{"id":"93bd9bb4-d5c6-4db6-acc7-7e93119cb9e2","arxiv_id":"1908.02194","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Oases of fixed points of augmented happy functions have bounded length, with exact maximal lengths of 2, 6, 5, 8, and 9 for even bases 2 through 20.","lead":"This paper studies \"oases\": runs of consecutive constants c for which an augmented happy function has at least one fixed point. It proves a general upper bound on oasis length and computes the exact maximum lengths for all even bases up to 20.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The upper-bound half of Theorem 11 depends on an exhaustive mirage search that is described but not shipped; without an independent replication of that finite enumeration the exact maximal oasis lengths for b≤20 are not fully verified.","rationale":"After reviewing the full manuscript, I find no internal inconsistency in the analytic proofs. The reduction from oases to mirages (Theorem 10) is valid, the digit-bounds in Theorems 5-7 check out, and the paper's small worked examples are consistent. The central numerical claim, Theorem 11, has two halves: examples of oases of the claimed lengths, supplied in Table 1, and the assertion that no longer oasis exists, supplied only by the unshown computer search. The second half is the load-bearing uncertainty, and the reader's weakest_assumption identifies exactly this missing artifact. I also noted a small, non-central issue: Table 1 calls {3,4} the minimal maximal-length oasis for base 2, but {0,1} is also a 2-oasis and is smaller; this needs a caveat but does not affect the length values. The proposed independent replication of the mirage enumeration would resolve the main concern. Therefore the reader's CONDITIONAL verdict stands unchanged.","tokens_in":7224,"tokens_out":26155,"duration_ms":268358,"concrete_test":"Independently implement the Section 3 search: for each even b = 2,4,...,20, enumerate r = 0,1,...,b^3-1, compute d = r - S_{[0,b]}(r), deduplicate d, sort, and find all maximal consecutive runs. Verify (a) the longest run length matches Theorem 11, and (b) at least one longest run consists entirely of positive integers, or otherwise explain why the oasis maximum is still the reported value. A second implementation in a different language, or a small checkable certificate listing the distinct d-values and maximal runs for each base, would settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 reduces the problem of bounding oasis length to finding the longest consecutive run in D_b = {r - S_{[0,b]}(r) : 0 ≤ r < b^3}, and Theorem 11 asserts exact maxima for all even b ≤ 20. The lower-bound half is checkable from Table 1: for each listed oasis one can verify that the displayed fixed points work. The upper-bound half, however, rests entirely on the assertion that the algorithm in Section 3 was run correctly and completely. The paper gives no code, no data, and no certificate; the sorted d-values and the longest runs are not shown. If the search missed a longer run, or if a longest run was wrongly assumed positive, Theorem 11's maximal lengths would be too small. I checked the surrounding analytic framework: Theorems 5, 6, 7, and 10 appear sound, and the algorithm's exclusion of r=0 is harmless because r=1 already gives d=0. Thus the single load-bearing gap is the unverified finite computation, not the mathematics.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a k-oasis base b as a set of k consecutive nonnegative augmenting constants c for which the augmented generalized happy function S_{[c,b]} has at least one fixed point. It proves that every base b has an oasis, that odd bases have only oases of length 1, that even b ≥ 6 have 5-oases, that the existence of one k-oasis implies infinitely many, and that fixed points belonging to the same oasis agree in all but the three least significant digits. It derives a general bound k ≤ b^3/2 + b^2/2 − b and, via a reduction to a finite search over values d = r − S_{[0,b]}(r) with 0 < r < b^3, reports exact maximal oasis lengths for all even bases b ≤ 20. The lower-bound constructions are explicit and checkable in Table 1.","tokens_in":7447,"tokens_out":9084,"duration_ms":83315,"significance":"The theoretical development is sound, elementary, and clean: the reduction of oasis length to mirage length (Lemma 9 and Theorem 10) is a nice contribution, and the general bound