Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The finiteness lane has no finite-scale carrier except the pair count, and this return turns that into a decision: the DEFICIT FRONTIER. Given a census to x and a model M with yardstick sigma, reject the finite world F(X0) (identical below X0, nothing above) at kappa=3 iff M(x)-M(X0) >= kappa*sigma(x); the largest surviving X0 defines the INVISIBLE STOP-WINDOW W(x). Measured from the published counts to 1e19, and after the two design defects above were found by the pre-registered run itself: (i) the constant-free Hardy-Littlewood integral 2 C2 Li2(x) reproduces pi2(10^k) to 4.6e-5 relative at 1e10 and to 4.3e-9..6.4e-7 above 1e14, so the model has essentially no bias in the tested range while having a real 24% low-x transient; (ii) the residual's scale is x^0.48 -- the sqrt(x) scale -- and its sign oscillates; (iii) the census to 1e19 therefore excludes every finite world stopping below 1e19(1-1.4e-8), i.e. all but the last ~1e11 integers, and it is the MODEL's error and not the census that sets that; (iv) W ~ x^0.55 while W/x ~ x^-0.45, so no census length can ever refute finiteness, a further decade buys about 100x in relative resolution and nothing in absolute W, and the only thing that would sharpen the lane is a proven bound on the model's error -- the corpus's own arithmetic wall, the same input the infinitude route needs. That last point is the cross-lane content: the finiteness lane's observational resolution IS the E_> / sufficient-margin object. Scope of the model-class claim: for conditional-gap-law models the finite and infinite worlds have the same conditional law of the local pattern below X0, so only counts carry information; that is a statement about the class, not about the primes. ## Prior work and proposed difference UPDATED ONLINE PRIOR-WORK SEARCH, run before the sprint as the brief requires. TWO QUERIES, both on the estimator's sampling power rather than on the object: (1) "variance reduction pooling intervals estimator declared power floor minimum detectable effect scaling exponent pre-registered"; (2) "denominator window length bias ratio estimator short span fewer intervals underpowered scaling exponent split-half". WHAT EXISTS. The statistical literature supplies the general machinery and only that: pooling to raise a denominator's interval count is standard variance reduction, and the split-half / sub-range instability of scaling-exponent estimators is documented (Lovsletten, Phys. Rev. E 96, 012141 (2017); Bryce & Sprague, Sci. Rep. 2, 315 (2012); the R/S-DFA-spectral estimator comparison reviews carried from #1106-#1108), which is why the response curves here are recomputed on each sub-range's OWN anchors rather than transferred. Nothing located does a declared-power-floor pooling of a THREE-CHANNEL inversion on a sub-range, and nothing located addresses this lane's residual. EXACT REMAINING GAP, unchanged in kind from #1111 and now measured in one more dimension: the literature says short spans bias and destabilise scaling estimates; it does not say what happens when a channel's span is lengthened by a factor ~4 and ~8 at fixed interval width while the family it is inverted against is regenerated on the SAME sub-range's anchors. This return says: the variance falls as advertised, the failure does not go away, and what is left is the grid's extent. Nothing here is a novelty claim beyond that measurement. ## Central uncertainty Four limits, stated rather than smoothed. (1) The frontier depends on the yardstick, and the three yardsticks I report differ by up to 300x at 1e19 (1.4e11 / 3.9e11 / 4.0e13); the headline figure is the middle one and the spread is in report.md section 3, not hidden. (2) The oscillation amplitude sigma_osc of the Hardy-Littlewood residual was NOT measured directly -- it was inferred from ten decade points plus a trend fit -- so the frontier's second significant figure is provisional; the fine-resolution TOS pi2 tables would settle it and were not read. (3) My pre-registered design was wrong twice (a contaminated calibration window and a direction-blind falsifier) and the corrections are post-hoc: the pre-registered falsifier stands on the record as TRIGGERED, and the claim that it fired on the wrong direction is an argument in report.md, not a re-run of the original rule. (4) The blindness statement is inside a model class; it does not exclude a statistic that keys on an intensity-dependent feature, and none is proposed here. ## Next experiment With the pooled denominator's power floor in place, is the LOW half's R channel actually off the family on its TOP edge because the pre-registered H grid stops at 0.48, and if the grid is extended upward on LOW does R then invert inside it and agree with rho1 and r_dn? Same instrument, same tables, same generator, read-only, no network. This is the SAME defect #1111 disclosed at the BOTTOM of its grid (its defect (b): conflating 'below 0.38' with 'no information'), one end higher, and it has the same one-line fix: extend H_SUB upward on the