PG-03 — One host, one pair: the limit and the correction exponent, read from successive sizes

Rewritten 2026-10-25 (records audit R28_reading/audits/2026-10-25-records.md, fix 7; second reader's corrections from R28_reading/distilled/PG-03-2026-10-25.check.md applied). It absorbs every run scored since the 2026-10-24 text, which took the record through 128: 129 (confirmed), 130 (confirmed), 133 (unanticipated), 134 (unanticipated) and 137 (withdrawn). The 20 runs under PG-03 (099–137): confirmed 8 (100, 102, 105, 111, 125, 128, 129, 130), falsified 6 (106, 110, 114, 117, 123, 124), unanticipated 2 (133, 134), withdrawn 4 (099 before any draw, 120, 121, 137). No run under PG-03 is sealed and open. It cites run files; no saved query set carries this reading.

What survived.

On H's pair. The free one-term form, fitted on five sizes, reached two sizes it had not seen. At 2187 on 1,500,000 samples, Q = 2.478 against a bar of 2.5, set by g₁ (run 102). At 19,683 on 48,000 samples, Q = 1.733, set by ln S (run 105). The pair it carries on run 100's seven new files is 1/ν = 0.5260 (0.5234 to 0.5284) and ω = 0.3009 (0.2810 to 0.3216), at χ² = 12.406 on 12 (run 100). The free ω stays between 0.30 and 0.36 in every fit since (runs 102, 105). A pure power law's ln S at 19,683 sits 3.86 to 4.37 combined errors from the drawn value; that is reported, not scored (run 105).

On H's invasion bursts: the comb. - The comb is the dual of the magnifications a host keeps. At L = 2187 on fresh seeds, H's odd teeth of the base-64 comb sit below a quarter of its even teeth (D_A = −0.00849 ± 0.00143), and the square carries none (D_B = −0.01368 ± 0.00173; S = 0.0046 against H's E = 0.073) (run 130). T and X at 729 carry odd teeth of 0.516 and 0.231, in the order the dual predicts (run 130, reported). - H's comb has its own period, and it is not 8. Fitted free, P̂ = 7.32 ± 0.04 on fresh draws and 7.34 on run 130's, so the bursts' dimension is D_b = ln P̂/ln 3 = 1.812 ± 0.005 against the carpet's 1.893; the residual sum of squares halves at P̂ (142.5 against 284.9) (run 133, reported beside an unanticipated clause). - A forbidden tooth stands. At P̂ the k = 3 tooth, at 3/2 of H's fundamental, is a₃ = −0.00675 ± 0.00131 (run 133). Grown from one point, H7 keeps it (a₃ = −0.0146 ± 0.0023, against the full side's −0.0066 ± 0.0005), so it is not the full-side source's (run 134, read beside an unscorable clause). - The period moves with burst size: P̂ = 7.00 ± 0.06 on bursts of 32 to 1172 and 7.94 ± 0.18 on 1172 to 48000 (a difference of 4.4 errors; the point source agrees, −1.16 ± 0.44), and τ depends on the source, 1.692 on the full side against 1.620 on a point (run 134, reported). A formation reader then showed that this split is not evidence of drift: a fixed comb at P = 3^1.84 = 7.55 with an unmodelled third harmonic reproduces 7.13 and 8.06 (§2, 2026-10-24, R28_reading/formation/drift/REPORT.md).