in Theorem 7 is explicit. If the computational half is verified, Theorem 11 settles the maximal oasis lengths for all bases up to 20 and shows the general bound is nearly sharp. The paper's lower-bound examples are concrete and machine-verifiable by hand. The main weakness is that the exact maximality claims in Theorem 11 rest on an exhaustive computer search that is described but not shipped, so the upper-bound half is not independently reproducible from the manuscript.","major_comments":[{"comment":"The exact maximal lengths asserted in Theorem 11 for bases 2 through 20 depend entirely on the claim that the finite enumeration over 0 < r < b^3 was performed correctly and completely. The manuscript provides the algorithm but no code, no sorted d-values, no longest-run output, and no certificate. A missed longer consecutive run, including one involving negative d-values, would invalidate the stated upper bounds and hence the exactness of Theorem 11. Please provide the implementation and its output (for example, the sorted distinct d-values and the longest runs for each base), or explicitly restate Theorem 11 as a conditional-on-search result. This is the load-bearing gap in the paper's central claim.","section":"Section 3, Algorithm steps 1–4, and Theorem 11"}],"minor_comments":[{"comment":"The proof contains a duplicated paragraph: the statement of the bound from [2, Theorem 4.2] appears twice, and in the second occurrence the subscript in \"mb,ˆn\" should read \"mb,¯n\". Please delete the duplicate and correct the typo.","section":"Section 2, proof of Theorem 5"},{"comment":"The example for optimality uses the notation \"10(15) (16)\"; consider writing it as (10 15)_16 or 1·16^2 + 15 to avoid confusion with multiplication.","section":"Section 2, after Theorem 5"},{"comment":"The algorithm restricts to 0 < r < b^3 and excludes r = 0; this is harmless because r = 1 gives d = 0, but the equivalence should be stated explicitly so that the reader sees no d-value is lost.","section":"Section 3, Algorithm step 1"},{"comment":"In the displayed equation following Eq. (2), the summation index is written as \"∑_{j=3}^n\" but should be \"∑_{i=3}^n\"; this is a typographical error that does not affect the derivation.","section":"Section 3, Theorem 10 proof"},{"comment":"The table lists \"Smallest fixed points\" for each maximal oasis but does not indicate which fixed point corresponds to which augmenting constant in the oasis; adding that correspondence, or at least stating the ordering of the oasis elements, would improve verifiability.","section":"Table 1"},{"comment":"The text says the table provides the \"minimal example\" of a maximal length oasis for each even base, but no ordering or search for minimality is described; if \"minimal\" is intended as a claim, it needs justification, and if it merely means \"an example found by the search,\" the wording should be relaxed.","section":"Section 3, discussion of Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper depends on several results from Part I, which is published and does not contain the target results; I regard this as ordinary self-citation rather than circularity. The decisive issue is the unshipped exhaustive computation behind Theorem 11. If the journal has a data/code policy, the authors should be asked to provide the implementation and raw output; without that, the exact maximality claims cannot be independently checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. This is a careful, modest follow-up to the authors' Part I. It defines k-oases and k-mirages, proves a general cubic upper bound on oasis length, and reduces the exact maximal length for even bases ≤ 20 to a tiny finite enumeration. The mathematics is elementary, the proofs are sound, and the main weakness is that the exact maxima for small bases rest on a described but unshipped exhaustive search.\n\nWhat's new: the oasis/mirage distinction is genuine, Theorem 7 gives a clean general bound, and Theorem 10's reduction from oases to mirages is a nice structural step. Theorems 5 and 6 provide the digit constraints that make the bound work. The lower-bound witnesses in Table 1 are explicit and checkable, and the upper-bound claim for general b is a real theorem, not a guess. This is honest progress in a niche subfield.