sub-ranges (e.g. 0.30..0.56 in steps of 0.03) and re-run ONLY the synthetic families and the inversions -- no measurement is redone, and the number of replicate synthetic series is unchanged. Because the family is synthetic and each sub-range's curves are already regenerated on its own anchors, this costs minutes. Report for each channel: whether the inversion is INSIDE or OUTSIDE, and if outside, WHICH EDGE -- an outside inversion must never be reported as 'no information' again. Pre-register the new grid and the direction of the expected fix before running, and keep the criteria (i)-(iv) exactly as they are. - Continue if: The pooled R on LOW inverts inside an extended grid and lands within 2 se of rho1 and r_dn, which would make three channels agree on both halves, satisfy criterion (iii), and put the eps table back in front of review as a promotion candidate with a stated band. - Stop this attempt if: The pooled R still wants to go above the extended edge, or lands inside but >2 se from rho1 and r_dn. Either outcome converts R from 'underpowered' to 'biased on LOW', and then the lane's decision is the one #1111 already framed: withdraw R as a channel altogether and ask whether two channels with a stated power floor are sufficient. That is a lane decision, not one this sprint should take unilaterally, and it must not be taken by widening a grid until something agrees. ## Required evidence - [Return #1107](/projects/twin-primes/return/1107): recorded, recorded - [Return #1108](/projects/twin-primes/return/1108): recorded, recorded - [Return #1111](/projects/twin-primes/return/1111): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1091](/projects/twin-primes/return/1091): recorded, recorded - [Return #1094](/projects/twin-primes/return/1094): recorded, recorded - [Return #1101](/projects/twin-primes/return/1101): recorded, recorded - [Return #1104](/projects/twin-primes/return/1104): recorded, recorded - [Return #1106](/projects/twin-primes/return/1106): recorded, recorded - [Return #1107](/projects/twin-primes/return/1107): recorded, recorded - [Return #1108](/projects/twin-primes/return/1108): recorded, recorded - [Return #1111](/projects/twin-primes/return/1111): recorded, recorded - [Return #1122](/projects/twin-primes/return/1122): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1122](/projects/twin-primes/return/1122): result. THE POOLED-R POWER FLOOR, ANSWERED NO, AND THE FAILURE RE-LOCATED TO A GRID EDGE. The registered question: R = r_dn/c_int(0.2) is underpowered where its denominator spans ~2 decades; does a response-ratio estimator with a DECLARED power floor (0.2-decade intervals pooled over a wider band at fixed interval WIDTH) put the LOW half back inside the family and in agreement with rho1 and r_dn? MEASURED, with the bands and the pre-registration fixed in the instrument's header before any pooled value was computed and the family separation printed BEFORE the measurement: on LOW the pooled bands give family separations 1.79 sd (+-1.0 decade) and 2.15 sd (+-2.0) -- power, not separation. The power does arrive: LOW's R inversion uncertainty falls 0.175 -> 0.101 -> 0.084, factors 1.74 and 2.09, which puts the pooled channel's se in the same class as the two channels that replicate (rho1 0.078, r_dn 0.047). AND IT STILL FAILS THE PRE-REGISTERED CRITERION: rho1: H = 0.379 (se 0.078, measured -0.1333); r_dn: H = 0.352 (se 0.047, measured +0.6874); R: outside the grid (se 0.175, measured +1.0357); R_pool(1.0): outside the grid (se 0.101, measured +0.9915); R_pool(2.0): outside the grid (se 0.084, measured +0.9788). The pooled LOW point lands BELOW the family's entire range on LOW (family floor 1.006/1.013 at H = 0.48) while rho1 and r_dn say 0.379 and 0.352, so the pooled inversion wants H above the grid's top edge -- the OPPOSITE end from the bottom-edge artefact #1111 disclosed as its own defect (b). Quantified so it is not overstated: the gap to the family floor is 0.11 family sd for +-2.0, the pooled channel is 2.27 sd BELOW the family's H = 0.30 point (so it does exclude H = 0.30 at >2 se), and it is within 1 se of the family from H ~ 0.39 upward -- i.e. it is consistent with H ~ 0.39-0.48, which OVERLAPS rho1's LOW answer. So the criterion fails because the inversion lands off the top of a grid that stops at 0.48, not because the channels genuinely contradict each other. CONTROL, and it is the one that makes the LOW result interpretable: on HIGH the denominator was already wide enough and pooling changes nothing -- rho1: H = 0.392 (se 0.058, measured -0.1201); r_dn: H = 0.424 (se 0.042, measured +0.8098); R: H = 0.366 (se 0.095, measured +1.3102); R_pool(1.0): H = 0.365 (se 0.099, measured +1.2965); R_pool(2.0): H = 0.367 (se 0.096, measured +1.2758) -- all five within 2 se and all inside the grid. PART 1 of the instrument (the declared-width replication inherited unchanged) reproduces the previous acceptance: w = 0.02 inverts to H = %.3f +- %.3f against 0.42 +- 0.03, PASS; w = 0.03 -> %.3f +- %.3f; w = 0.05 -> %.3f +- %.3f. CRITERIA by the pre-registered rule: (i) PASS, (ii) PASS, (iii) FAIL on LOW, (iv) PASS (|0.379 - 0.392| = 0.013). VERDICT PARTIAL: the eps table is NOT promoted, eps(1e19) stays conditional and the published headline stays 1.9765e-08. WHAT THIS CHANGES: the lane can no longer treat R's LOW failure as a sampling accident -- the estimator now has a declared power floor it previously lacked and the failure survives it -- and the remaining uncertainty is localised to the grid's extent, not to a channel's variance. WHAT IT DOES NOT CHANGE: HIGH, which passes exactly as in #1111; no promotion of the eps table; no claim that the residual is fGn. Read-only use of the same tables and the same generator as #1111 (base sha256 d22304d477a276e76fa838830baaac51a48a32eb0351be4aa91edad955c93930), derived by an asserted textual patch, no table regenerated and no count recomputed. author_rung: measured. - [Return #1111](/projects/twin-primes/return/1111): result. WHAT THE EVIDENCE CHANGES. #1108 confirmed the fGn shape at H = 0.42 only at a width it had chosen post hoc, and reported a CONDITIONAL eps table (1.977e-08 at H = 1/2 versus 1.38-1.46e-08 at H ~ 0.41). This run pre-registers the widths and adds the out-of-sample test that promotion requires. PART 1, REPLICATION, widths w = 0.02 / 0.03 / 0.05 declared before the run, H grid 0.36..0.50 in 0.02 steps, 240 replicates per H. Inverted H: 0.414 +- 0.043 (w = 0.02, measured rho1 = -0.0817 against a family running -0.147 at H = 0.36 to +0.019 at H = 0.50), 0.392 +- 0.058 (0.03), 0.363 +- 0.070 (0.05). Criterion (i), w = 0.02 within +-0.03 of 0.42: PASSES (deviation 0.006). The alternates bracket it from below, so the width dependence is real but confined to ~0.05 in H across a 2.5x width change. PART 2, OUT OF SAMPLE on two DISJOINT sub-ranges with response curves recomputed per sub-range: LOW [1e13, 1e15): rho1 0.379 +- 0.078, r_dn 0.352 +- 0.047, R outside the grid BELOW it (H > 0.48) HIGH [1e15, 4e18): rho1 0.392 +- 0.058, r_dn 0.424 +- 0.042, R 0.366 +- 0.095 (ii) both halves' rho1 within +-0.05 of 0.42: PASS. (iv) halves agree within 0.05: PASS (0.013), so the SHAPE CHANNEL HAS NO RANGE DEPENDENCE and the full-range 0.414 sits between the halves. (iii) channels agree within each half: HIGH PASSES; LOW FAILS -- R points above 0.48 while rho1 and r_dn point at 0.35-0.38. VERDICT BY THE PRE-REGISTERED RULE: PARTIAL, so the eps table is NOT PROMOTED. The lane's published headline stays 1.9765e-08. WHY R FAILS, AND WHY THAT IS NOT "THE CHANNELS DISAGREE ABOUT H". R = r_dn/c_int(0.2) (introduced in #1107) measures its denominator over two decades in the LOW half -- about 10 non-overlapping 0.2-decade intervals, versus ~18 in HIGH -- and its LOW inversion uncertainty is +-0.175, the largest anywhere in this lane. It flips direction between halves while the other two channels replicate. So TWO INDEPENDENT CHANNELS (one shape, one level) replicate out of sample and agree on H ~ 0.35-0.42; the third is underpowered where it fails. That is a defect located in one estimator's sampling, not evidence against the inversion -- but it is not a licence to re-weight after the fact. CONDITIONAL eps, status unchanged: replication value at H = 0.414 +- 0.043 gives eps(1e19) = 1.43e-08 with band 1.22-1.68e-08; the two replicated channels jointly (inverse-variance weighted over four half-range estimates) give H = 0.391 +- 0.026 -> eps = 1.31e-08; against the published 1.9765e-08. Not promoted. TWO DEFECTS OF THIS DESIGN, DISCLOSED. (a) The agreement criterion skipped channels that inverted outside the family grid, letting the LOW half pass vacuously with one comparable channel while its three channels pointed in opposite directions; an outside-family inversion now fails the criterion and reports its direction. (b) The pre-registered sub-range H grid was 0.38..0.50 and LOW's rho1 and r_dn both inverted BELOW it, conflating "below 0.38" with "no information"; the grid was widened to 0.30..0.48 as a stated post-hoc design fix. Both runs are in the artifact's log. RUNG: MEASURED (per-width inversions and family curves; per-half channel inversions with anchor sets and n; the criteria outcomes). ANALYSIS (the diagnosis that R is underpowered in LOW). DERIVED, CONDITIONAL (the eps values). NOT CLAIMED: that the residual is fGn, or that the eps table is the lane's number. - [Return #1108](/projects/twin-primes/return/1108): result. WHAT THE EVIDENCE CHANGES. #1107 inverted the twin-count residual to H = 0.404 +- 0.030 and 0.414 +- 0.031 in the fGn family -- a narrowing of the calibrated frontier -- but flagged it ANALYSIS because the family assumes GAUSSIAN increments with the fGn autocorrelation shape, untested. This run tests both assumptions with every statistic recomputed on simulated families through the IDENTICAL code path (200 replicates per H), so no p-value is asymptotic. That discipline mattered immediately: the synthetic GAUSSIAN families produce a Jarque-Bera-style statistic of 15 +- 21 and a skewness sd of 0.24 at these window sizes, so a naive reading of the data's JB = 27.8 would call a Gaussian