On the ladder, where answers are exact. - On the diamond lattices every correction to the threshold moments is a power series in λ^(−n), with the next term at λ^(−3n) (run 106). - λ(b, s) = λ(s, b) on six pairs, to 80 digits (runs 110, 111). The swapped lattice is the dual's two maps composed in the other order, and a multiplier at a fixed point does not change under cyclic reordering (run 111, reading). - No rule in the box has period three at k ≤ 6, where the box holds 1,053,696 rules. Band rules first have it at k = 7, the not-all-equal rule at k = 8 (run 114). - Every period-3 rule at k ≤ 12, 124 of them, stretches more than 1.85 at its fixed point (run 117; smallest 1.854). - The sign of the Li–Yorke slack O₂ separates period three on all 680 named rules at k = 2 … 16 (runs 120 and 121, both withdrawn, reported beside). It is a theorem for unimodal maps, and the slack's sign over all of [0, 1] is the period-3 test for any continuous map (CASE-32, closed before a panel). - At the crisis of the continuous band family, the stable-born 3-cycle's multiplier μ₋ is at or below −8 at all 5,700 crisis points found for k = 6 … 30. The largest is −8.034242, at k = 30 (run 125). - Over k = 32 … 64 the gap d₋ = −8 − μ₋ falls like k^(−2.089) and the excess d₊ = μ₊ − 8 like k^(−1.121). The local exponents drift toward −2 and −1. The rates were predicted with no data past k = 30 (run 128). - The four constants of the crisis, derived in closed form without data past k = 64, hold at k = 72 … 128: d₋k² → 27.1732 against 27.2532 (window 0.15), d₊k → 8.27126 against 8.28684 (0.05), the window width 0.93068 against 0.931582 (0.02), and its centre 0.68366 against 0.683325 (0.01) (run 129). L₁'s residuals curve, which a 1/k² term would do; not scored. - Since then, on the record and not as a run: the barycentre law is proved, and Q3 at large k is a derivation, μ₋ < −8 for every p₀ ≥ ½, with run 129's constants as C₋ and C₊ and the closest approach as C₋'s minimiser (§2, 2026-10-24, R28_reading/formation/proof_f3/ and amp_q3/). The back-reading marks runs 125, 128 and 129 as forced in that language, and O, O₂'s size and its correlations (117, 123, 124) as artefacts of the old coordinates (§2, 2026-10-24, backread_q3).

What died. - The tied form on H. With ω = 1/ν the one-term form fails on fresh draws: T(new) = 96.743 against a bar of 25 (run 100). Its predictions miss the new sets in the mean by +6.84 combined errors at 2187 (run 102, reported) and by −7.11 at 19,683 (run 105, reported). The failure needs the two smallest sizes together. Without 81 the tie costs 5.441, under the falsify line of 6.63 (run 100), and 7.90 on eight files (run 105). Without 243 it costs 13.361 (READING-PG03-LOO.md). - "One pair fits all five sizes," as written. Adding the three values at 81 costs the free form 10.58 on 3 (p about 0.014) on the new files, and 1/ν moves from 0.5111 without 81 to 0.5260 with it (run 100). - The tie as an exact identity on the diamond. Falsified as written, by a tolerance the statistic could not reach: ln S's ratio at n = 16 is 0.656167 against 1/λ = 0.654508 (run 106). - The sign rule "the lean is positive iff b < s." (3,4) has g₁ = −0.060585 at level 24 (run 110). - Hump height as what period three needs (run 114). - O, the one-sided distance to period three. It is pinned near 0⁻ at the repelling fixed point, and it matched its dual in 0 of 148 pairs (run 117). - Stretch as a scale past the threshold. ρ is 0.24 to 0.41 against O (run 117), and against O₂ it is 0.71 at k = 7 and at most 0.10 in size beyond (run 120). - O₂'s size as an invariant. It is not entropy: ρ(O₂, h) is 0.51 to 0.80 against a sealed 0.9 (run 123). It is not the multiplier split: ρ(O₂, Δμ) is 0.85 at k = 8, falling to 0.51 at k = 16, against 0.95 (run 124). O₂ stays as a sign test. The margin is now defined by axioms on the two 3-cycles' multipliers (docs/design/2026-10-23-period-three-margin-axioms.md, M2). - H's comb at period 8. At P = 8 the odd residue is χ²_odd = 165.48 on fresh draws; freed, it falls to 33.11 (run 133). The residue run 130 left (O = 0.0098, 6.5 errors from zero) was mostly the period's. - Not confirmed: "H carries no odd tooth at its own period." Run 133's clause landed between confirm (20) and falsify (60), so it was unanticipated, not falsified; the k = 3 tooth that remains is the reason, and run 134 kept it on a point source. "The tooth is the full side's" died as a reading: the clause was unscorable by 1.8e−5 on its error bar, and χ²₃ = 41.07 would have falsified it (run 134). - A control on a value the arithmetic cannot carry. The dual-equality check withdrew run 120 (11 of 220 pairs) and run 121 (4 pairs at 50 digits), both on rounding (OPS, instance 12). Run 137 was withdrawn the same way from the other side: an integer target, 34,585, against a 1e−6 tolerance in ln m_d, where one unit is 2.9e−5 (OPS, instance 25). Its numbers are readings, not scores. - A premise's count taken from memory. Run 134's point source carried 16 times fewer bursts per draw than the full side (504 against 8,088); the premise assumed 3 to 5 times fewer, from run 095 by memory, and that set the power the clause missed (run 134; OPS, instance 19).

