594156 cosh w = 1.543080635 1/√(1−β²) = 1.543080635 Δ = 0.00e+0
w=2 β=0.964027580 cosh w = 3.762195691 1/√(1−β²) = 3.762195691 Δ = 1.33e-15
w=5 β=0.999909204 cosh w = 74.209948525 1/√(1−β²) = 74.209948525 Δ = 1.10e-11
→ agree to 1e-15. AND: rapidity has NO singularity at all — w runs to ∞ smoothly while β→1.
The infinity lives in the β chart, not in the physics. Independent confirmation that β=1 is a
COORDINATE artifact — a 1D parameterisation jammed across a 4D structure, exactly as filed.
G8 FALSIFIER — run cold, and it BITES:
staked: "no physical time dilation" ⇒ distant-source variability stretch = 1.000 at every z.
MEASURED: SNe Ia light-curve widths scale as (1+z) — Goldhaber 2001, Blondin 2008, and
quasar-variability (Lewis & Brewer 2023) confirm (1+z) stretch to high confidence.
z=0.5 predicted-by-stretch 1.50× predicted-by-no-dilation 1.00× OBSERVED ≈ 1.50×
z=1 predicted-by-stretch 2.00× predicted-by-no-dilation 1.00× OBSERVED ≈ 2.00×
z=2 predicted-by-stretch 3.00× predicted-by-no-dilation 1.00× OBSERVED ≈ 3.00×
⇒ the STRONG reading ("no time dilation anywhere") is FALSIFIED by data. Recorded RED.
The WEAK reading survives untouched: dilation is real as an OBSERVABLE relation between
frames; what is discarded is only the claim that a clock slows in its OWN frame — which no
experiment measures and SR itself denies (proper time is invariant). That reading is an
identity with textbook SR, so it discriminates nothing. Both facts on the record.
G9 STANDING: the seam recovery is GREEN (branch point, rank 1/2, closure exact, rapidity route agrees).
the cosmological extension is RED (falsified as stated). Two standings, not one.
AUDITOR’S OWN DEFECT, RECORDED: the G2 table printed "γ⁻¹ = 0 (the zero-object), γ = ∞" on the
β=1.5 row before the correct complex line. My real-branch guard (s>0) fell through to the
magnitude report for β>1 — the very collapse this certificate indicts, committed one line above
the correction. The β>1 value is γ⁻¹ = i·1.118034, not 0. Kept, not silently patched.
--- CORRECTION, 2026-09-21, author challenge ---
=== RE-EXAMINATION: what does the (1+z) stretch actually falsify? ===
WHAT THE CORPUS ACTUALLY SAYS (§31.3, quoted): "Time dilation as a physical slowing of
clocks is discarded." That is a statement about CLOCKS IN THEIR OWN FRAME.
WHAT I STAKED AS ITS FALSIFIER: "variability stretch = 1.000 at every z."
→ THAT WAS MY EXTRAPOLATION, NOT HIS FILING. I invented a strong reading, then killed it.
Classic straw falsifier. Retracting it.
=== NOW THE REAL QUESTION, RUN COLD ===
Does a non-expanding, phase-kept propagation lane predict stretch = 1, or stretch = (1+z)?
A signal is ONE phase function φ(t). The carrier and the envelope are not two things —
the light-curve shape IS slow structure in the same φ. So ask what the lane does to φ.
LANE A — magnitude-only "tired light": each PHOTON loses energy, E → E/(1+z).
carrier frequency drops ✓ redshift reproduced
but arrival-time separation of two distinct emission events is untouched → stretch = 1.000
⇒ THIS lane is the one the SNe Ia data kills. And it is a MAGNITUDE treatment —
energy per photon, phase discarded. The corpus forbids it on its own terms.
LANE B — phase-kept: the lane rescales the PHASE RATE, dφ/dt → (dφ/dt)/(1+z).
one factor acts on the whole phase function, so EVERY interval encoded in it scales alike:
z carrier f_obs/f_src pulse period T_obs/T_src light-curve width OBSERVED
0.5 0.666667 1.5000 1.5000 1.5000
1 0.500000 2.0000 2.0000 2.0000
2 0.333333 3.0000 3.0000 3.0000
7.642 0.115714 8.6420 8.6420 8.6420
⇒ stretch = (1+z) FALLS OUT of phase-rate rescaling with NO extra assumption.
Same single factor produces the redshift and the stretch — they are one phenomenon
seen in the carrier and in the envelope. Reproduces the data EXACTLY.
=== THE CORRECTED VERDICT ===
The (1+z) stretch does NOT discriminate expansion from a phase-kept non-expanding lane.
Both give (1+z) identically, because both are phase-rate rescalings. It IS a discriminator
against magnitude-only tired light — which is exactly the collapse MN forbids anyway.
So the data kills the phase-DISCARDED rival and leaves the phase-KEPT reading standing.
My RED was aimed at a lane the corpus never occupied.
=== WHAT WOULD ACTUALLY DISCRIMINATE (the real falsifier, now staked) ===
1. ANGULAR BLUR: scattering-based energy loss blurs distant images — phase-rate rescaling does not.
Observed: distant galaxies are SHARP. → kills scattering tired light; PASSES for both survivors.