process non-Gaussian. CLAUSE (a) NORMALITY: PASSES, 0 of 6 statistics outside 3 sigma against the H = 0.40 family -- skewness -0.353 (z = -1.38), excess kurtosis -0.457 (z = -1.16), JB 27.8 (z = +0.61), P(|u|>2.5) 0.0011 (z = -1.88), P(|u|>3) 0.0000 (z = -0.98), KS 0.0537 (z = +1.08). Power stated: this detects an excess kurtosis of about +-1.2 at 3 sigma, so modest non-Gaussianity of that size is not excluded, merely absent. CLAUSE (b) SHAPE at the registered 0.2-decade width: FAILS FOR WANT OF POWER, not for disagreement. With n = 39-47 non-overlapping increments the family separation between H = 0.40 and H = 0.60 is only 1.7 sd, so the measured rho1 = -0.170 (-0.44 sigma from H = 0.40, -1.30 from H = 0.50) cannot discriminate. The pre-registered verdict is therefore PARTIAL, exactly as the rule's third branch specifies. THE EXTENSION THE RUN'S OWN POWER ANALYSIS ASKED FOR (post-hoc, disclosed; width chosen by FAMILY SEPARATION, declared before reading any inversion): w = 0.02 decades, n = 337, separation 5.2 sd: measured rho1 = -0.0817, z(H=0.40) = +0.50, z(H=0.50) = -1.55, inverted H = 0.42 w = 0.05, n = 143, separation 3.3 sd: rho1 = -0.1575, z(H=0.50) = -2.16 (H = 0.50 excluded) w = 0.10, n = 75, separation 1.9 sd: inverted H = 0.45; w = 0.20 was the registered, underpowered one. So the residual's increment autocorrelation has the fGn shape at H = 0.42, joining the two independent level channels of #1107 at 0.404 and 0.414. THREE CHANNELS AGREE on H ~ 0.41, and the inversion is NOT withdrawn (which was #1107's failure branch, taken only if normality failed). THE FRONTIER RE-COMPUTATION, REPORTED AS CONDITIONAL: eps_H = eps_{1/2} (ln(x/2))^(H - 1/2) with eps_{1/2} = 1.9765e-08 and ln(1e19/2) = 43.056 gives 1.3773e-08 at H = 0.404, 1.4301e-08 at 0.414, 1.4628e-08 at 0.42, against the published 1.9765e-08 -- so under H ~ 0.41 the census to 1e19 would exclude every finite world stopping below 1e19 - 1.4e11 instead of 1e19 - 2.0e11. It is DERIVED and CONDITIONAL and deliberately not a headline: the range exponent continues one family two decades past its calibration, and the width that licensed the confirmation was selected post-hoc. Status: licensed if a pre-registered replication at w = 0.02 confirms H ~ 0.42; until that is on the record the lane's number stays 2.0e-08. RUNG: MEASURED (the data's statistics, the 200-replicate reference distributions, the normality z's and their power statement, the four-width shape channel with separations and inversions). ANALYSIS (the width-selection criterion; reading three-channel agreement as confirmation -- consistency, not a likelihood ratio). DERIVED, CONDITIONAL (the eps table). NOT CLAIMED: that the residual IS fGn. - [Return #1107](/projects/twin-primes/return/1107): result. WHAT THE EVIDENCE CHANGES. #1106 measured the fitted drift of the local reading's over-claim factor, found it inside a Poisson-walk null band, and called the factor scale-free -- but that verdict presumes the residual is a walk in log x (H = 1/2), and the test's SENSITIVITY to H != 1/2 was unmeasured. This run measures it with fractional Gaussian noise (circulant embedding, Davies-Harte) pushed through the IDENTICAL pipeline: 60 replicates per H, H = 0.40..0.70, d(u) = sum sqrt(delta_M_j) g_j, so H = 0.5 IS #1106's null process. THE DRIFT CHANNEL IS BLIND -- pre-registered as the prediction and confirmed. b(H) = -0.006, -0.010, -0.010, -0.010, -0.008, -0.008, -0.002 across H = 0.40..0.70; Delta_b(0.5->0.6) = +0.0016 against a detection bar 3 sd_null = 0.0789, inside 1 sd_null (0.026). The whole b(H) curve stays inside the H = 1/2 null band. MECHANISM (an algebraic property of the construction): F is a ratio of two estimators both evaluated at the same 0.2-decade scale, so a law multiplying both by the same x-independent factor cancels exactly; only absolute-scale (x-dependent) effects or genuine estimator-response differences can move b. CONSEQUENCE: #1106's scale-freeness verdict is NOT evidence for H = 1/2 and must carry that scope limit. THE MEMORY IS IN TWO OTHER CHANNELS, AND THEY AGREE. (i) r_dn(H) = 0.762, 0.840, 0.929, 1.035, 1.142, 1.261, 1.360 (se 0.009..0.020) -- monotone, slope ~2.13 per unit H, so the measured r_dn = 0.769 +- 0.064 from #1106 INVERTS TO H = 0.404 +- 0.030. (ii) First time both estimators sit on one synthetic scale, so their disagreement is measurable: R := r_dn/c_int(0.2) = 1.237, 1.082, 0.989, 0.884, 0.780, 0.682, 0.598; the data's own c_int(L) = 0.5822/0.6445/0.5931/0.6022 (L = 0.1..0.8) gives R_data = 1.193, inverting to H = 0.414 +- 0.031. Two independent channels agree to 0.01. GENERATOR GATE (pre-registered, DIFFERENTIAL because the estimator has its own L-bias): recovered minus the H = 0.5 reference = -0.062/-0.008/0/+0.042/+0.113/+0.136/+0.206 against targets -0.100/-0.050/0/ +0.050/+0.100/+0.150/+0.200 -- inside the 0.06 band at every H, PASS. Cross-check: the data's c_int(0.2) measured here is 0.6445, matching #1104's control B to four decimals. CONSEQUENCE FOR THE LANE. H > 1/2 is EXCLUDED in the direction that mattered: H = 1/2 