The one object. It has moved, and the record holds two.

The frontmatter object is H's pair (1/ν, ω). Runs 100 to 105 rest on it. On eight files it stands at 1/ν = 0.5248 and ω = 0.3134 with 81, and 0.5181 and 0.3585 without (run 105, reported). Run 100's seal set aside whether ω is H's own number or 1/ν with a large next analytic term, as not separable at these sizes. Nothing since has separated them. The comb runs (130, 133, 134; 137 withdrawn) are on the same host and read a log-periodic correction in burst size, the kind of factor §1's blind spot hides in ln L, where it can be seen: its teeth are fixed by the magnifications H keeps, and its period in burst size (7.32, not 8) gives the bursts a dimension of their own. The blind spot itself, the period ln 3 in ln L on threshold against L, stays where it was (§2, run 130 sealed).

Runs 106 to 129 rest on another object: the multiplier of a hierarchical rule's renormalisation map. First it was λ = R′(p*) on the diamond. Then it was μ₋ of the stable-born 3-cycle on hump maps, read at the crisis, and now derived there in closed form (run 129; Status, 2026-10-24). A monotone (percolation) rule gives a non-decreasing map, which has no period three, so the hump rules are not percolation (R28_reading/formation/ lattice-meaning.md). The split this raised under docs/PROGRAMMES.md, One object, is ruled: kept together. By the owner's test (10:41), a formation reader given the stripped band map found it a known kind that yields to the tools in hand (Status, 2026-10-24, R28_reading/formation/bandmaps/REPORT.md). The ladder and the comb are carried by the runs' line: fields. They are not numbered lines in §1, and the frontmatter lists L1 to L6 only.

Next, and why. Nothing is sealed and open. Three seals are named in the record. TODO places two of them: the Q3 seal comes before E4 proper (TODO:658), and the H8 reseal is drafted after the D4/I2 premise chat (TODO:651). The g₁-residual seal is not in TODO; it lives in Status (2026-10-13) and older Distilleds. On TODO's order the Q3 seal is next: it waits on nothing, while the H8 reseal waits on a premise chat. The record does not place the g₁ seal against either. - The Q3 band line. An exact seal on the back-reading's predictions P1 to P3, with the finite-k check of μ₊ ≥ 8 and μ₋ ≤ −8 as a reading (TODO; R28_reading/formation/backread_q3/REPORT.md). It is the first seal on the ladder since the closed-form derivation, and it tests that derivation's predictions exactly. - The H8 comb reseal. Run 137's two questions, with control (iii)'s targets at full precision from h8_premise.log and fresh seeds, after a seal-lint check that a target's digits can carry its tolerance (TODO, the H8 reseal; OPS instance 25). 137's readings are glimpses now and are declared as such: T read present, D1 moved, D2 gated out with margins nearer P, and two of five groups put a step near 8⁵ (Status, 2026-10-25). It is the comb line's only open question, and nothing else on the line can move until the reseal says what is real at 6561. - The sign of H's g₁ residual as a seal of its own (Status, 2026-10-13; the 2026-10-20 Distilled, Next (2)), before L4's pure-power seal with 2187 refitted. Every g₁ comparison so far falls below the free form, each inside its bar: −2.478 scored and −2.38 pooled at 2187 (run 102), and −0.999 at 19,683 (run 105). It is the oldest named seal on the pair still standing.

