Exclude birth-lowered P104 markets without paid deletions (no κ)

research/p104-prefix-birth-obstruction/PROOF.md

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+# A birth-lowered market needs a paid deletion
+
+This is a scoped accounting obstruction, with new finite continuation fixtures.
+It does not re-prove or audit the source construction.
+
+## Contract
+
+Fix the PR6030 word stored by DreamingOfClouds's PR6170 toolkit. Write
+`t = 1-C`, with `0<t<1`. Use the source recursive ledger
+
+    F(t) = 18 sum_helpers span(h)^t - 18 sum_MOVEs rank(MOVE)^t
+           + unchanged outer terms.
+
+A public MOVE ascends strictly through nested binary subspaces. Its rank is
+the difference of their dimensions. A helper's productive span is its final
+dimension minus its initial dimension. These are inherited pricing rules,
+not a claim that all physical work is priced by this equation.
+
+For each of the fourteen market birth frames G in the pinned rebuild manifest,
+let O=G^perp under the literal binary dot product. Select one or more public
+helpers initially at G and finally at the full nine-dimensional frame.
+Replace each selected birth G by B contained in O intersect G. Preserve:
+
+- every original MOVE, with its register, source and destination frames;
+- the final frame of every selected helper;
+- all other productive initial/final frames and their credits.
+
+Arbitrary scalar resynthesis, including hub/market coupling, is allowed.
+Additional paid work is allowed and charged. No additional productive credit
+is granted. The selected registers must still reach G to execute their first
+original MOVE. Multiple changed registers are charged separately.
+
+The theorem is conditional on a candidate satisfying this contract. It does
+not assert that an arbitrary scalar resynthesis is realizable or correct.
+
+## General lower bound
+
+Suppose a helper's original productive span is s>0 and lowering its birth
+adds dimension d>0. Before its first original MOVE it must ascend from the
+new birth to the old birth. Let the positive ranks of those new MOVEs be
+`d_1,...,d_k`. Nesting gives `sum_i d_i=d`.
+
+For `0<t<1`, strict subadditivity gives
+
+    sum_i d_i^t >= d^t,
+    (s+d)^t < s^t+d^t.
+
+Its target-credit increase is `18((s+d)^t-s^t)`. Every original paid MOVE
+remains, and the extra ascents cost at least `18d^t`. Hence
+
+    Delta F(t) <= -18 [s^t+d^t-(s+d)^t] < 0.                 (1)
+
+Summing (1) over selected helpers proves the same strict loss for any
+nonempty simultaneous choice. Extra paid work can only decrease F further.
+This argument does not inspect scalar coefficients, row contents, event
+order between different registers, or independence of the proposed targets.
+It already rejects the optimistic invoice. Correctness and native delivery
+cannot make that invoice better without changing the stated contract.
+
+For completeness, strict subadditivity follows by fixing s>0 and observing
+that `g(d)=(s+d)^t-d^t` has negative derivative for d>0, with limiting value
+`s^t` at zero. Iterating the non-strict form proves the path inequality.
+
+## Exact P104 specialization and sharpness
+
+The checker enumerates the fourteen manifest markets. Each G has dimension
+five; its orthogonal O has dimension four; and I=O intersect G has dimension
+two. The helpers born at these G and ending at the full frame form a disjoint
+set of 62 roles. These extracted source facts select the new candidate family;
+they are not a replay of the original arithmetic or invoice.
+
+A binary plane has exactly five subspaces: zero, its three lines, and itself.
+Thus there are 310 single-helper choices (h,B). Their added dimensions are
+five, four and three respectively, and every old helper span is four.
+For fixed s and t, the deficit
+
+    D(s,d,t)=s^t+d^t-(s+d)^t
+
+strictly increases with d, since its d derivative is
+`t[d^(t-1)-(s+d)^(t-1)]>0`. Consequently every changed helper loses at least
+
+    18 [4^t+3^t-7^t],                                   (2)
+
+and a nonempty m-helper change loses at least m times (2).
+The bound holds throughout the open power interval, including the selected
+complex supplier's power, without a numerical root search.
+
+The optimistic lower bound is attained as an invoice difference: take
+B=I, change only h's initial frame, and prepend the single MOVE B->G.
+This MOVE changes no logical row. At the old entry boundary, every register
+again has precisely its old frame and arbitrary unit row. Keeping the entire
+old program therefore preserves its arbitrary-input continuation identity,
+conditional on the source's identity. No arithmetic replay of that source is
+needed or claimed. The new recursive bill is exactly (2) worse. Scalar or
+physical overhead can add cost; attainment here concerns this recursive bill.
+
+The checker constructs all 310 such continuation splices, checks their
+new prefix and literal unchanged suffix, and derives their changed target and
+paid entries. It does not claim to enumerate every possible scalar coupling.
+
+## What this excludes and what remains
+
+The result excludes obtaining a gain solely by lowering market births and
+resynthesizing scalars while preserving the original paid transitions and
+other target endpoints. A prospective coupled word must remove an original
+paid transition or obtain another justified target/boundary gain worth more
+than its strict preparation deficit. Replacing, splitting or deleting old
+MOVEs, changing helper final frames, adding productive targets, owned/native
+interfaces and actual consumer costs require a new comparison.
+
+No universal helper-split barrier, supplier admission, complete multiplication
+theorem, selected-frontier change or new kappa is proved. No source result was
+independently audited. The source's arbitrary-input identity and recursive
+pricing convention are cited premises. Independent human and formal
+proof-kernel review are unclaimed.
+
+The argument uses classical concavity and monotone dimension accounting.
+The contribution is this quantified changed-boundary obstruction, its sharp
+P104 preparation bill and the explicit new continuation family. Credit
+DreamingOfClouds for PR6170's geometry, Lemma 11 opening, word and ledger, Ali
+Romman and the retained P104/native/helper/caller/compiler/BIT contributors.
+This contribution was produced by Nick Petalas.