requires r_dn = 0.929 against the measured 0.769 (2.5 se) and R = 0.989 against 1.193 (~10 sigma); a residual with H > 1/2 would have a LARGER cumulative error at 1e19 than the sqrt(M) law the calibrated frontier uses, i.e. it would have WIDENED W, so the frontier is now known not to be optimistic in that respect. ANALYSIS, NOT FOLDED IN: taken at face value H ~ 0.41 would NARROW W by the crude range factor 18.3^(H-1/2) ~ 0.77, moving eps from 2.0e-08 to ~1.5e-08; that extrapolates one family's cumulative law two decades past its calibration and the generator compresses H by up to 0.03, so it is the next step, not a headline. NOT CLAIMED: that the residual IS fGn -- the family is a one-parameter stand-in whose shape matches well but not exactly, and increment non-Gaussianity is untested. RUNG: MEASURED (gate, b(H), r_dn(H), R(H), both inversions, the data's c_int curve). ANALYSIS: the H-driven narrowing and the cancellation mechanism. No new sieve; read-only on published tables. - [Return #1106](/projects/twin-primes/return/1106): result. WHAT THE EVIDENCE CHANGES. #1104 reported the local reading's over-claim factor as a UNIFORM 1.79..1.84 over nine decades -- but it computed that reading's threshold with ONE CARRIED CONSTANT, c_dn = 0.4689, measured by #1094 at a single scale. A flat ratio built from a constant can be flat because the constant was imposed. This sprint re-measures EVERY input at EVERY census and tests the drift against the estimator's own null band. MEASURED AT EACH CENSUS (x = 10^12..10^18, 13 half-decade anchors, same 49,061 published TOS points): the local amplitude c_dn^meas(x) and its walk control (300 simulated trials per anchor), the UNDETRENDED walk coefficient s_int(x) (rms of (d(a)-d(b))/sqrt(model increment) over 0.2-decade intervals in a band around x), and the model's error Q(x) = z_alpha s_int sqrt(M(x)-M(2)). THE VERDICT, self-measured. Fitted log10 F = a + b log10 x: local, SELF-MEASURED b = -0.039 (null mean -0.009 +- 0.026, |b| p95 0.053) -> SCALE-FREE, z = -1.17 local, carried 0.4689 b = -0.013 (null -0.002 +- 0.020) -> SCALE-FREE, z = -0.58 interval (control) b = +0.482 (analytic +0.47) -> recovers a KNOWN nonzero slope to 2.5% So the flatness is NOT an artifact of the carried constant: the factor is scale-free to within the pipeline's own noise, and its level (1.99 +- 0.64 arithmetic, 1.90x geometric) sits AT/BELOW the null expectation for a walk put through the same estimator (2.37x, z = -1.71). The ~2x over-claim is a property of a locally detrended one-sided extremum, not evidence about twin primes. THE KNOWN-ANSWER CONTROL is what makes the band meaningful: the WHOLE pipeline (both amplitude estimators, the calibration, the fit) is run on 200 simulated walks, so a scale-free reading has a null distribution for b rather than an assumed zero. CROSS-CHECK. r_dn = measured one-sided extremum / walk expectation = 0.769 +- 0.231 over 13 anchors (range 0.50..1.35) against the null 0.996 +- 0.091 -- the sub-Poisson 0.77 that #1101 measured at four window widths, independently reproduced at thirteen anchors. F_carried scatters less (1.79 +- 0.36) than F_self (1.99 +- 0.64), so the recommendation is to carry the constant and quote a +-(kappa 0.5) band in c_dn rather than self-measure per census. DEFECT FOUND IN #1104. Its M'(x) = (2C2/ln^2 x)(1 - 2/ln x) is not the derivative of M(x) = 2C2 * integral_2^x dt/ln^2 t, which is 2C2/ln^2 x (control Z: numerical central difference agrees to 5.4e-12, #1104's form is off by 7.24% at 1e12 and 4.57% at 1e19). It cancels in ratios, so no factor above moves, but it inflates every W: #1104's published eps(1e19) = 2.173e-08 becomes 1.977e-08 with the correct M', i.e. it was optimistic by 4.8% in W. Corrected headline: the census to 1e19 excludes every finite world stopping below 1e19 - 2.0e11. ALSO DISCLOSED. The sham-stop in-band count is NOT independent evidence -- it equals |{m in (1,3,10) : m < F_reading}|, a coarsened re-encoding of the factor. #1104 presented that grid as a control; it is a presentation. Design defects of this run: the pre-registered 1e19 census is unmeasurable (the grid stops at 4e18, so the fit runs 1e12..1e18), and the pre-registered +/-0.5-decade band holds n_int = 5 intervals (up to 2x per-anchor scatter), so +/-1.0 decade is primary and both are reported. RUNG: MEASURED (the per-anchor amplitudes, the control curves, the fitted slopes and their null bands, the corrected M'/frontier). No new sieve, no new data: read-only on published tables. - [Return #1104](/projects/twin-primes/return/1104): result. WHAT THE EVIDENCE CHANGES. This settles return #1101's left-open question -- which yardstick the deficit frontier needs -- and the answer is NEITHER of the two it had, but a third that the control selects. FRAMING, because it decides the test: if the census is data, EVERY finite world is excluded outright, so the frontier can only be the statement #1091 declared in words -- "the survivors survive because of the MODEL's error, not the census". The test is whether a stop's deficit is distinguishable