What would end the programme. - Closed. L4 is confirmed (run 105). What remains is the rule written back on how an exponent is quoted: its sizes, its last local slope, and its limit under the host's pair, with the census requoted under it or shown not to need it (the first reading, 2026-10-04). Section 4 holds one entry (OPS.md, e55905e, 2026-10-04), under a note that it was empty on filing, with twenty runs sealed under the programme. These came out of this programme's runs and are not entered there: OPS, A check is written against what the instrument can resolve (adopted 2026-10-20 from run 106's candidate), and its instances 11, 12, 14, 15, 16, 19, 21 and 25 (runs 117; 120 and 121; CASE-32; 123; the formation before 125; 134; 133 and 134's premises; 137); and the theme rows of runs 111, 130, 133 and 134 in themes/symmetry.md §2.1 and §2.1f. OPS, A limit is sealed with its approach, came from the owner's ruling after Euler's fountain (OPS:272), not from this programme's runs; it was first applied in run 128 (Status, erratum, 2026-10-24). - Falsified. The conjecture's falsifier (3), a turn in the width's local slope or in g₁ on H at or beyond 729; or the free form missing a fresh size. - Dissolved. All lines closed and nothing written back. - The ladder. The split was ruled "kept together"; the ladder ends here if PG-03 returns to monotone rules.

Vocabulary. The §1 block gained the period-three line's terms on 2026-10-24, and multiplier and comb (with teeth) on 2026-10-25, the definitions GLOSSARY's Promoted 2026-10-24 table points at. On 2026-10-25 it also gained the clause terms in (1), and the rulings of 2026-10-24 and of this text were carried out there.

(1) Words the programme used and the block lacked. Each of these stands in a sealed clause of a scored run and in this Distilled, and each was added to the §1 block on 2026-10-25 (docs/PROGRAMMES.md, Rules: a term enters the block the day it enters a sealed clause): O₂, the Li–Yorke slack (runs 123, 124; also in the clauses of 120 and 121, both withdrawn); band rule [a, b] and NAE (114, 117); rung (106, 110, 111); crisis, R²(c) = q (125); D_A, D_B (130); point source (134). μ₋, μ₊, the closest approach, the odd and even teeth, O, E and P̂ are already in the block. Used but not yet in a scored clause: hump (114's question; 121's, withdrawn); stretch (117's question; the clause writes |λ|max); lean (110's question; GLOSSARY: "lean" waits); core [R²(c), R(c)] (the questions of 120 and 121, both withdrawn; O₂ is computed on it); D_b (133, reported beside the clause); full-side source (134's question; the clause writes "the side's"); step, F̂ (137's clauses; 137 withdrawn); margin M (the axioms note); drift of the period (134, reported).

(2) Words the block had that the programme could not use. "Band", the sealed interval, was never written into a clause; L4 was sealed on Q and z. "The null inside it" was tested under "tied form". "The pair" stands in no clause; the clauses write 1/ν and ω. "Host" now stands in run 130's sealed question and falsifier and is kept (GLOSSARY, Promoted 2026-10-24: "kept; the proposal to retire it is withdrawn").

(3) Where docs/rung-vocabulary-v5.md lacks the word. - O₂'s size: §1's number kinds have no kind for a magnitude that changes under a smooth change of coordinates. - μ₋ and the margin: §1's RG exponents are eigenvalues at a fixed point. A cycle's multiplier has no row. - Crisis, core and the log 2 ceiling: §5's behaviour form, marked thin, has fixed points and stability, and no invariant sets or crises. - Comb, teeth, P̂: §1 has no complex exponents and no log-periodic amplitude. - O₂'s sign has a home: §8.2's "sign is the smallest piece of size", and certificate direction.