\n\nSoft spots. The upper-bound half of Theorem 11 depends entirely on the assertion that the Section 3 algorithm was run correctly and completely. The search space is tiny — about 24,000 integers total across the even bases up to 20 — and the algorithm is straightforward. But no code, raw output, or certificate is provided. A reader who wants to verify the exact maxima has to re-implement the search. I consider this a minor-to-moderate issue: the chance of a missed longer run is small, and the claimed maxima look consistent with the explicit examples. Still, for any computational claim, shipping the script would settle it. There is also a harmless duplicated sentence in the proof of Theorem 5 and a small typo in the notation; neither affects the argument.\n\nWho should read this: people working on digit sums, fixed points of functions, or recreational number theory. It will not change broader mathematics, but it completes the desert/oasis picture for small bases. It deserves a serious referee, because the reasoning is sound and the gap is fixable.\n\nMy recommendation: send it to review, and ask the authors to provide the code or dataset that verifies Theorem 11.","headline":"A clean, modest sequel that completes the oasis picture for small bases; the math is sound but the exact maxima for b≤20 depend on an asserted search that should be shipped.","tokens_in":7971,"tokens_out":3432,"would_cite":false,"duration_ms":32967,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63"],"pacs":[],"model":"deepseek-v4-flash","headline":"Oasis lengths for augmented happy functions are bounded, with exact maxima through base 20.","keywords":["happy number","fixed point","iteration","augmented generalized happy functions","oasis","mirage","consecutive augmenting constants","base-b digit sum"],"falsifier":"Run an independent implementation of the algorithm for each even base $2\\le b\\le20$: compute and sort all $d=r-S_{[0,b]}(r)$ for $0<r<b^3$ and inspect the longest run of consecutive $d$ values; if any run is longer than the maximum reported in Theorem 11 ($2,6,5,8,8,8,8,8,8,9$ for $b=2,4,6,8,\\dots,20$), the theorem is false.","tokens_in":7050,"feed_emoji":"🌴","tokens_out":11927,"duration_ms":110145,"temperature":0.7,"pith_summary":"This paper studies augmented generalized happy functions $S_{[c,b]}$, which add an augmenting constant $c$ to the sum of the squares of the base-$b$ digits of an integer. Earlier work showed that the values of $c$ for which $S_{[c,b]}$ has no fixed point contain arbitrarily long consecutive blocks, called deserts. The paper considers the complementary blocks, called oases: consecutive $c$ for which $S_{[c,b]}$ does have a fixed point. Its central assertion is that, unlike deserts, oases have bounded length for each fixed base, and it computes those exact maximum lengths for every even base $2 \\le b \\le 20$. This completes the desert picture by showing that the two phenomena are asymmetric: deserts can be arbitrarily long, oases cannot.","feed_headline":"Happy-function oases are bounded; exact maxima found up to base 20","feed_subtitle":"Consecutive constants with fixed points never run longer than 2, 6, 5, 8, or 9, up to base 20.","key_machinery":"The load-bearing object is the $k$-mirage base $b$: a set of $k$ consecutive integers of the form $d = r - S_{[0,b]}(r)$, with $r$ ranging over non-negative integers having at most three base-$b$ digits. The paper shows that two fixed points lying in the same oasis must agree in every digit beyond the third (Theorems 5 and 6), so the augmenting constant of a fixed point is determined entirely by its last three digits. Using $c = a - S_{[0,b]}(a)$ (Equation (1)), every oasis is shown to produce a mirage, and every mirage consisting only of positive integers is an oasis (Lemma 9, Theorem 10). Searching all $0<r<b^3$ for the longest consecutive run of $d$-values therefore gives an upper bound on oasis length, and in the cases $2\\le b\\le 20$ the search also attains the bound.","core_discovery":"On its own terms, the paper establishes that a $k$-oasis base $b$ exists only for $k \\le b^3/2 + b^2/2 - b$ when $b$ is even, while for odd $b$ every oasis has length $1$. It then proves the bound is attained in the small even cases by displaying the exact maximal lengths: $2$ for $b=2$, $6$ for $b=4$, $5$ for $b=6$, $8$ for each of $b=8,10,12,14,16,18$, and $9$ for $b=20$. Table 1 records, for each of these bases, the smallest example of a maximal-length oasis and the smallest fixed