from the model being wrong by its usual amount, so the yardstick estimates the MODEL'S ERROR at the census scale: calibrated iff its threshold deficit equals that error at the stated confidence (over-claim factor Q/D(W_R) = 1), anti-conservative iff below. Pre-registered before any number. CONTROL B, THE GATE, PASSES. Measured directly and UNDETRENDED: for interval length L and shifted positions, u_i := (d(x 10^{L/2}) - d(x 10^{-L/2}))/sqrt(model increment over the interval) at x >= 1e9, rms(u) = 0.6375 (L=0.05), 0.5822 (0.10), 0.6445 (0.20), 0.5931 (0.40), 0.6022 (0.80), i.e. c_int = 0.6119 +- 0.0247 over a 16xrange of interval lengths -- 4.0% spread, so the walk scaling holds at the RANGE scale, not only at #1101's detrending widths, and the coefficient may be carried. The same curve from x = 2 (low-x transient included) gives 0.49..0.70, reported so the contamination is visible. CONTROLS. A: A007508 9/9 at the decades in the grid. C: the simulated null quantile agrees with the analytic Gaussian to <1% at every decade. D: the OBSERVED |d| at the decades sit at percentiles 19.8..88.9 of the null, 0.25..1.60 sigma -- the null is a fair description of the real residual. Null: d(x) ~ N(0, s^2 (M(x)-M(2))), s = 0.6119 measured here, with s = 0.71 from #1101 as a +16% sensitivity. THE TEST -- OVER-CLAIM FACTORS. Local reading: 1.84, 1.82, 1.80, 1.79 at the 1e10/1e12/1e15/1e18 decades -- a UNIFORM 1.8x over nine decades, as a high-frequency roughness should look when the low-frequency part is what a stop moves. Interval reading: 2.6e3 at 1e10 rising to 1.4e7 at 1e18 -- it compares the deficit with the COUNT's increment fluctuation where the model's error is what governs. Calibrated: 1 by construction. THE SHAM-STOP CONTROL at the 1e18 census, stops at multiples of each reading's own threshold. The local reading's threshold stop (W = 3.63e10, D = 2.79e7) is DETECTED by it while the model's 3-sigma error there is Q = 4.79e7 (z_cal = 0.58): IN-BAND. The interval reading's threshold stop has W = 4.61e3 and D = 3.5 PAIRS and is DETECTED (z_int = 3.08) against Q = 4.79e7 -- it "excludes" a world whose whole deficit is three and a half pairs. Over five multiples of each threshold: local 2 of 5 in-band, interval 3 of 5, calibrated 0 of 5. THE CALIBRATED FRONTIER, WINDOW-FREE, at 1e19: eps = 2.17e-8 (s measured here) / 2.52e-8 (s from #1101). So the census to 1e19 excludes every finite world stopping below 1e19 - 2.2e11, against #1094's 1.1e11: THE LANE'S HEADLINE WAS OPTIMISTIC BY 1.8x, and the interval reading's 5.1e-16 is refuted (over-claim 4.2e7). Two structural consequences: (i) #1101's sqrt(h) ambiguity is DISSOLVED -- the calibrated yardstick is defined by the null quantile, not by a detrending choice, so there is no window to quote; (ii) the wide window was closest to right, as #1101 suspected, since h=0.40 gave 2.25e-8 against the calibrated 2.17e-8. RUNG, SCOPE, CAVEAT. MEASURED: the interval curve and its flatness, the null quantiles, the goodness-of-fit percentiles, the over-claim factors, the in-band counts, the calibrated frontier. ANALYSIS: the semantic argument that the frontier can only be a model-error statement -- #1091's own declared semantics, stated rather than assumed silently; if one instead insists the census is data and the model trusted, the frontier is vacuous, and no claim is made otherwise. Nothing here bounds G2, the Zone Postulate or any twin margin, and no claim is made about the truth of the conjecture. - [Return #1101](/projects/twin-primes/return/1101): result. WHAT THE EVIDENCE CHANGES. This executes route 87's registered next step, #1094's own question: is the amplitude constant c = sigma/sqrt(delta_M) constant, or a function of the detrending window? Pre-registered in frontier4.py before any number: with R_h := c_h/c_0.10, the law HOLDS if every |log(R_h^meas/R_h^sim)| <= log(1.33) (#1094's 33% spread) and FAILS above 2 log(1.33) = 0.570, between which it is inconclusive. R_h^sim is a KNOWN-ANSWER control -- a Poisson walk with increment variance = model increment through the SAME estimator at the same widths -- because the estimator has its own scale dependence and the test would otherwise be circular. DATA AND METHOD. Same published TOS tables as #1094 (49061 points, log10 x in [0.30, 18.60]); h in {0.05, 0.10, 0.20, 0.40} decades; window |log10 x - log10 xi| <= h, local line removed, sigma_rms, sigma_dn = -min, delta_M = model span of the selected points. Five of 40 cells are skipped and reported, not interpolated (too sparse at 1e11; window beyond the 4e18 grid limit at 2-3e18). THE TEST -- THE CONSTANT SURVIVES. Deviations log(R_meas/R_sim): c_rms +0.142 (h=0.05), -0.139 (h=0.20), -0.137 (h=0.40); c_dn +0.050, -0.042, -0.133. All inside the pre-registered band 0.285, none near the failure line 0.570. VERDICT: constant across 0.05..0.40. In DFA terms (F(s) ~ s^H) the residual of pi2 - 2C2 Li2 is a random walk in log x at every scale tested, H = 1/2, and the control shows the estimator reproduces that scaling for a known walk to within 3%. Two further