research/p104-prefix-birth-obstruction/README.md

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+# A paid-deletion requirement for birth-lowered P104 markets
+
+Claim type: scoped accounting proof with finite continuation certificates.
+No new kappa, supplier admission or selected-frontier change.
+
+For the PR6030 geometry retained in DreamingOfClouds's PR6170 toolkit, lowering
+a market helper's birth into its associated hub cannot improve the recursive
+moment if every old paid MOVE and all other productive endpoints remain.
+This holds even with arbitrary scalar resynthesis. Each changed helper loses
+at least
+
+    18 [4^t + 3^t - 7^t] > 0,  for every 0<t<1.
+
+The lower bound is sharp for the inherited recursive ledger. A single
+three-rank ascent attains it. Thus a coupled proposal needs an old paid
+transition deletion or another justified target/boundary gain. Merely moving
+births downward and using scalar coupling does not pay.
+
+The [proof](PROOF.md) gives the exact contract and the general deficit
+`18[s^t+d^t-(s+d)^t]`. Classical concavity is credited background. The new
+contribution is the changed-boundary screen, its sharp P104 bill and its
+explicit continuation family. It addresses a subfamily of PR6170 Lemma 11's
+untested bottom-to-market opening. It does not audit that lemma or settle the
+whole opening. The initial claim's construction search stops at this negative
+budget gate.
+
+## Evidence and reproduction
+
+The standard-library checker binds the source word, manifest and lemma bytes.
+It extracts fourteen markets and the associated two-dimensional intersections,
+then constructs all 310 single-helper birth choices for the 62 eligible roles.
+The new MOVE prefixes restore the exact original entry frames and leave all
+arbitrary logical values unchanged. Every old program event remains literal.
+The source continuation identity is inherited, not replayed.
+
+Added-rank counts are 62 at rank three, 186 at rank four and 62 at rank five.
+Twenty-eight integer path compositions corroborate the sharp transport bound
+with exact square-root intervals. Seven malformed-boundary controls reject.
+The paid-deletion control has positive target gain, which prevents applying the
+obstruction after its unchanged-paid-inventory hypothesis is removed.
+
+From the repository root:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-prefix-birth-obstruction/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-prefix-birth-obstruction/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-prefix-birth-obstruction
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-prefix-birth-obstruction --check-source-history --check-modes
+```
+
+Elsewhere, substitute a CPython 3.12 interpreter. Expected direct marker:
+`PASS_P104_BIRTH_OBSTRUCTION splices=310 compositions=28 rejections=7 no_new_kappa`.
+Every run emits a generated SHA256-bound JSON receipt and writes no files.
+The registered checks have 120-second timeouts.
+
+## Limits
+
+The universal claim rests on the written proof. The finite checker corroborates
+geometry and new splice identities; it does not exhaust scalar programs.
+The source pricing rules and arbitrary-input identity are cited premises.
+Sharpness concerns the recursive bill, not a complete physical cost optimum.
+The word is the selected PR6030 snapshot. Later supplier proposals need their
+own source binding, even when the general inequality applies.
+
+No new native/caller/compiler/precision/stock/outer replay, complete multiplier,
+Lean, Docker or proof-kernel verification is claimed. The five selected
+framework assumptions remain. Independent human and formal proof-kernel review
+are unclaimed. A candidate that removes or replaces an old MOVE, changes another
+productive endpoint or obtains new independent target credit needs a new bill.
+
+Python allocates dynamically. The checker bounds its live input to the pinned
+770-role, at-most-1,200-frame and fewer-than-26,000-event word. Each continuation
+fixture is released before the next; no unbounded search or recursive routine
+runs. Functions remain below sixty lines and explicit checks remain active
+under optimization. No external dependency or compiler is required.
+
+## Open next questions
+
+Can one coupled bottom-to-market trajectory delete an old paid transition
+while preserving every arbitrary helper input? Start with a bound market role,
+name the exact MOVE to remove, and solve its changed scalar continuation with
+the full signed invoice. That scalar word, native realization and complete
+consumer bill remain missing. No additional search is reserved.
+
+## Sources and credit
+
+Base `a5bff3bf133e482541c09eba4e773884bdb2b1b7`, priority revision 8.
+[SOURCES.json](SOURCES.json) pins the source bytes and original task card.
+Credit DreamingOfClouds for PR6170's word, geometry, ledger and Lemma 11 opening;
+Ali Romman, Cinna Davis and the retained P104, owned-root, helper, native,
+caller, compiler and BIT contributors retain their predecessor credit.
+The source construction and original invoice were not audited or replayed.
+
+This contribution was produced by Nick Petalas. Apache-2.0.

research/p104-prefix-birth-obstruction/SOURCES.json

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+  "base_commit": "a5bff3bf133e482541c09eba4e773884bdb2b1b7",
+  "files": [
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+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
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+    },
+    {
+      "path": "research/p104-designer-toolkit-v2/P104-LEMMAS.md",
+      "usage": "input",
+      "git_blob": "129f28dd10c9e1c43d29a9dd280e022070288a73",
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+    {
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+}