(4) What the vocabulary enforced. Run 117's O was defined on a one-sided ordering, which broke under an orientation-reversing dual; O became O₂ (run 120). Run 106's "exact" was scored against a tolerance the word implied and the statistic could not reach. Run 124's premise called the −8 cycle "stable", renamed "stable-born" at scoring. Run 137's control target was written as an integer, "34,585", and the integer could not carry the tolerance written beside it.

(5) Symbols. Collisions inside the programme: - λ is R′(p*) on the diamond (106 to 111), the stretch |λ|max (117, 120), and the window width 0.931582 in units of √k (129); λ* is run 129's λ, named as a minimiser (the closest approach, amp_q3). - B is the correction amplitude on H and the beta function B(α, β) (125). - k is the number of children (114 on), the sample column of perc's output (run 100, control (ii)), and the tooth index (130 on: k = 1 … 4). - c is the median's correction amplitude (§1), the critical point (120 on), c_j, the Bernstein coefficients (114), and the window centre (γ = 2c − 1, amp_q3). - h is the window variable in amp_q3 and the entropy in run 123. - q is the dual variable (111) and the smallest nonzero fixed point (125). - s is the series length (110, 111), the slack s(a) (120), and the deformation parameter (proof_f3); s² = c(1 − c) in amp_q3. S is the standard deviation and the square's teeth (130). - a and b are the band's edges (114 on), the tooth coefficients aₖ, bₖ (130 on), and, on the diamond, b is also the number of branches (110, 111). - P is the comb's period (133 on) and the period-3 predictions P1 to P3 of the back-reading. - Q is run 102's and 105's statistic, the logistic map Q of proof_f3, and the question labels Q1 and Q3. - E is H's even teeth (130) and the symmetry theme's E1 to E4. - F̂, the step's shift factor (137), sits beside PG-01's fresh masks F01–F16 and F₈/F₆₄ (095). Collisions across programmes, as the glossary lists them (Collisions open): core is also PG-01's ring core (core / pendant / island) and its network-reliability core (132); hump and ladder / rung k are PG-01's; band is also the glossary's band · reading · sensor (owner docs/BANDS.md); λ: the operator's second eigenvalue λ₂ (§1, the codifier's lead beside the lines) sits beside PG-01's λ₂ of the Laplacian (108); Q: run 105's statistic sits beside PG-01's Q, T_op on non-empty subsets; orbit: the periodic orbit (114 on) against PG-01's group orbit (107, 119). Not in the glossary's list: T₁, the block's own addition (§1, 2026-10-24), which sits beside run 100's T(new) and pattern T; and D_A, which is also PG-01's p_med(#22) − p_med(#24) (run 098).

(6) Definitions that drifted. - "Band" on 2026-10-04 was the sealed interval at 19,683. From 2026-10-22 it is a rule that accepts weights in [a, b] (run 114). - "Host" on 2026-10-04 was a graph family. On 2026-10-22 it was also a rule ("Li–Yorke's own host on the ladder", Status). In the comb runs it is a graph family again (130). - "Ladder" on 2026-10-16 was exactly solvable toy models on the way to the carpet (run 106). From 2026-10-22 it is non-monotone rules, which are not percolation. - "O" became "O₂" on 2026-10-23. "The comb's period" was 8 by construction (130) and is fitted from 133.

(7) Prose only. The bend, blind spot, the null inside it, the ladder (as the line's name), the margin (until a clause uses M), the ledger's "real system", and the drift (until a clause fits it). Ruled to prose only today: core [R²(c), R(c)], hump and lean (Entry criterion and ceiling).