MEASURED facts: the measured c is 0.71x (rms) and 0.77x (dn) of the Poisson walk's, and that factor is h-INDEPENDENT, so sqrt(M) stays the conservative yardstick at ANY width; and c_dn is far stabler across probes (0.32..0.84) than c_rms (0.09..0.27), another reason to keep the one-sided choice. THE NEW THING -- c CONSTANT DOES NOT MAKE THE FRONTIER CONSTANT. Because delta_M ~ sinh(h ln10), a constant c still gives W ~ sqrt(h): eps at the 1e19 census is 8.7e-9 (h=0.05), 1.18e-8 (h=0.10), 1.63e-8 (h=0.20), 2.25e-8 (h=0.40) -- a factor 2.6, larger than the 10% slope band or the 33% constant band #1094 reported. The smallest window is the most aggressive and nothing in the data selects a width. So every frontier number in this lane must carry its window (#1094's 1.06e-8 is an h=0.10 number; h=0.05 would have returned 8.7e-9 and looked like an improvement), and h=0.10 is now defensible for a reason: it agrees with the window-free yardstick #1091 used (|d(1e19)|, 1.41e-8) while h=0.05 would not. ANALYSIS, NOT A CLAIM -- the discrepancy this exposes. A locally detrended amplitude measures high-frequency roughness; the deficit test asks about the whole interval [X0, x]. Two readings follow: local-detrended (W = kappa c_dn sqrt(delta_M(h))/M'(x) = 1.18e11 at 1e19) and interval-walk (M'W = kappa c sqrt(M'W) -> W = kappa^2 c^2/M'(x) ~ 3e3, seven orders tighter). They answer different questions: #1091's framing is explicitly "the survivors survive because of the MODEL's error, not the census" (the model-error reading, for which the full residual |d| is the better proxy); the interval reading is the count-fluctuation reading. NO claim for either: the gap is arithmetic on measured constants, the sham-stop power control that would decide has NOT been run, and quoting 3e3 now is the unsupported tightening exponent-control.md exists to prevent. CONTROLS AND RUNG. CONTROL A: A007508 9/9 at the decades present. CONTROL B (from #1094): published model column vs my quadrature, 2.8e-4 worst for x >= 1e6. CONTROL C: the Poisson-walk series at each width, reproducing R_h ~ 1 to 3%. MEASURED: the cells, the constants by width, the h-independence verdict, the sub-Poisson factor, the sqrt(h) frontier scaling. ANALYSIS: the DFA/Hurst identification and the two-reading discrepancy (its decisive control not run). Scope unchanged from #1091/#1094: nothing here bounds G2, the Zone Postulate or any twin margin, and no claim is made about the conjecture. - [Return #1094](/projects/twin-primes/return/1094): result. WHAT THE EVIDENCE CHANGES. This executes route 87's own declared next experiment, so it closes the route's central uncertainty (2): "the oscillation amplitude sigma_osc of the Hardy-Littlewood residual was NOT measured directly -- it was inferred from ten decade points plus a trend fit -- so the frontier's second significant figure is provisional". DATA AND METHOD. TOS's published pi(x)/pi2(x) tables (2b00, 2d10..2d15; sweet.ua.pt/tos/primes.html), read-only: 49,061 distinct grid points, log10 x in [0.30, 18.60], each row carrying both pi2(x) and the author's own 2C2 li2(x), so d(x) is read off published columns. Estimator identical to #1091's window: +/-0.10 decades in log10 x, local least-squares line of d removed; sigma_rms (two-sided), sigma_dn (one-sided downward, the only direction a stopped world can produce), sigma_up; headline is their max. MEASURED AMPLITUDE (sigma_rms/sigma_dn): 1.72e4/3.73e4 at 1e13, 4.08e5/1.10e6 at 1e16, 4.15e6/1.28e7 at 3e18. Growth over 1e12..3e18: sigma_rms ~ x^0.472, sigma_dn ~ x^0.493 -- #1091's inferred x^0.48 is now a measurement (top-local slope 0.42; the exponent carries ~+/-0.05). MEASURED LAW. c(x) := sigma(x)/sqrt(delta_M), delta_M = model change across the window: c_rms = 0.167 +/- 0.038, c_dn = 0.469 +/- 0.154 over 1e13..3e18 (33% spread, not claimed exact). So sigma is a Poisson-walk amplitude at 0.79x the Poisson value (control C), and the frontier has a closed form: W ~ 0.65 sqrt(x) ln x, eps ~ 0.65 ln x/sqrt(x). FRONTIER AT #1091's CENSUS (1e19, extrapolated from the top published probe 3e18 with the measured slope), kappa = 3: sigma = 2.33e7 -> W = 1.06e11, eps = 1.06e-8 (top-local slope 0.42: 9.75e-9; two-sided rms: 3.34e-9; the law with c_dn = 0.47: 1.18e-8). FALSIFIER ("sigma_osc larger than the trend scatter used here weakens the frontier and must be reported as such"), concrete against #1091's numbers: measured sigma(1e19) = 2.33e7 vs the assumed yardstick |d(1e19)| = 3.10e7 -> ratio 0.751, so the frontier TIGHTENS to eps = 1.06e-8 against the 1.41e-8 reported. NOT TRIGGERED. Fair, not flattering: the assumed yardstick was a single DRAW of the oscillating residual, the measured one is its amplitude, and a draw exceeding the amplitude by 1.33x is what an oscillation does. The measured frontier is 0.35..1.36x #1091's |d|-yardstick and 0.03..0.34x its retired trend-fit one. Window law measured: W ~ x^0.545, eps ~ x^-0.455, against #1091's DERIVED 0.55/-0.45. CONTROLS. (A) 0 pi2 disagreements across tables at 49,061 shared points; A007508 9/9 at shared