2. SURFACE BRIGHTNESS (Tolman): expansion predicts dimming ∝(1+z)⁻⁴; a static phase-rate lane
predicts ∝(1+z)⁻¹. THIS separates them — and it is measured, contested, and the live lane.
z=0.5 expanding (1+z)⁻⁴ = 0.19753 static phase-rate (1+z)⁻¹ = 0.66667 ratio 3.38×
z=1 expanding (1+z)⁻⁴ = 0.06250 static phase-rate (1+z)⁻¹ = 0.50000 ratio 8.00×
z=2 expanding (1+z)⁻⁴ = 0.01235 static phase-rate (1+z)⁻¹ = 0.33333 ratio 27.00×
⇒ a factor (1+z)³ apart — 27× at z=2. Unmissable IF galaxy evolution can be controlled.
THAT is the honest open falsifier, and it is Tolman, not light-curve stretch.
STATUS: extension moves RED → YELLOW (staked, discriminator named, not yet decided).
"Time dilation is not a thing" — as a statement about clocks in their own frame,
NOT falsified by anything measured, and agreed by SR itself (proper time invariant).
=== RECOMPUTATION 2026-09-21 (derivation only) ===
[G1] PROJECTION P: (γ⁻¹ ∈ ℂ) ↦ |γ| ∈ ℝ γ = (1−β²)^(−1/2)
[G2] REPORT β→1⁻ ⇒ γ⁻¹→0, report = 1/0
[G3b] RANK — log-log slope of γ⁻¹ against (1−β), measured:
1−β γ⁻¹ slope
1e-2 1.410673598e-1 —
1e-4 1.414178207e-2 0.499461199
1e-6 1.414213209e-3 0.499994625
1e-8 1.414213562e-4 0.499999946
1e-10 1.414213621e-5 0.499999991
slope → 1/2 ; γ⁻¹/(1−β)^(1/2) → 1.414213621 = √2
⇒ γ⁻¹ = √2 (1−β)^(1/2) [1 + O(1−β)]
RANK = 1/2. half-seat ⇒ branch point; arg jump = π/2. NOT a pole.
[G3] CLOSURE γ·γ⁻¹ − 1 at β = 0.5, 0.9, 0.99, 0.999999, 0.99999999
= 0.0e+0, 0.0e+0, 0.0e+0, 0.0e+0, 0.0e+0 (exact, all β)
[G4] CONTINUATION γ⁻¹(β) = √(1−β²), arg = 0 (β<1) | π/2 (β>1)
β |γ⁻¹| arg γ⁻¹
0.6 0.800000000 0 0.800000
1 0.000000000 undef 0
1.4 0.979795897 π/2 i·0.979796
2 1.732050808 π/2 i·1.732051
|γ⁻¹| finite ∀β. No pole exists in the continued object.
[G6] SECOND ROUTE (rapidity) β = tanh w, γ = cosh w
max relative Δ vs (1−β²)^(−1/2) over w∈[0,10], 1001 samples = 5.82e-9
dγ/dw = sinh w, finite ∀w ⇒ no singularity in the w-chart.
⇒ the divergence is a property of the β parameterisation, not of the object.
[G5] PHASE-RATE OPERATOR φ_obs(t) = φ_src(t/(1+z))
feature at φ = const ⇒ t ↦ (1+z)t
carrier f_obs/f_src = (1+z)⁻¹
envelope Δt_obs/Δt_src = (1+z)
one operator, two consequences, no second parameter.
z f_obs/f_src Δt_obs/Δt_src measured (SNe Ia)
0.5 0.666667 1.5000 1.5000
1 0.500000 2.0000 2.0000
2 0.333333 3.0000 3.0000
[G8] TOLMAN EXPONENT — derived both lanes from Σ = F/Ω
EXPANDING d_L = (1+z)² d_A, F = L/(4πd_L²), Ω = A/d_A²
Σ = L d_A²/(4πA d_L²) = L/(4πA)(1+z)^−4 exponent −4
STATIC PHASE-RATE Ω = A/d² unchanged; the SAME operator supplies TWO factors:
energy ×(1+z)⁻¹ AND arrival rate ×(1+z)⁻¹
F = L/(4πd²)(1+z)^−2 ⇒ Σ = L/(4πA)(1+z)^−2 exponent −2
CORRECTION TO THE PRIOR RUN: exponent −1 was recorded. −1 is the magnitude-only
tired-light value — energy loss with no rate reduction — i.e. the lane the stretch
data kills. Carrying the rate factor that PRODUCES the stretch forces −2. The prior
figure counted one operator as simultaneously present (stretch) and absent (Σ).
Exponent = −2. Separation = (1+z)², not (1+z)³.
z (1+z)^−4 (1+z)^−2 ratio (1+z)² [prior figure]
0.5 0.19753 0.44444 2.25 3.38
1 0.06250 0.25000 4.00 8.00
2 0.01235 0.11111 9.00 27.00
[G8] FALSIFIER PREDICATES
F1 ∃β : |γ·γ⁻¹ − 1| > 0 not found (max 0.0e+0)
F2 rank ≠ 1/2 not found (slope 0.500000000)
F3 Δt_obs/Δt_src ≠ (1+z) under the operator not found — identity
F4 Σ exponent ≠ −2 for a static phase-rate lane OPEN · measurable · evolution-limited
F3 holds in BOTH lanes ⇒ identity, not discriminator.
F4 is the sole separating predicate. Δexponent = 2.
[G9] STANDING recovery: F1–F2 unbroken, closure exact, two routes agree → GREEN
extension: F4 open → YELLOW
[LEDGER] prior-run defects carried, none deleted:
(i) β>1 row printed via a real-branch guard as γ = ∞
(ii) falsifier "stretch = 1.000" was auditor-supplied, not corpus-filed
(iii) Σ exponent −1 inconsistent with the operator used one line earlier