point for each augmenting constant in it.","pith_inferences":["Inference: The same finite mirage search should determine exact maximal oasis lengths for even bases beyond $20$, with the practical limitation being the $O(b^3\\log b)$ cost of enumerating and sorting all $r-S_{[0,b]}(r)$ values; the paper stops at $b=20$.","Inference: The gap between the general upper bound $\\sim b^3/2$ and the computed maxima (all $\\le9$) suggests that the true maximal oasis length grows much more slowly than the bound, and the asymptotic growth rate is left open.","Inference: The reduction to the last three digits is tied to the square in the digit-sum function; for sums of higher powers of digits, the analogous fixed-point sets would have a different finite cut-off, so oasis lengths for higher-power happy functions need not follow the same pattern."],"forward_implications":["For any fixed base $b$, oasis length is finite: no matter how large the augmenting constants become, a block of more than the maximum number of consecutive fixed-point-bearing $c$ values cannot occur.","In odd bases every oasis has length exactly $1$, so only even bases can exhibit nontrivial oases.","For every even base $2\\le b\\le20$, a block of $c$ values of length $2$, $6$, $5$, $8$, or $9$ (according to base) can be found starting at the minimal example in Table 1, and any longer block would contain a gap.","Whenever a $k$-oasis base $b$ exists, infinitely many $k$-oases exist, so the maximal examples found are not unique; they repeat at larger values of $c$.","Computing the maximum oasis length reduces to a finite check over $r<b^3$; no search over all fixed points is needed."],"supporting_citations":[{"why":"Part I of this work; supplies the parity lemma, digit-interval bounds, and fixed-point symmetry lemmas used in Theorems 5, 6, 7, and 10.","marker":"[2]"},{"why":"Defines the augmented generalized happy function $S_{[c,b]}$ and establishes the fixed-point equation on which all later arguments build.","marker":"[1]"},{"why":"Introduces generalized happy numbers, the base-$b$ digit-square iteration that $S_{[c,b]}$ augments by a constant $c$.","marker":"[3]"}],"fun_headline_variants":["Oasis bound proven; exact maxima for even bases to 20","Happy-function oases: length limit, exact sizes to base 20","Maximal oasis lengths found for happy functions up to base 20","Oases in augmented happy functions: bound and computed maxima","Even-base oasis lengths bounded and maximized to base 20"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claimed exact maxima rest on the finite computer search being complete and correct; a missed longer run of candidate values would make a listed maximum too small.","fun_headline_variants_meta":{"raw":{"variants":["Oasis bound proven; exact maxima for even bases to 20","Happy-function oases: length limit, exact sizes to base 20","Maximal oasis lengths found for happy functions up to base 20","Oases in augmented happy functions: bound and computed maxima","Even-base oasis lengths bounded and maximized to base 20"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00083,"raw_usage":{"total_tokens":3599,"prompt_tokens":893,"completion_tokens":2706,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":2617}},"tokens_in":509,"tokens_out":2706,"duration_ms":16859,"temperature":1.0,"reasoning_tokens":2617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:51:17.636621+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent implementation of the algorithm for each even base $2\\le b\\le20$: compute and sort all $d=r-S_{[0,b]}(r)$ for $0<r<b^3$ and inspect the longest run of consecutive $d$ values; if any run is longer than the maximum reported in Theorem 11 ($2,6,5,8,8,8,8,8,8,9$ for $b=2,4,6,8,\\dots,20$), the theorem is false.","supporting_citations":[{"cited_title":"Baker Swart, K","cited_arxiv_id":null,"evidence_quote":"Part I of this work; supplies the parity lemma, digit-interval bounds, and fixed-point symmetry lemmas used in Theorems 5, 6, 7, and 10."},{"cited_title":"Baker Swart, K","cited_arxiv_id":null,"evidence_quote":"Defines the augmented generalized happy function $S_{[c,b]}$ and establishes the fixed-point equation on which all later arguments build."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces generalized happy numbers, the base-$b$ digit-square iteration that $S_{[c,b]}$ augments by a constant $c$."}],"review_version":1}