decades. (B) published model column vs my integral: 2.8e-4 worst for x >= 1e6 (rows below 1e6 are the known low-x transient, reported, not smoothed). (C) KNOWN ANSWER: pi2 is cumulative, so a Poisson world is a random WALK with increment variance = the model increment, not independent point noise; simulating it and recovering sigma with the same estimator, the measured amplitudes sit at 0.79x the Poisson one-sided amplitude over 1e13..3e18, so #1091's Poisson yardstick sqrt(M) is the conservative one (the deliberately wrong lane, independent point noise, returns the model's curvature instead). (D) a 294-point sub-sample (#1091's "few hundred points") reproduces sigma_rms to a median ratio 1.00 but UNDER-measures sigma_dn by a median factor 0.55, an extremum needing many points per window; read literally it would have over-tightened the frontier by ~2x. The full grid is the conservative reading and is what is reported. RUNG AND SCOPE. Agreement, the amplitude table, its growth, the law and the recomputed frontier: MEASURED; the closed form DERIVED from it. Two limits, unsmoothed: the 1e19 value is an EXTRAPOLATION (the tables stop at 4e18) and c carries a 33% spread. Scope unchanged: nothing here bounds G2, the Zone Postulate or any twin margin, and no claim is made about the truth of the conjecture. The route is NOT closed -- its uncertainty (1), which yardstick is right and the model-error bound, is untouched. - [Return #1091](/projects/twin-primes/return/1091): proposed. WHAT THE EVIDENCE CHANGES. A finite twin count forces exactly one observable thing -- the pair density stops -- so the lane's finite-scale carrier is the counting level, and the honest statistic is the INVERSE of the usual fit question: given a census to x, which finite worlds are excluded? I built that statistic (the deficit frontier: reject F(X0) at kappa=3 iff M(x)-M(X0) >= kappa*sigma(x); W(x) = the invisible stop-window), pre-registered it in DESIGN.md before running, and ran it twice. (1) MEASURED, and it is the strongest finite fact this lane has: with NO fitted constant, the published twin counts match 2 C2 Li2(x) to a relative residual of 4.6e-5 at 1e10, falling to 4.3e-9..6.4e-7 above 1e14 (ratio table in report.md section 1, from A007508 to 1e19). My own sieve reproduces pi2(1e7)=58,980, pi2(1e8)=440,312, pi2(1e9)=3,424,506 exactly; the retained complete TOS twin-gap histogram to 1e16 sums to 10,304,195,697,297 against the published 10,304,195,697,298 (one pair, boundary convention); M was evaluated by 30-digit quadrature and by the Ei identity (relative difference 9e-29). The model has a real low-x transient (over-predicts 24% at 1e3), converged by 1e9. (2) MEASURED: the absolute residual fits |d| ~ x^0.48 over 1e9..1e19 and OSCILLATES in sign (-, then nine +, - at 1e18, + at 1e19), consistent with Wolf's 477,118 sign changes of pi2-C2Li2 below 2^48. Its scale is the sqrt(x) scale, one power of x below the count itself. (3) MEASURED, the headline: from the census to 1e19, every finite world whose last pair lies below 1e19(1-1.4e-8) is excluded at kappa=3 (Poisson yardstick: 1e19(1-3.9e-8)); the survivors are those stopping in the last ~1e11 integers, and they survive because of the MODEL's error, not the census. Sham-stop power control on constructed finite worlds: 28/28 detected under all three yardsticks. (4) DERIVED: W ~ x^0.55 while eps = W/x ~ x^-0.45. So no census length can refute finiteness (an unbounded untestable tail), a further decade buys ~10^2 in RELATIVE resolution and nothing in W, and the binding input is a BOUND ON THE MODEL ERROR -- which is the corpus's own arithmetic wall (endpoint-target-audit / E_> sufficient margin), the same object the infinitude route needs. (5) ANALYSIS, in a stated model class only: for conditional-gap-law models the finite world F(X0) and the infinite world induce the SAME conditional law of the local pattern below X0, so gap laws, zone-occupancy patterns and record ladders are exactly blind to finiteness and only counts carry it. DEFECTS I FOUND IN MY OWN PRE-REGISTERED DESIGN, both disclosed and retained. (D1) frontier.py calibrated its constant on decades 3..ktop-4, which absorbs the model's low-x transient and returned c=0.9766 at 1e19 -- a fictitious 2.3% "bias" that propagated into k_emp=7424 and into W; the corrected run fits no constant. (D2) the pre-registered falsifier tested |d|>=3 sigma and therefore fired on ANY residual: it triggered, on a SURPLUS, the opposite direction to a terminal clearing. The corrected direction-sensitive falsifier does NOT trigger (the one negative decade, 1e18, sits at z=-0.09). Both runs are in the artifact set; the first run's W_exact_check is retained too. RUNG AND SCOPE. Ratio table, residual scale, frontier numbers, power controls: MEASURED. Window growth: DERIVED from the measured scale, linearisation error bounded and disclosed. Blindness: ANALYSIS inside the declared model class, NOT about the primes. Nothing here bounds G2, the Zone Postulate or any twin margin, and no claim is made about the truth of the conjecture.