(8) Names that carry an open claim. - "The margin": M1 is open, one orbit or the pair (the axioms note). - "The comb": that the teeth are the Fourier dual of the dilations a host keeps. Run 130 confirmed it on H and the square; the k = 3 tooth on H (133, 134) is the one place it does not hold as stated. - "Stable-born": that the two 3-cycles are born together as a saddle-node pair; the band words LLR and LRR replace it (§1, 2026-10-24). - "The forbidden tooth": that H forbids it, which holds for a comb of fixed period; under a drifting period it may be a leak (Status, 2026-10-24, the drift formation).

(9) Metaphors. - The bend: L1 died on it; it is kept for data only. - The tie: a hypothesis, named as one. - The ladder and its rungs: these import a climb from solvable rungs toward the carpet. That is untested. No rung has reached a carpet, and the hump rungs have left percolation. - The comb and its teeth: a Fourier picture of discrete scale invariance. Tested by run 130 and held; the teeth's count under a drifting period is open. - The margin: imports a distance to a boundary. The axioms replaced it with multipliers. - Friction, or the dropped term (Euler's fountain): imports the idea that the first correction carries the new structure. Tested in run 128, and its constants derived (129).

(10) Quantifiers weakened since 2026-10-24. - "H's comb has only even teeth" became: only even teeth at a fixed period of 8, to a quarter (130); at its own period one odd tooth, k = 3, remains (133, 134). - "The comb's period in burst size is 8" became: 7.32 over the whole window, and 7.00 to 7.94 by half (133, 134). - The earlier weakenings stand, in their weakened forms: the tied form is rejected with 81 and 243 in, not without (100, 105); μ₋ ≤ −8 at every crisis point found on a b grid of step 0.01 (125); the rates are about −2 and −1, and the windows admit −2.2 (128); O₂'s separation of period three is a theorem on unimodal rules (CASE-32).

Entry criterion and ceiling. The 2026-10-24 text ruled each block term in or out. One of its proposals is superseded: host is kept, not retired (GLOSSARY, Promoted 2026-10-24). Multiplier and comb entered the glossary separately, with this programme as owner; they were not proposals of that text. The rest stand as proposed there and were carried out in the §1 block on 2026-10-25: methods programme, estimator, local slope, pair slope, the moments reading, S and W80 leave the block; tied form, the pair (as a term), band (the interval), the null inside it and blind spot are retired.

The 2026-10-24 text told its candidates to wait one rewrite. This is that rewrite, and the ceiling allows no second wait (docs/PROGRAMMES.md, Distilled, fifth paragraph: "enters or is retired, with the reason; there is no third option"). Each is ruled here. A term enters only if it names a computation (a function, a column or a query) and stands in a sealed clause of a scored run (Rules, A programme owns its vocabulary; the fold-in's part (1)). One that does not is retired, to the prose-only list where that rule allows it ("a prose-only term may stand in a lead or in Distilled and not in a clause"). None of those entering meets the glossary's third condition (a second programme with the same meaning, or a panel reaching the object blind), so none is proposed for the glossary. - Enter: O₂ (core120.O2; the clauses of 123 and 124); band rule [a, b] (rule114.band; 114, 117); NAE (rule114.nae; 114, 117); rung (b, s) (the diamond map R_{b,s} of diamond106.py, diamond110.py and diamond111.py; 106, 110, 111; written "rung (b, s)" beside PG-01's rung k and the dictionary's rung); crisis (crisis125.py, the zeros in b of R²(c) − q; 125). - Entered already: μ₋, μ₊, under multiplier (§1, 2026-10-25; GLOSSARY, Promoted 2026-10-24). - Retire to prose only: core [R²(c), R(c)] (in no clause; O₂ is computed on it, and the word collides with PG-01's ring core and network-reliability core); hump (in no clause; it collides with PG-01's hump, itself prose only from 2026-10-25); lean (in run questions only; the clauses write g₁'s sign). - Retire, the symbol kept: stretch (117's clause writes |λ|max, and the symbol stays).

The candidates new since 2026-10-24 wait one rewrite: D_A, D_B and point source are in the block from 2026-10-25. Point source already meets all three conditions, since PG-01's run 095 uses it in the same sense in a scored clause, so it is proposed for the glossary now.